http://www.elab.tecnico.ulisboa.pt/wwwelab/wiki/api.php?action=feedcontributions&user=Ist128595&feedformat=atomwwwelab - User contributions [en]2021-12-03T11:02:13ZUser contributionsMediaWiki 1.34.2http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_Planck%27s_Constant&diff=1427Determination of Planck's Constant2013-11-06T20:11:38Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Espectro_dos_leds.png|thumb|alt=Led's spectrum|Figure 1: Led's spectrum.]]<br />
The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light.<br />
The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).<br />
<br />
After each experiment the photocell is connected to ground to discharge.<br />
<br />
The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.<br />
<br />
The graph in figure 1 shows each led spectrum, table 1 shows more details.<br />
<br />
Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift. <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.<br />
|-<br />
!Color<br />
!Frequency (THz)<br />
!Wavelegth (nm)<br />
!Espectros dos leds<br />
|-<br />
|Blue.ab<br />
|638.7<br />
|469.70<br />
|[[File:Espectro_Azul.ab.txt|Blue.ab]]<br />
|-<br />
|Blue<br />
|684.6<br />
|438.20<br />
|[[File:Espectro_Azul.txt|Blue]]<br />
|-<br />
|Red<br />
|482.2<br />
|622.21<br />
|[[File:Espectro_Vermelho.txt|Red]]<br />
|-<br />
|Yellow<br />
|514.4<br />
|583.16<br />
|[[File:Example.txt|Yellow]]<br />
|-<br />
|Green<br />
|530.8<br />
|565.22<br />
||[[File:Espectro_Verde.txt|Green]]<br />
|}<br />
<br />
=Protocol=<br />
The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).<br />
<br />
#Choose a led to light upon the photocell <br />
#Measure the stopping potential. Note the time necessary to reach the maximum potential.<br />
#Repeat step 2 for different intensities.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Color #1 __________(name)<br />
!Intensity (%)<br />
!Stop Potential (V)<br />
!Time (s)<br />
|-<br />
| <br />
|100<br />
| <br />
| <br />
|-<br />
| <br />
|80<br />
| <br />
| <br />
|-<br />
| <br />
|60<br />
| <br />
| <br />
|-<br />
| <br />
|40<br />
| <br />
|<br />
|-<br />
| <br />
|20<br />
| <br />
|<br />
|}<br />
<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Planck's constant|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]<br />
<br />
The kinetic energy of the photoelectron depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.<br />
<br />
#Obtain the stop potentials for different colour leds.<br />
#Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.<br />
<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Colour (name)<br />
!Frequency (THz)<br />
!Stop Potential (V)<br />
|-<br />
| <br />
|<br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
| <br />
|<br />
|<br />
|}<br />
<br />
=Advanced Protocol=<br />
# Study the photocell charging process for different intensities.<br />
# Find the wavelength expected value using the led spectra.<br />
# Use those values for a new graphical fitting and compare the results.<br />
# Redo the voltage vs. frequency graph, this time with error bars. <br />
<br />
Note: This setup uses a 12bit ADC from 0V to 5V.<br />
<br />
<br />
=Theoretical Principles=<br />
<br />
==Photoelectric effect==<br />
The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed.<br />
When a photon with energy \( E \) collides with a metal, it transfers its energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface. <br />
For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons.<br />
The light intensity changes only the number of emitted photoelectrons, not its energy (this goes against what is expected in the classical theory of radiation).<br />
<br />
Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)): <br />
<br />
\[<br />
E = h \nu <br />
\]<br />
<br />
Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all its energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound to the crystal structure of the metal), the electron is only freed if the photon energy is higher than the difference between the energy of the electron in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the energy of the photon minus the amount "spent" when the electron leaves the metal:<br />
<br />
\[<br />
E = h \nu - e \phi<br />
\]<br />
<br />
where \( e \) is the electron charge and \( \phi \) is the diference in ''workfunction''. As the light frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emitted. In this case, \( E_{max} = 0 \) and we can write:<br />
<br />
\[<br />
h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi<br />
\]<br />
<br />
==In This Setup==<br />
A photoelectric cell is a device where light shines upon a metalic surface (cathode) and excites the electrons that will be collected by a concentric metalic surface (anode), as shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If you connect the cathode and anode through an external circuit, you can measure the current created by the photoelectrons. In a more generic sense, the maximum kinetic energy of the electron is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the anode. This way, when current no longer flows, we will know the value of the current.<br />
<br />
[[File:Plank-celula1.png|thumb]]<br />
<br />
We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the cell. This creates an electric current through the cell. As the capacitor charges, the voltage between its terminals increases, reducing the volltage between the terminals of the cell(because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell voltage reaches \( V_c = \frac{h \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Voltage (stopping power) vs. Maximum in the frequency of light]]<br />
<br />
Knowing two or more voltage values for certain frequencies, the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \). <br />
<br />
<br />
=Historical Elements=<br />
In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante de Planck | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_Planck%27s_Constant&diff=1426Determination of Planck's Constant2013-11-06T18:50:49Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Espectro_dos_leds.png|thumb|alt=Led's spectrum|Figure 1: Led's spectrum.]]<br />
The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light.<br />
The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).<br />
<br />
After each experiment the photocell is connected to ground to discharge.<br />
<br />
The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.<br />
<br />
The graph in figure 1 shows each led spectrum, table 1 shows more details.<br />
<br />
Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift. <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.<br />
|-<br />
!Color<br />
!Frequency (THz)<br />
!Wavelegth (nm)<br />
!Espectros dos leds<br />
|-<br />
|Blue.ab<br />
|638.7<br />
|469.70<br />
|[[File:Espectro_Azul.ab.txt|Blue.ab]]<br />
|-<br />
|Blue<br />
|684.6<br />
|438.20<br />
|[[File:Espectro_Azul.txt|Blue]]<br />
|-<br />
|Red<br />
|482.2<br />
|622.21<br />
|[[File:Espectro_Vermelho.txt|Red]]<br />
|-<br />
|Yellow<br />
|514.4<br />
|583.16<br />
|[[File:Example.txt|Yellow]]<br />
|-<br />
|Green<br />
|530.8<br />
|565.22<br />
||[[File:Espectro_Verde.txt|Green]]<br />
|}<br />
<br />
=Protocol=<br />
The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).<br />
<br />
#Choose a led to light upon the photocell <br />
#Measure the stopping potential. Note the time necessary to reach the maximum potential.<br />
#Repeat step 2 for different intensities.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Color #1 __________(name)<br />
!Intensity (%)<br />
!Stop Potential (V)<br />
!Time (s)<br />
|-<br />
| <br />
|100<br />
| <br />
| <br />
|-<br />
| <br />
|80<br />
| <br />
| <br />
|-<br />
| <br />
|60<br />
| <br />
| <br />
|-<br />
| <br />
|40<br />
| <br />
|<br />
|-<br />
| <br />
|20<br />
| <br />
|<br />
|}<br />
<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Planck's constant|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]<br />
<br />
The kinetic energy of the photoelectron depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.<br />
<br />
#Obtain the stop potentials for different colour leds.<br />
#Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.<br />
<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Colour (name)<br />
!Frequency (THz)<br />
!Stop Potential (V)<br />
|-<br />
| <br />
|<br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
| <br />
|<br />
|<br />
|}<br />
<br />
=Advanced Protocol=<br />
# Study the photocell charging process for different intensities.<br />
# Find the wavelength expected value using the led spectra.<br />
# Use those values for a new graphical fitting and compare the results.<br />
# Redo the voltage vs. frequency graph, this time with error bars. <br />
<br />
Note: This setup uses a 12bit ADC from 0V to 5V.<br />
<br />
<br />
=Theoretical Principles=<br />
<br />
==Photoelectric effect==<br />
The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed.<br />
When a photon with energy \( E \) collides with a metal, it transfers its energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface. <br />
For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons.<br />
The light intensity changes only the number of emitted photoelectrons, not its energy (this goes against what is expected in the classical theory of radiation).<br />
<br />
Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)): <br />
<br />
\[<br />
E = h \nu <br />
\]<br />
<br />
Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all its energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound to the crystal structure of the metal), the electron is only freed if the photon energy is higher than the difference between the energy of the electron in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the energy of the photon minus the amount "spent" when the electron leaves the metal:<br />
<br />
\[<br />
E = h \nu - e \phi<br />
\]<br />
<br />
where \( e \) is the electron charge and \( \phi \) is the diference in ''workfunction''. As the light frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emitted. In this case, \( E_{max} = 0 \) and we can write:<br />
<br />
\[<br />
h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi<br />
\]<br />
<br />
==In This Setup==<br />
A photoelectric cell is a device where light shines upon a metalic surface (cathode) and excites the electrons that will be collected by a concentric metalic surface (anode), as shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If you connect the cathode and anode through an external circuit, you can measure the current created by the photoelectrons. In a more generic sense, the maximum kinetic energy of the electron is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the anode. This way, when current no longer flows, we will know the value of the current.<br />
<br />
[[File:Plank-celula1.png|thumb]]<br />
<br />
We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the cell. This creates an electric current through the cell. As the capacitor charges, the voltage between its terminals increases, diminuishing the volltage between the terminals of the cell(because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell voltage reaches \( V_c = \frac{h \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Voltage (stopping power) vs. Maximum in the frequency of light]]<br />
<br />
Knowing two or more voltage values for certain frequencies, the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \). <br />
<br />
<br />
=Historical Elements=<br />
In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante de Planck | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_Planck%27s_Constant&diff=1425Determination of Planck's Constant2013-11-06T18:44:09Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Espectro_dos_leds.png|thumb|alt=Led's spectrum|Figure 1: Led's spectrum.]]<br />
The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light.<br />
The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).<br />
<br />
After each experiment the photocell is connected to ground to discharge.<br />
<br />
The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.<br />
<br />
The graph in figure 1 shows each led spectrum, table 1 shows more details.<br />
<br />
Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift. <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.<br />
|-<br />
!Color<br />
!Frequency (THz)<br />
!Wavelegth (nm)<br />
!Espectros dos leds<br />
|-<br />
|Blue.ab<br />
|638.7<br />
|469.70<br />
|[[File:Espectro_Azul.ab.txt|Blue.ab]]<br />
|-<br />
|Blue<br />
|684.6<br />
|438.20<br />
|[[File:Espectro_Azul.txt|Blue]]<br />
|-<br />
|Red<br />
|482.2<br />
|622.21<br />
|[[File:Espectro_Vermelho.txt|Red]]<br />
|-<br />
|Yellow<br />
|514.4<br />
|583.16<br />
|[[File:Example.txt|Yellow]]<br />
|-<br />
|Green<br />
|530.8<br />
|565.22<br />
||[[File:Espectro_Verde.txt|Green]]<br />
|}<br />
<br />
=Protocol=<br />
The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).<br />
<br />
#Choose a led to light upon the photocell <br />
#Measure the stopping potential. Note the time necessary to reach the maximum potential.<br />
#Repeat step 2 for different intensities.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Color #1 __________(name)<br />
!Intensity (%)<br />
!Stop Potential (V)<br />
!Time (s)<br />
|-<br />
| <br />
|100<br />
| <br />
| <br />
|-<br />
| <br />
|80<br />
| <br />
| <br />
|-<br />
| <br />
|60<br />
| <br />
| <br />
|-<br />
| <br />
|40<br />
| <br />
|<br />
|-<br />
| <br />
|20<br />
| <br />
|<br />
|}<br />
<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Planck's constant|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]<br />
<br />
The kinetic energy of the photoelectron depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.<br />
<br />
#Obtain the stop potentials for different colour leds.<br />
#Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.<br />
<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Colour (name)<br />
!Frequency (THz)<br />
!Stop Potential (V)<br />
|-<br />
| <br />
|<br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
| <br />
|<br />
|<br />
|}<br />
<br />
=Advanced Protocol=<br />
# Study the photocell charging process for different intensities.<br />
# Find the wavelength expected value using the led spectra.<br />
# Use those values for a new graphical fitting and compare the results.<br />
# Redo the voltage vs. frequency graph, this time with error bars. <br />
<br />
Note: This setup uses a 12bit ADC from 0V to 5V.<br />
<br />
<br />
=Theoretical Principles=<br />
<br />
==Photoelectric effect==<br />
The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed.<br />
When a photon with energy \( E \) collides with a metal, it transfers its energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface. <br />
For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons.<br />
The light intensity changes only the number of emitted photoelectrons, not its energy (this goes against what is expected in the classical theory of radiation).<br />
<br />
Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)): <br />
<br />
\[<br />
E = h \nu <br />
\]<br />
<br />
Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all its energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound to the crystal structure of the metal), the electron is only freed if the photon energy is higher than the difference between the energy of the electron in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the energy of the photon minus the amount "spent" when the electron leaves the metal:<br />
<br />
\[<br />
E = h \nu - e \phi<br />
\]<br />
<br />
where \( e \) is the electron charge and \( \phi \) is the diference in ''workfunction''. As the light frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emitted. In this case, \( E_{max} = 0 \) and we can write:<br />
<br />
\[<br />
h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi<br />
\]<br />
<br />
==In This Setup==<br />
A photoelectric cell is a device where light shines upon a metalic surface (cathode) and excites the electrons that will be collected by a concentric metalic surface (anode), as shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If you connect the cathode and anode through an external circuit, you can measure the current created by the photoelectrons. In a more generic sense, the maximum kinetic energy of the electron is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the anode. This way, when current no longer flows, we will know the value of the current.<br />
<br />
[[File:Plank-celula1.png|thumb]]<br />
<br />
We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the former. This creates an electric current through the cell. As the capacitor charges, the voltage between it's terminals increases, diminuishing the volltage between the cell's terminals (because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell's voltage reaches \( V_c = \frac{h \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Voltage (stopping power) vs. Maximum in the frequency of light]]<br />
<br />
Knowing two or more voltage values for certain frequencies the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light's frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \). <br />
<br />
<br />
=Historical Elements=<br />
In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante de Planck | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_Planck%27s_Constant&diff=1424Determination of Planck's Constant2013-11-06T18:25:22Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Espectro_dos_leds.png|thumb|alt=Led's spectrum|Figure 1: Led's spectrum.]]<br />
The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light.<br />
The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).<br />
<br />
After each experiment the photocell is connected to ground to discharge.<br />
<br />
The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.<br />
<br />
The graph in figure 1 shows each led spectrum, table 1 shows more details.<br />
<br />
Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift. <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.<br />
|-<br />
!Color<br />
!Frequency (THz)<br />
!Wavelegth (nm)<br />
!Espectros dos leds<br />
|-<br />
|Blue.ab<br />
|638.7<br />
|469.70<br />
|[[File:Espectro_Azul.ab.txt|Blue.ab]]<br />
|-<br />
|Blue<br />
|684.6<br />
|438.20<br />
|[[File:Espectro_Azul.txt|Blue]]<br />
|-<br />
|Red<br />
|482.2<br />
|622.21<br />
|[[File:Espectro_Vermelho.txt|Red]]<br />
|-<br />
|Yellow<br />
|514.4<br />
|583.16<br />
|[[File:Example.txt|Yellow]]<br />
|-<br />
|Green<br />
|530.8<br />
|565.22<br />
||[[File:Espectro_Verde.txt|Green]]<br />
|}<br />
<br />
=Protocol=<br />
The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).<br />
<br />
#Choose a led to light upon the photocell <br />
#Measure the stopping potential. Note the time necessary to reach the maximum potential.<br />
#Repeat step 2 for different intensities.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Color #1 __________(name)<br />
!Intensity (%)<br />
!Stop Potential (V)<br />
!Time (s)<br />
|-<br />
| <br />
|100<br />
| <br />
| <br />
|-<br />
| <br />
|80<br />
| <br />
| <br />
|-<br />
| <br />
|60<br />
| <br />
| <br />
|-<br />
| <br />
|40<br />
| <br />
|<br />
|-<br />
| <br />
|20<br />
| <br />
|<br />
|}<br />
<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Planck's constant|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]<br />
<br />
The photoelectron's kinetic energy depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.<br />
<br />
#Obtain the stop potentials for different colour leds.<br />
#Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.<br />
<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Example of a table <br />
|-<br />
!Colour (name)<br />
!Frequency (THz)<br />
!Stop Potential (V)<br />
|-<br />
| <br />
|<br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
|<br />
| <br />
|<br />
|-<br />
| <br />
|<br />
|<br />
|}<br />
<br />
=Advanced Protocol=<br />
# Study the photocell's charging process for different intensities.<br />
# Find the expected value for wavelength using the led spectra.<br />
# Use those values for a new graphical fitting and compare the results.<br />
# Redo the voltage vs. frequency graph, this time with error bars. <br />
<br />
Note: This setup uses a 12bit ADC from 0V to 5V.<br />
<br />
<br />
=Theoretical Principles=<br />
<br />
==Photoelectric effect==<br />
The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed.<br />
When a photon with energy \( E \) collides with a metal, it transfers it's energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface. <br />
For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons.<br />
The light's intensity changes only the number of emitted photoelectrons, not it's energy (this goes against what is expected in the classical theory of radiation).<br />
<br />
Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)): <br />
<br />
\[<br />
E = h \nu <br />
\]<br />
<br />
Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all it's energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound in the metal's crystal structure), the electron is only freed if the photon's energy is higher than the difference between the electron's energy in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the photon's energy minus the amount "spent" when the electron leaves the metal:<br />
<br />
\[<br />
E = h \nu - e \phi<br />
\]<br />
<br />
where \( e \) is the electron charge and \( \phi \) is the diference in ''workfunction''. As the light's frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emited. In this case, \( E_{max} = 0 \) and we can write:<br />
<br />
\[<br />
h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi<br />
\]<br />
<br />
==In This Setup==<br />
A photoelectric cell is a device where light shines upon a metalic surface (cathode) and exites electrons that will be collected by a concentric metalic surface (anode) like what's shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If we connect the cathode and anode through an external circuit we can measure the current created by the photoelectrons. In a more generic sense, the electron's maximum kinetic energy is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the former. This way, when current no longer flows, we will know the the value of the current.<br />
<br />
[[File:Plank-celula1.png|thumb]]<br />
<br />
We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the former. This creates an electric current through the cell. As the capacitor charges, the voltage between it's terminals increases, diminuishing the volltage between the cell's terminals (because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell's voltage reaches \( V_c = \frac{h \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.<br />
<br />
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Voltage (stopping power) vs. Maximum in the frequency of light]]<br />
<br />
Knowing two or more voltage values for certain frequencies the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light's frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \). <br />
<br />
<br />
=Historical Elements=<br />
In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante de Planck | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Paschen_Curve&diff=1418Paschen Curve2013-11-04T23:53:52Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
<!--[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]--><br />
<br />
Being a different state of matter, there must be conditions at which a gaseous material progresses into plasma state. This experiment explores the conditions at which this transition occurs by studying the breakdown voltage, that is the voltage at which the plasma is formed, between parallel plates as a function of pressure and the distance between plates. <br />
Since the plasma is a good conductor, unlike the gas, it is fairly easy to verify this transition by watching the current drawn from power supply.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/paschen.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Paschen Curve<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
=Experimental Apparatus=<br />
<br />
The state of matter transition from gas into plasma is investigated in this experiment, allowing the study of Paschen's law. This is accomplished through an apparatus similar to the one used by Paschen, in which a voltage is applied to two parallel electrodes surrounded by low pressure gas. The breakdown voltage is determined as a function of pressure and distance separating the two plates.<br />
<br />
The breakdown phenomenon can be interpreted as a chain reaction, where one charged particle collides with a neutral one and generates an extra charged particle. If this collision process has a net gain, then there will be a discharge across the electrodes, otherwise the process will decay and the ionization will stop. It is therefore required that energy of the impacting particle exceeds the ionization energy of the neutral particle, so that there is no recombination and this is a function of both the gas pressure and the plates distance.<br />
<br />
Using the model for a chain reaction, it can be shown that the breakdown voltage under certain conditions is described by the equation:<br />
<br />
\[ V = \dfrac{a.pd}{\ln(pd) + b} \]<br />
<br />
Where (i) \(V\) is the breakdown voltage, (ii) \(p\) is the pressure, and (iii) \(d\) is the gap distance. The constants (iv) \(a\) and (v) \(b\) depend upon the composition of the gas.<br />
<br />
Taking into account the fact that there is no remote control over the distance between electrodes, the first step for the user should be to devise the pressure range through which the experiment will be performed. This will determine the number of times you will have to do the experiment. To configure this parameter, you must use the "Gas Pressure" slider. While the experiment is not in use, the vacuum pump will be connected to the main reaction chamber since the last time the experiment was performed until the moment it was activated. Considering that by the end of the experiment the pump down is monitored as it goes down until it reaches a limit of \(0.05 mBar\), the pressure inside the vessel before the experiment starts will always be at least that one. After the experiment begins, the chamber will be filled with Argon gas until the "Gas Pressure" is reached.<br />
<br />
After that, the user can choose the voltage sweeping characteristics by selecting its "Maximum", "Minimum" and "Increase Step", which affect the signal as shown in figure <!--\ref{paschen_protocol:paschen_config}-->. Choosing a bigger step might hide the details but will allow a faster determination of the region at which the breakdown occurs, while a smaller step will provide a higher detail on the determination of the data. It is recommended that the step value remains constant, since the quickly varying signal of the steps makes the breakdown easier and if it changes throughout the experiment the data will not be usable.<br />
<br />
Finally the data acquisition is also configured by setting the "Maximum", "Minimum" and "Increase Step", since the ADCs performs a data acquisition every time the voltage is increased. Therefore the number of data points is equal to the number of times the voltage has to be updated.<br />
<br />
As mentioned in the beginning, this setup performs the experiment under constant pressure, so this will only result in one data point. To get the Paschen Curve, the user should go through a large range of pressures.<br />
<br />
=Data Analysis=<br />
<br />
The data from a single experiment gives only one point, so the experiment should be repeated several times for different pressures. As with the other e-lab experiment, when the user activates the experiment and the pressure is set, a voltage ramp starts sweeping the electrodes, and the ADC is used to get both current and voltage applied to the electrodes. The software client prints the values in the interface of the voltage, current and pressure during the whole experiment. After that, the user will be able to see a clear transition from 0 to saturation in the current graph. The corresponding point in the voltage is that pressure breakdown. After gathering the points of many different pressures, the user will have a data set similar to the one displayed in figure <!--\ref{paschen_protocol:fit1}-->. Although there is a very high precision in the determination of the voltage values in the chamber, multiple runs of the experiment under the same conditions will show that there is often a window of about 50V under which the disruption can occur. Therefore this value was used for errors in the fit instead of the error with the ADC.<br />
<br />
As it is, the data is still in raw form. The values in pressure must be multiplied by the gap distance and only after this the experimental data is ready to be adjusted to the equation <!--\ref{paschen_protocol:paschen}-->, as seen in figure <!--\ref{paschen_protocol:fit2}-->. In order to adjust to a systematic error in the measurements, you should add an extra fitting parameter:<br />
<br />
\[ V = \dfrac{a.(pd + c)}{\ln(pd + c) + b} \]<br />
<br />
From the fit \(a\) and \(b\) can be extracted, which can allow the determination of the gas inside the chamber. On the contrary, taking the information regarding the gas inside the chamber, in this case Argon, the data points can be used to determine the distance between the plates by fitting the data to the equation <!--\ref{paschen_protocol:paschen}-->.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Paschen_Curve&diff=1417Paschen Curve2013-11-04T23:37:39Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
<!--[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]--><br />
<br />
Being a different state of matter, there must be conditions at which a gaseous material progresses into plasma state. This experiment explores the conditions at which this transition occurs for a gas by studying the breakdown voltage, that is the voltage at which the plasma is formed, between parallel plates as a function of pressure and the distance between plates. <br />
Since the plasma is a good conductor, as opposed to the gas, it is fairly easy to verify this transition by watching the current drawn from power supply.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/paschen.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Paschen Curve<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
=Experimental Apparatus=<br />
<br />
The state of matter transition from gas into plasma is investigated in this experiment, allowing the study of the Paschen's law. This is done through an apparatus similar to the one used by Paschen, in which a voltage is applied to two parallel electrodes surrounded by low pressure gas. The breakdown voltage is determined as a function of pressure and distance separating the two plates.<br />
<br />
The breakdown phenomenon can be interpreted as a chain reaction where one charged particle collides with a neutral one and generates an extra charged particle. If this collision process has a net gain, then there will be a discharge across the electrodes, otherwise the process will decay and the ionization will stop. It is therefore required that energy of the impacting particle exceeds the ionization energy of the neutral particle, so that there is no recombination and this is a function of both the gas pressure and the plates distance.<br />
<br />
Using the model for a chain reaction, it can be shown that the breakdown voltage for given conditions is described by the equation:<br />
<br />
\[ V = \dfrac{a.pd}{\ln(pd) + b} \]<br />
<br />
Where (i) \(V\) is the breakdown voltage, (ii) \(p\) is the pressure, and (iii) \(d\) is the gap distance. The constants (iv) \(a\) and (v) \(b\) depend upon the composition of the gas.<br />
<br />
Taking into account the fact that there is no remote control over the distance between electrodes, the first step for the user should be to devise the range of pressure trough which the experiment will be performed. This will determine the number of times you will have to do the experiment. To configure this parameter, you must use the "Gas Pressure" slider. While the experiment is not in use the vacuum pump will be connected to the main reaction chamber, since the last time the experiment was performed until the moment it was activated. Considering that by the end of the experiment the pump down is monitored as it goes down until it reaches a limit of \(0.05 mBar\), the pressure inside the vessel before the experiment starts will always be at least that one. After the experiment begins, the chamber will be filled with Argon gas until the "Gas Pressure" is reached.<br />
<br />
After that, the user can choose the voltage sweeping characteristics by selecting its "Maximum", "Minimum" and "Increase Step", which affect the signal as shown in figure <!--\ref{paschen_protocol:paschen_config}-->. Choosing a bigger step might hide the details but will allow a faster determination of the region at which the breakdown occurs, while a smaller step will provide a higher detail on the determination of the data. It is recommended that the step value remains constant, since the quickly varying signal of the steps makes the breakdown easier and if it changes throughout the experiment the data will not be usable.<br />
<br />
Finally the data acquisition is also configured by setting the "Maximum", "Minimum" and "Increase Step", since the ADCs preforms a data acquisition every time the voltage is increased. Therefore the number of data points is equal to the number of times the voltage has to update.<br />
<br />
As was mentioned in the beginning, this setup performs the experiment under constant pressure, so this will only result in one data point. To get the Paschen Curve, the user should go trough a large range of pressures.<br />
<br />
=Data Analysis=<br />
<br />
The data from a single experiment gives only one point, so the experiment should be repeated several times for different pressures. As with the other e-lab experiment, when the user activates the experiment and the pressure is set, a voltage ramp starts sweeping the electrodes, and the ADC is used to get both current and voltage applied to the electrodes. The software client prints the values in the interface of the voltage, current and pressure during the whole experiment. After that, the user will be able to see a clear transition from 0 to saturation in the current graph. The corresponding point in the voltage is the that pressure breakdown. After gathering the points of many different pressures, the user will have a data set similar to the one displayed in figure <!--\ref{paschen_protocol:fit1}-->. Although there is a very high precision in the determination of the voltage values in the chamber, multiple runs of the experiment under the same conditions will show that there is often a window of about 50V under which the disruption can occur. Therefore this value was used for errors in the fit instead of the error with the ADC.<br />
<br />
As it is, the data is still in raw form. The values in pressure must be multiplied by the gap distance and only after this the experimental data is ready to be adjusted to the equation <!--\ref{paschen_protocol:paschen}-->, as seen in figure <!--\ref{paschen_protocol:fit2}-->. In order to adjust to a systematic error in the measurements, you should add an extra fitting parameter:<br />
<br />
\[ V = \dfrac{a.(pd + c)}{\ln(pd + c) + b} \]<br />
<br />
From the fit \(a\) and \(b\) can be extracted, which can allow the determination of the gas inside the chamber. On the contrary, taking the information regarding the gas inside the chamber, in this case Argon, the data points can be used to determine the distance between the plates by fitting the data to the equation <!--\ref{paschen_protocol:paschen}-->.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1416Langmuir Probe2013-11-04T23:15:55Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma in accordance with its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data of the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half of the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Being \(j^+_{sat}\) the ion current density, \(A_s\) the contact surface of the probe, \(e\) the electron charge, \(n\) the ion density in the plasma, and \(c_s\) the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and the electrons will be able to reach the probe. Taking a Maxwell distribution of the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Being \(T_e\) the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate of the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively you can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, you want to get the density, so you have to make another correction, which corresponds to adding the value of current at the point where you know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case you use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented above (as seen in Figure 4). From the fit the \(T_e\) can be extracted, as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\), which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, you can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and given that the probe is \(10mm\) and has a diameter of \(0.2mm\) you can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1415Langmuir Probe2013-11-04T23:13:39Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma in accordance with its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half of the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Being \(j^+_{sat}\) the ion current density, \(A_s\) the contact surface of the probe, \(e\) the electron charge, \(n\) the ion density in the plasma, and \(c_s\) the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and the electrons will be able to reach the probe. Taking a Maxwell distribution of the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Being \(T_e\) the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate of the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively you can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, you want to get the density, so you have to make another correction, which corresponds to adding the value of current at the point where you know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case you use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented above (as seen in Figure 4). From the fit the \(T_e\) can be extracted, as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\), which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, you can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and given that the probe is \(10mm\) and has a diameter of \(0.2mm\) you can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1414Langmuir Probe2013-11-04T23:12:15Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma in accordance with its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half of the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Being \(j^+_{sat}\) the ion current density, \(A_s\) the contact surface of the probe, \(e\) the electron charge, \(n\) the ion density in the plasma, and \(c_s\) the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and the electrons will be able to reach the probe. Taking a Maxwell distribution of the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Being \(T_e\) the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate of the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively you can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, you want to get the density, so you have to make another correction, which corresponds to adding the value of current at the point where you know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case you use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented above (as seen in Figure 4). From the fit the \(T_e\) can be extracted, as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\), which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, you can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(10mm\) and has a diameter of \(0.2mm\) you can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1413Langmuir Probe2013-11-04T20:49:23Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma in accordance with its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half of the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Being \(j^+_{sat}\) the ion current density, \(A_s\) the contact surface of the probe, \(e\) the electron charge, \(n\) the ion density in the plasma, and \(c_s\) the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and electrons will be able to reach the probe. Taking a Maxwell distribution of the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Being \(T_e\) the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate of the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively we can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, we want to get the density, so we have to make another correction, which corresponds to adding the value of current at the point where we know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case we use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented in above (as seen in Figure 4). From the fit the \(T_e\) can be extracted as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\) which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, you can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(10mm\) and has a diameter of \(0.2mm\) we can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1389Langmuir Probe2013-10-26T12:27:55Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma according to its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Being \(j^+_{sat}\) the ion current density, \(A_s\) the contact surface of the probe, \(e\) the electron charge, \(n\) the ion density in the plasma, and \(c_s\) the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and electrons will be able to reach the probe. Taking a Maxwell distribution of the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Being \(T_e\) the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate of the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively we can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, we want to get the density, so we have to make another correction, which corresponds to add the value of current in point where we know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case we use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented in above (as seen in Figure 4). From the fit the \(T_e\) can be extracted as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\) which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, one can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(10mm\) and has a diameter of \(0.2mm\) we can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1388Langmuir Probe2013-10-26T12:23:24Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma, which either attracts or repels the electrons in the plasma according to its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma, acting as a Langmuir probe. This central filament has a ground reference around the probe with approximately 200mm, in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) that ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons, the probe will attain a floating potential,\(V_f\), negative in relation to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half the density far away from the probe.<br />
<br />
The probe voltage, \(V_s\) is sweeped in relation to the ground winding filament, using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough, all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma, this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Where \(j^+_{sat}\) is the ion current density, \(A_s\) is the contact surface of the probe, \(e\) is the electron charge, \(n\) is the ion density in the plasma, \(c_s\) is the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and electrons will be able to reach the probe. Taking a Maxwell distribution for the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Where \(T_e\) is the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate for the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct for this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively we can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, we want to get the density, so we have to make another correction, which corresponds to add the value of current in point where we know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case we use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented in above (as seen in Figure 4). From the fit the \(T_e\) can be extracted as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\) which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, one can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(10mm\) and has a diameter of \(0.2mm\) we can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Langmuir_Probe&diff=1387Langmuir Probe2013-10-25T17:12:07Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:Langmuir_with_plasma.jpg|318|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
Plasmas have different characteristics from other states of matter and many diagnostic tools have been developed in order to measure their parameters . <br />
This experiment allows the measurement of some of these characteristics using one of the most simple methods, an electrical probe, the Langmuir probe. It consists of a thin filament made of conductive material placed inside the plasma which either attracts or repels the electrons in the plasma according to its polarization. Measuring the probe I-V characteristic, that is the relationship between the polarization voltage and the respective collected current, you can determine the electron temperature and density of the plasma.<br />
<br />
<table style="background-color:white"><tr><td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/langmuir.sdp<br />
*Laboratory: Advanced in e-lab.ist.utl.pt[http://elab.ist.eu]<br />
*Control room: Langmuir Probe<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
</td><br />
<td style="vertical-align:top"><br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:450px"><br />
'''Videos'''<br />
<div class="mw-collapsible-content"><br />
{{#evp:youtube|EwK4Bn_7IVc|Glow discharche in Langmuir Probe experiment.|center}}<br />
<br />
</div><br />
</div><br />
</td><br />
</tr></table><br />
<br />
=Experimental Apparatus=<br />
<!--[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]--><br />
<br />
The vacuum chamber consists of a Pyrex cylinder on which a glow discharge is produced by RF power. A tungsten filament with a diameter of 0.2mm is placed in the middle of the chamber partially isolated by an alumina cover and only a 10mm tip is exposed to the plasma acting as a Langmuir probe. This central filament as a ground reference around the probe with approximate 200mm in order to be much longer than the probe itself.<br />
<br />
The RF generator (3.5W, 50kHz, 1kV) which ionizes the gas and generates the plasma is connected to two parallel molybdenum plates similar to a capacitor.<br />
<br />
=Protocol=<br />
<br />
==Determination of the electron temperature and density==<br />
<br />
[[File:Langmuir_Data_untreated_nograph.png|200|thumb|Figure 2: Raw Data from the experiment]]<br />
[[File:Langmuir_Data_untreated.png|200|thumb|Figure 3: Linear fit to the ion saturation region]]<br />
[[File:Langmuir_Data_treated.png|200|thumb|Figure 4: Fit to probe characteristic equation]]<br />
<br />
The Langmuir probe uses one or more electrodes placed inside the plasma to measure the current collected while applying a voltage. This will provide the I-V characteristic for that plasma. Normally a triangular or sawtooth wave is used to get equi-separated points.<br />
<br />
When the probe is electrically isolated, a plasma sheath is formed in the interface between the plasma and the probe. To compensate for the higher mobility of the electrons the probe will attain a floating potential,\(V_f\), negative with respect to the plasma potential itself, \(V_p\). The density at the sheath entrance is roughly half of the density away from the probe.<br />
<br />
The probe voltage, \(V_s\), is sweeped with respect to the ground winding filament using a variable voltage source. If the polarization of the probe compared to the plasma potential is negative enough all the electrons will be repelled and the ion flux to the probe is independent of the potential applied. In a totally ionized plasma this ion saturation current is described by the following expression:<br />
\[ i^+_{sat} = j^+_{sat} A_s = \frac{e \, n \, c_s \, A_s}{2} \]<br />
<br />
Where \(j^+_{sat}\) is the ion current density, \(A_s\) is the contact surface of the probe, \(e\) is the electron charge, \(n\) is the ion density in the plasma, \(c_s\) is the ion sound speed.<br />
<br />
When the polarization applied to the probe is increased, the voltage drop in the sheath is reduced and electrons will be able to reach the probe. Taking a Maxwell distribution for the speed of the electrons, the relation between current and tension will be:<br />
\[ i = -i^+_{sat} \left( 1- e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
Where \(T_e\) is the electron temperature. This expression assumes that there is only one probe and that it is non-perturbative.<br />
<br />
From the last equation it is possible to extract an estimate for the floating potential, \(V_f\). This is done by taking the value at which the characteristic crosses zero.<br />
<br />
It will also be easily seen that the data does not follow the regular characteristic on the ion saturation side, it should be constant instead of having a slope. This has to do with the fact that the sheath thickness expands with the applied voltage. To correct for this, a linear function is fitted on this side and the slope is subtracted (as seen in Figure 3). Alternatively we can include this contribution in the equation:<br />
<br />
\[ i = -i^+_{sat} \left( 1- \alpha\ (V-V_f) - e^{\dfrac{e}{k T_e} (V_s - V_f)} \right) \]<br />
<br />
<br />
However, we want to get the density, so we have to make another correction, which corresponds to add the value of current in point where we know the current is completely due to ions, namely to voltage values much lower than the floating potential (in this case we use two times the floating potential, hence the need for our initial estimate).<br />
<br />
After that, the experimental data has to be adjusted to the equation presented in above (as seen in Figure 4). From the fit the \(T_e\) can be extracted as well as the ion saturation current, \(i^+_{sat}\) and the floating potential \(V_f\) which should have a value close to the one we determined empirically.<br />
<br />
By knowing the area of the probe and the speed of sound in the ions of the plasma, one can determine the electron density in the plasma. <br />
Since \(c_s = \sqrt{\dfrac{k T_e}{M}}\), and the fact that the probe is \(10mm\) and has a diameter of \(0.2mm\) we can use the \(i^+_{sat}\) to determine the density.<br />
<br />
=Links=<br />
<br />
Principles of Plasma Diagnostics, Hutchinson, Cambridge</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Dielectric_effect_in_a_Cilindric_Capacitor&diff=1386Dielectric effect in a Cilindric Capacitor2013-10-24T21:55:20Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:condensador-fotografia.png|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
This purpose of this experiment is the determination of the capacity of a variable coax cylindrical capacitor. It has two sections, a first one with a Polystyrene dielectric, and a second one with air, allowing the determination of the relative dielectric constant of Polystyrene. <br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Ligações'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/condensador.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Condensador Cilíndrico<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]<br />
<br />
The capacitor is made of a 200mm long copper tube with an outer diameter of 12mm (inner plate), having been outfitted with a Polystyrene dielectric to a certain length. This is enclosed in a second copper tube with an inner diameter of 16mm (outer plate), that moves along the first, thus acting as the plates of a variable capacitor. This way, the area of the capacitor corresponds only to the parts where the two cylinders overlap.<br />
<br />
Note that, because of set-up constraints, there is a "minimum" capacitor of 26mm. In other words, x=0 actually corresponds to a 26mm long cylindrical capacitor and Polystyrene dielectric. <br />
<br />
<br />
=Protocol=<br />
<br />
==Determination of the relative dielectric constant==<br />
Take two sets of experimental data, one covering the Polystyrene area and the other covering air. The end-points of the sweep should be chosen in a way that allows a precise determination of the slope of the graphical representation. This slope will give the relation between the capacity and the length of the capacitor (each set referring to its corresponding dielectric). The ratio between the two slopes will give the relative dielectric constant of Polystyrene.<br />
<br />
<br />
=Advanced Protocol=<br />
<br />
==Determination of the dielectric constant of air==<br />
Considering Gauss's law, it is possible to determine the capacity of the cylindrical capacitor, using the formula:<br />
<br />
\[<br />
C = \frac{ 2 \pi \epsilon _0 }{ ln(\frac{b}{a}) } L<br />
\]<br />
<br />
By doing a linear regression with the data from the section of the capacitor without the dielectric (which means the air acts as the dielectric) it is possible to accurately determine the value of the capacity. From this, reversing the formula, you can find the permittivity of the air (close to vacuum, i.e. \( \epsilon _0 \)).<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante Dieléctrica num Condensador Cilíndrico | Portuguese Version (Versão em Português)]]<br />
*[http://en.wikipedia.org/wiki/Polystyrene Wikipedia page for Polystyrene]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Dielectric_effect_in_a_Cilindric_Capacitor&diff=1385Dielectric effect in a Cilindric Capacitor2013-10-24T21:50:20Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
[[File:condensador-fotografia.png|thumb|Figure 1: Photo of the experimental setup]]<br />
<br />
This purpose of this experiment is the determination of the capacity of a variable coax cylindrical capacitor. It has two sections, a first with a Polystyrene dielectric, and a second with air, allowing the determination of the relative dielectric constant of Polystyrene. <br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Ligações'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/condensador.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Condensador Cilíndrico<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:condensador-seccao.png|thumb|Figure 2: Sectioned schematic of the capacitor, where a=12mm and b=16mm]]<br />
<br />
The capacitor is made of a copper tube with an outer diameter of 12mm and 200mm long (inner plate), having been outfitted with a Polystyrene dielectric to a certain length. This is enclosed in a second copper tube with an inner diameter of 16mm (outer plate), that moves along the first, thus acting as the plates of a variable capacitor. This way, the area of the capacitor corresponds only to the parts where the two cylinders overlap.<br />
<br />
Note that, because of set-up constraints, there is a "minimum" capacitor of 26mm. In other words, x=0 actually corresponds to a 26mm long cylindrical capacitor and Polystyrene dielectric. <br />
<br />
<br />
=Protocol=<br />
<br />
==Determination of the relative dielectric constant==<br />
Take two sets of experimental data, one covering the Polystyrene area and the other covering air. The end-points of the sweep should be chosen in a way that allows a precise determination of the slope of the graphical representation. This slope will give the relation between the capacity and the length of the capacitor (each set referring to its corresponding dielectric). The ratio between the two slopes will give the relative dielectric constant of Polystyrene.<br />
<br />
<br />
=Advanced Protocol=<br />
<br />
==Determination of the dielectric constant of air==<br />
Considering Gauss's law, it is possible to determine the capacity of the cylindrical capacitor, using the formula:<br />
<br />
\[<br />
C = \frac{ 2 \pi \epsilon _0 }{ ln(\frac{b}{a}) } L<br />
\]<br />
<br />
By doing a linear regression with the data from the section of the capacitor without the dielectric (which means the air acts as the dielectric) it is possible to accurately determine the value of the capacity. From this, reversing the formula, the permittivity of the air can be found (close to vacuum, i.e. \( \epsilon _0 \)).<br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante Dieléctrica num Condensador Cilíndrico | Portuguese Version (Versão em Português)]]<br />
*[http://en.wikipedia.org/wiki/Polystyrene Wikipedia page for Polystyrene]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_the_Adiabatic_Constant&diff=1384Determination of the Adiabatic Constant2013-10-24T20:32:56Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is the determination of the ratio between the specific heat of air (constant pressure and constant volume), through the use of adiabatic oscillations of an embolus of known dimensions.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/gamma.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Cp/Cv<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|BWd4R-ud81I|Slow motion video of the piston performing the damped oscillation motion.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is composed of a syringe, which embolus weighs 26.4 gram and has a diameter of 18.9 mm. The embolus has reduced friction due to graphite lubrication and the fact that the apparatus is in the vertical position.<br />
<br />
<br />
=Protocol=<br />
Ruchhardt’s method (see bellow) is a way to determine the specific heat of a gas in a very precise way, but it is very sensitive to the measurement of the oscillations period. Because of this, extra care in this measurement is recommended and thus, two methods are used to determine this quantity: the waveform recorded by the pressure transducer and the average period, digitally determined. The data must be used judiciously, exploring all the information that it can give. <br />
After a reference volume is selected, the embolus is agitated so that it oscillates freely around the equilibrium position. <br />
\( \gamma \) can be inferred from the oscillation period.<br />
<br />
<br />
=Advanced Protocol=<br />
By redoing the experiment for several volumes, a better adjustment can be achieved between the experimental data and the theoretical function. When adjusting the experimental data, allowing the parameter \( \gamma \) to be free as well as the volume and pressure, the measure precision can be increased, since atmospheric pressure can have variations of up to 1% and because the volume measured will have a systematic error due to the various external connections to the syringe. It should be noted that the piston mass and the diameter have a 0.5% precision.<br />
<br />
<br />
=Data Analysis=<br />
By using [[Fitteia]], you can plot the experimental results and adjust a theoretical function with certain parameters. This [http://www.elab.tecnico.ulisboa.pt/anexos/2012outros/gamma.sav file] is an example of a fit of this experiment (right-click on the link and "Save As").<br />
<br />
<br />
=Theoretical Principles=<br />
With this method, it is possible to determine the ration between the specific heat of a gas through experimentation. If the gas in study is the atmospheric air (mostly diatomic), this ratio should be 1.4.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px"><br />
'''Ruchhardt's Method'''<br />
<div class="mw-collapsible-content"><br />
<br />
If we consider a piston without friction, oscillating freely in a cylinder of volume \( V_0 \), with pressure \( p \), then the force exerted upon the piston ( \( m \ddot{y} \) ) equals the force of gravity minus the variation of pressure upon the piston( \( A \Delta p \) ).<br />
<br />
\[<br />
-mg+A \Delta p = m \ddot{y}<br />
\]<br />
<br />
The variation of pressure for small oscillations in volume is:<br />
<br />
\[<br />
\Delta p = \frac{\partial p}{\partial V} | _{V = V_0}\Delta V<br />
\]<br />
<br />
if we consider a fast enough process so that no exchange in heat occurs (adiabatic process)<br />
<br />
\[<br />
pV^{\gamma} = p_0 V_0 ^{\gamma}, \quad p = \frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma} } <br />
\]<br />
<br />
From the above equation we have:<br />
<br />
\[<br />
\frac{\partial p}{\partial V} | _{V = V_0} = - \gamma \frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma +1} } | _{V = V_0} = - \gamma \frac{p_0}{V_0} <br />
\]<br />
<br />
and<br />
<br />
\[<br />
-mg+ A (- \gamma \frac{p_0}{V_0} \Delta V) = m \ddot{y} , \text{ where } \Delta V = Ay<br />
\]<br />
<br />
simplifying<br />
<br />
\[<br />
\ddot{y} + \gamma \frac{p_0 A^2}{m V_0} y+g = 0<br />
\]<br />
<br />
We make<br />
<br />
\[<br />
\gamma \frac{p_0 A^2}{m V_0} = \omega ^2, \text{ so that } \ddot{y} + \omega ^2 y + g = 0<br />
\]<br />
<br />
Changing the point of origin to the equilibrium position of the piston, we can easily see that this is the equation for the motion of a frictionless harmonic oscillator<br />
<br />
\[<br />
\ddot{y}' + \omega ^2 y' = 0 \text{ with } y = y' - \frac{g}{\omega ^2} \text{ and } \omega ^2 = (\frac{2 \pi}{T})^2 = \gamma \frac{p_0 A^2}{m V_0}<br />
\]<br />
<br />
Measuring the oscillation period, \( T \), we can determine \( \gamma \)<br />
<br />
\[<br />
\gamma = \frac{4mV_0}{p_0 r^4 T^2}<br />
\]<br />
<br />
where \( r \) is the cylinder radius.<br />
A more precise estimation can be achieved using the differential equation considering the dumping effect caused by friction. Is such a situation you could consider friction being proportional to velocity leading to:<br />
\[<br />
\ddot{y} + 2\lambda\omega \dot{y}+\omega ^2 y + g = 0<br />
\]<br />
Considering again the change in the origin, the result of such an equation leads to:<br />
\[<br />
y' = y'_{0} e^{-\lambda \omega t}cos( \sqrt{1 - \lambda^2}\omega t + \phi)<br />
\]<br />
where the period leads to a slight correction due to the dumping factor.<br />
</div><br />
</div><br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante Adiabática do Ar | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_the_Adiabatic_Constant&diff=1383Determination of the Adiabatic Constant2013-10-24T20:31:34Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is the determination of the ratio between the specific heat of air (constant pressure and constant volume), through the use of adiabatic oscillations of an embolus of known dimensions.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/gamma.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Cp/Cv<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|BWd4R-ud81I|Slow motion video of the piston performing the damped oscillation motion.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is composed of a syringe, which embolus weighs 26.4 gram and has a diameter of 18.9 mm. The embolus has reduced friction due to graphite lubrication and the fact that the apparatus is in the vertical position.<br />
<br />
<br />
=Protocol=<br />
Ruchhardt’s method(see bellow) is a way to determine the specific heat of a gas in a very precise way, but it is very sensitive to the measurement of the oscillations period. Because of this, extra care in this measurement is recommended and thus, two methods are used to determine this quantity: the waveform recorded by the pressure transducer and the average period, digitally determined. The data must be used judiciously, exploring all the information that it can give. <br />
After a reference volume is selected, the embolus is agitated so that it oscillates freely around the equilibrium position. <br />
\( \gamma \) can be inferred from the oscillation period.<br />
<br />
<br />
=Advanced Protocol=<br />
By redoing the experiment for several volumes, a better adjustment can be achieved between the experimental data and the theoretical function. When adjusting the experimental data, allowing the parameter \( \gamma \) to be free as well as the volume and pressure, the measure precision can be increased, since atmospheric pressure can have variations of up to 1% and because the volume measured will have a systematic error due to the various external connections to the syringe. It should be noted that the piston mass and the diameter have a 0.5% precision.<br />
<br />
<br />
=Data Analysis=<br />
By using [[Fitteia]], you can plot the experimental results and adjust a theoretical function with certain parameters. This [http://www.elab.tecnico.ulisboa.pt/anexos/2012outros/gamma.sav file] is an example of a fit of this experiment (right-click on the link and "Save As").<br />
<br />
<br />
=Theoretical Principles=<br />
With this method, it is possible to determine the ration between the specific heat of a gas through experimentation. If the gas in study is the atmospheric air (mostly diatomic), this ratio should be 1.4.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px"><br />
'''Ruchhardt's Method'''<br />
<div class="mw-collapsible-content"><br />
<br />
If we consider a piston without friction, oscillating freely in a cylinder of volume \( V_0 \), with pressure \( p \), then the force exerted upon the piston ( \( m \ddot{y} \) ) equals the force of gravity minus the variation of pressure upon the piston( \( A \Delta p \) ).<br />
<br />
\[<br />
-mg+A \Delta p = m \ddot{y}<br />
\]<br />
<br />
The variation of pressure for small oscillations in volume is:<br />
<br />
\[<br />
\Delta p = \frac{\partial p}{\partial V} | _{V = V_0}\Delta V<br />
\]<br />
<br />
if we consider a fast enough process so that no exchange in heat occurs (adiabatic process)<br />
<br />
\[<br />
pV^{\gamma} = p_0 V_0 ^{\gamma}, \quad p = \frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma} } <br />
\]<br />
<br />
From the above equation we have:<br />
<br />
\[<br />
\frac{\partial p}{\partial V} | _{V = V_0} = - \gamma \frac{ p_0 V_0 ^{\gamma} }{ V^{\gamma +1} } | _{V = V_0} = - \gamma \frac{p_0}{V_0} <br />
\]<br />
<br />
and<br />
<br />
\[<br />
-mg+ A (- \gamma \frac{p_0}{V_0} \Delta V) = m \ddot{y} , \text{ where } \Delta V = Ay<br />
\]<br />
<br />
simplifying<br />
<br />
\[<br />
\ddot{y} + \gamma \frac{p_0 A^2}{m V_0} y+g = 0<br />
\]<br />
<br />
We make<br />
<br />
\[<br />
\gamma \frac{p_0 A^2}{m V_0} = \omega ^2, \text{ so that } \ddot{y} + \omega ^2 y + g = 0<br />
\]<br />
<br />
Changing the point of origin to the equilibrium position of the piston, we can easily see that this is the equation for the motion of a frictionless harmonic oscillator<br />
<br />
\[<br />
\ddot{y}' + \omega ^2 y' = 0 \text{ with } y = y' - \frac{g}{\omega ^2} \text{ and } \omega ^2 = (\frac{2 \pi}{T})^2 = \gamma \frac{p_0 A^2}{m V_0}<br />
\]<br />
<br />
Measuring the oscillation period, \( T \), we can determine \( \gamma \)<br />
<br />
\[<br />
\gamma = \frac{4mV_0}{p_0 r^4 T^2}<br />
\]<br />
<br />
where \( r \) is the cylinder radius.<br />
A more precise estimation can be achieved using the differential equation considering the dumping effect caused by friction. Is such a situation you could consider friction being proportional to velocity leading to:<br />
\[<br />
\ddot{y} + 2\lambda\omega \dot{y}+\omega ^2 y + g = 0<br />
\]<br />
Considering again the change in the origin, the result of such an equation leads to:<br />
\[<br />
y' = y'_{0} e^{-\lambda \omega t}cos( \sqrt{1 - \lambda^2}\omega t + \phi)<br />
\]<br />
where the period leads to a slight correction due to the dumping factor.<br />
</div><br />
</div><br />
<br />
<br />
=Links=<br />
*[[Determinação da Constante Adiabática do Ar | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Acoustic_Standing_Waves&diff=1382Acoustic Standing Waves2013-10-23T11:01:53Z<p>Ist128595: </p>
<hr />
<div><br />
=Description of the Experiment=<br />
The purpose of this experiment is to explore basic concepts of standing waves, using sound.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/statsound.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: statsound<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="290" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/StatSound.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is a 1458 millimiters long PVC tube (sometimes referred to as "Kundt's tube"). On one end there is a fixed speaker that can produce an audio sine, a triangular or a single pulse wave. On the opposite side there is a movable piston, changing the effective length of the tube. Along the tube there are several microphones to register the sound intensity at fixed points.<br />
<br />
The following table shows the positions of the microphones in relation to the source (speaker):<br />
<br />
{| border="1"<br />
|-<br />
! Designation<br />
! Distance to source (mm)<br />
|-<br />
| Mic 1 (reference) <br />
| 36<br />
|-<br />
| Mic 2 (center) <br />
| 746<br />
|-<br />
| Mic 3 (extremity) <br />
| 1246<br />
|-<br />
| Mic 4 (embolus surface) <br />
| between 1269 and 1475<br />
|-<br />
| Tube's extremity <br />
| 1455<br />
|+ Table 1 – Microphones distance to the sond source (membrane of the speaker )<br />
|}<br />
<br />
<br />
The reference mic (Mic 1) should be used to verify that the emitted sound is the desired one (i.e., there is no distortion caused by the speaker). On the piston surface there is another microphone (Mic 4) capable of moving between 1269mm and 1475mm.<br />
The sound is acquired through 2 channels of a sound card: the left channel (CH 1) is always bounded to the reference microphone (Mic 1); the other channel (CH 2) can be connected to one of the other three microphones.<br />
<br />
The experimental data is captured by the soundcard and processed online (normalization).<br />
<br />
<br />
=Protocol=<br />
This assembly is also used for the stationary wave experiment, and thus has two modes of operation: in "Standing" mode (position and frequency sweep), the rms value of the soundwave is registered in power decibels. This is done by averaging the tms amplitude of each 200ms sample. These values should be thoroughly analysed because of the non-liniarity in frequency of the speaker(this can lead to a wrong interpretation of results).<br />
<br />
The standing waves occur when the length of the tube is a multiple of half the wavelength.<br />
<br />
\[<br />
L = n \times \frac{\lambda}{2} = n \times \frac{v_{sound}}{2 \times f} \quad n=1,2,3,...<br />
\]<br />
<br />
However, if one of the ends is open (speaker) and the other is closed (piston) the condition changes to: <br />
<br />
\[<br />
L = (2n-1) \times \frac{v_{sound}}{4 \times f} \quad n=1,2,3,...<br />
\]<br />
<br />
In this case, the wave reflected by the piston reaches the speaker just as another wave is being generated (same phase), so there is a high increase in the sound intensity (constructive interference). The membrane of the speaker, as opposed to what might be sugested by common sense, does not "close" the tube. This happens because the membrane oscillates with the air (in the same way that a person pushing a swing is not considered an obstacle). This type of interference occurs when a plane exceeds the speed of sound, creating shock-waves.<br />
<br />
In this situation, the intensity recorded by the microphones is very high and allows a clear view of the phenomenon at hand. The following table shows some typical values of the experiment: <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 2 – Typical experimental values<br />
|-<br />
! scope="col" | Harmony <br />
! scope="col" | Frequency (Hz)<br /> with piston at 1.45m<br />
! scope="col" | Resonance distance (m)<br /> with frequency at 740Hz<br />
|-<br />
|1<br />
|117.24<br />
|0.23<br />
|-<br />
|2<br />
|234.48<br />
|0.46<br />
|-<br />
|3<br />
|351.72<br />
|0.69<br />
|-<br />
|4<br />
|468.97<br />
|0.92<br />
|-<br />
|5<br />
|586.21<br />
|1.15<br />
|-<br />
|6<br />
|703.45<br />
|1.38<br />
|-<br />
|7<br />
|820.69<br />
|1.61<br />
|-<br />
|8<br />
|937.93<br />
|1.84<br />
|-<br />
|9<br />
|1055.17<br />
|2.07<br />
|-<br />
|10<br />
|1172.41<br />
|2.30<br />
|}<br />
<br />
=Advanced Protocol=<br />
There are several experiments that can be done, like trying to understand why the insensity changes between microphones. <br />
To be precise, the standing wave is a sum of several waves that travel in opposing directions but have the same phase, which causes the intensity to be high in the anti-nodes (the piston is always an anti-node pressure when in ressonance). <br />
<br />
However, there are some parts of the tube, called nodes, where the amplitude is zero (or close to zero). An interesting challenge is to calculate the position and frequency (of the piston and the wave, respectively) that create a node on one of the microphones and try to understand why there is still a reading or, alternatively, determine the maxima of the standing wave. The experiment can also be "reversed", i.e. determine the frequency/distance relations that make the wave null, which means a destructive interferance between reflected and emitted waves.<br />
<br />
A phenomenon that also occurs is the existence of standing waves with an open tube (position at 1450mm - piston on the outside). In this case, the sound wave finds an infinite at the tube's exit where the pressure is constant and thus behaves as a pressure node for the standing wave. Here, the resonance condition is different but, because there is partial reflection on the piston, it is harder to interpret.<br />
<br />
<br />
=Links=<br />
*[[Estudo de Estacionárias | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Acoustic_Standing_Waves&diff=1381Acoustic Standing Waves2013-10-23T10:54:05Z<p>Ist128595: </p>
<hr />
<div><br />
=Description of the Experiment=<br />
The purpose of this experiment is to explore basic concepts of standing waves, using sound.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/statsound.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: statsound<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="290" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/StatSound.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is a 1458 millimiters long PVC tube (sometimes referred to as "Kundt's tube"). On one end there is a fixed speaker that can produce an audio sine, a triangular or a single pulse wave. On the opposite side there is a movable piston, changing the effective length of the tube. Along the tube there are several microphones to register the sound intensity at fixed points.<br />
<br />
The following table shows the positions of the microphones in relation to the source (speaker):<br />
<br />
{| border="1"<br />
|-<br />
! Designation<br />
! Distance to source (mm)<br />
|-<br />
| Mic 1 (reference) <br />
| 36<br />
|-<br />
| Mic 2 (center) <br />
| 746<br />
|-<br />
| Mic 3 (extremity) <br />
| 1246<br />
|-<br />
| Mic 4 (embolus surface) <br />
| between 1269 and 1475<br />
|-<br />
| Tube's extremity <br />
| 1455<br />
|+ Table 1 – Microphones distance to the sond source (speaker's membrane)<br />
|}<br />
<br />
<br />
The reference mic (Mic 1) should be used to verify that the emitted sound is the desired one (i.e., there is no distortion caused by the speaker). On the piston surface there is another microphone (Mic 4) capable of moving between 1269mm and 1475mm.<br />
The sound is acquired through 2 channels of a sound card: the left channel (CH 1) is always bounded to the reference microphone (Mic 1); the other channel (CH 2) can be connected to one of the other three microphones.<br />
<br />
The experimental data is captured by the soundcard and processed online (normalization).<br />
<br />
<br />
=Protocol=<br />
This assembly is also used for the stationary wave experiment, and thus has two modes of operation: in "Standing" mode (position and frequency sweep), the rms value of the soundwave is registered in power decibels. This is done by averaging the tms amplitude of each 200ms sample. These values should be thoroughly analysed because of the non-liniarity in frequency of the speaker(this can lead to a wrong interpretation of results).<br />
<br />
The standing waves occur when the length of the tube is a multiple of half the wavelength.<br />
<br />
\[<br />
L = n \times \frac{\lambda}{2} = n \times \frac{v_{sound}}{2 \times f} \quad n=1,2,3,...<br />
\]<br />
<br />
However, if one of the ends is open (speaker) and the other is closed (piston) the condition changes to: <br />
<br />
\[<br />
L = (2n-1) \times \frac{v_{sound}}{4 \times f} \quad n=1,2,3,...<br />
\]<br />
<br />
In this case, the wave reflected by the piston reaches the speaker just as another wave is being generated (same phase), so there is a high increase in the sound intensity (constructive interference). The membrane of the speaker, as opposed to what might be sugested by common sense, does not "close" the tube. This happens because the membrane oscillates with the air (in the same way that a person pushing a swing is not considered an obstacle). This type of interference occurs when a plane exceeds the speed of sound, creating shock-waves.<br />
<br />
In this situation, the intensity recorded by the microphones is very high and allows a clear view of the phenomenon at hand. The following table shows some typical values of the experiment: <br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Table 2 – Typical experimental values<br />
|-<br />
! scope="col" | Harmony <br />
! scope="col" | Frequency (Hz)<br /> with piston at 1.45m<br />
! scope="col" | Resonance distance (m)<br /> with frequency at 740Hz<br />
|-<br />
|1<br />
|117.24<br />
|0.23<br />
|-<br />
|2<br />
|234.48<br />
|0.46<br />
|-<br />
|3<br />
|351.72<br />
|0.69<br />
|-<br />
|4<br />
|468.97<br />
|0.92<br />
|-<br />
|5<br />
|586.21<br />
|1.15<br />
|-<br />
|6<br />
|703.45<br />
|1.38<br />
|-<br />
|7<br />
|820.69<br />
|1.61<br />
|-<br />
|8<br />
|937.93<br />
|1.84<br />
|-<br />
|9<br />
|1055.17<br />
|2.07<br />
|-<br />
|10<br />
|1172.41<br />
|2.30<br />
|}<br />
<br />
=Advanced Protocol=<br />
There are several experiments that can be done, like trying to understand why the insensity changes between microphones. <br />
To be precise, the standing wave is a sum of several waves that travel in opposing directions but have the same phase, which causes the intensity to be high in the anti-nodes (the piston is always a pressure anti-node when in ressonance). <br />
<br />
However, there are some parts of the tube, called nodes, where the amplitude is zero (or close to zero). An interesting challenge is to calculate the position and frequency (of the piston and the wave, respectively) that create a node on one of the microphones and try to understand why there is still a reading or, alternatively, determine the maxima of the standing wave. The experiment can also be "reversed", i.e. deternime the frequency/distance relations that make the wave null, which means a destructive interferance between reflected and emitted waves.<br />
<br />
A phenomenon that also occurs is the existence of standing waves with an open tube (position at 1450mm - piston on the outside). In this case, the sound wave finds an infinite at the tube's exit where the pressure is constant and thus behaves as a pressure node for the standing wave. Here, the condition for resonance is different but, because there is partial reflection on the piston, it is harder to interpret.<br />
<br />
<br />
=Links=<br />
*[[Estudo de Estacionárias | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Light_Polarization&diff=1380Light Polarization2013-10-22T22:19:08Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment allows you to measure the cross-polarization result of light. Two polarizers with variable angles are used in series over a white LED and the light intensity is measured.<br />
<br />
Light can be described as an electromagnetic wave with a property called polarization due to the electric field oscillation in the plane orthogonal to the propagation direction. When the oscillation is just over a single direction, the light is said to be polarized. Certain materials have the property to block the wave except along this precise direction. The objective of this experiment is to reveal how this occurs regarding the light intensity.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/polaroide.sdp<br />
*Laboratory: Intermediate [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control Room: Polaroide<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus consists on a light source (high bright white LED) passing a collimator, which focuses the light rays into a parallel beam of light. At the beginning of the optical path, a vertical light polariser can be interposed.<br />
<br />
In the optical path, light travels through two polarized lenses without graduation, having the angle of one of them been pre-set and being the other one free to rotate around the axis of propagation.<br />
<br />
The light is finally collected through a converging lens into a photo-diode that measures the incident radiation intensity. This intensity is obviously the result of attenuation introduced by polarizing systems brought into its optical path.<br />
<br />
<br />
=Protocol=<br />
In this control room we can measure the attenuation of a light beam caused by the cross-rotation of two polarised lenses. This beam can be selected from the light source or can be previously polarized.<br />
<br />
The supervisor of the experiment can choose two sweep limits for one polarizer and set the angle of the second polarizer acquiring the value of the transmitted power in a photo-diode.<br />
<br />
The resolution (angle increment between two samples) can be chosen according to the interest of the control room supervisor.<br />
<br />
<br />
=Links=<br />
*[[Polarização da Luz | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Light_Polarization&diff=1379Light Polarization2013-10-22T22:12:37Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment allows to measure the cross-polarization result of light. Two polarizers with variable angles are used in series over a white LED and the light intensity is measured.<br />
<br />
Light can be described as an electromagnetic wave with a property called polarization due to the electric field oscillation in the plane orthogonal to the propagation direction. When the oscillation is just over a single direction, the light is said to be polarized. Certain materials have the property to block the wave except along this precise direction. The objective of this experiment is to reveal how this occurs regarding the light intensity.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/polaroide.sdp<br />
*Laboratory: Intermediate [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control Room: Polaroide<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus consists on a light source (high bright white LED) passing a collimator, which focuses the light rays into a parallel beam of light. At the beginning of the optical path, a vertical light polarisercan be interposed.<br />
<br />
In the optical path light travel through two polarized lenses without graduation, the angle of one of them is pre-set and the other one is free to rotate around the axis of propagation.<br />
<br />
The light is finally collected through a converging lens into a photo-diode that measures the incident radiation intensity. This intensity is obviously the result of attenuation introduced by polarizing systems brought into its optical path.<br />
<br />
<br />
=Protocol=<br />
In this control room we can measure the attenuation of a light beam caused by the cross-rotation of two polarises lenses. This beam can be selected from the light source or can be previously polarized.<br />
<br />
The supervisor of the experiment can choose two sweep limits for one polarizer and set the angle of the second polarizer acquiring the value of the transmitted power in a photo-diode.<br />
<br />
The resolution (angle increment between two samples) can be chosen according to the interest of the control room supervisor.<br />
<br />
<br />
=Links=<br />
*[[Polarização da Luz | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Linear_Momentum_Conservation&diff=1378Linear Momentum Conservation2013-10-22T22:01:52Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The pedagogic purpose of this experiment is to teach users about concepts like '''reference frame''', '''center of mass''' and '''conservation of linear momentum'''.<br />
<br />
Besides conservation of momentum, this experiment also shows the collision in four different reference frames: those of the center of mass, of each vehicle and of Earth.<br />
<br />
To achieve this, two cars are launched and will collide with each other and, while that happens, the control system calculates the speed at which the camera should move to film the reference frame chosen by the user.<br />
<br />
By recording and reviewing the video, the user can see all the physical phenomena. The control room gives the inicial and final speeds of the cars.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [unavailable] <!-- rtsp://elabmc.ist.utl.pt/scuba.sdp --><br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: [unavailable]<br />
*Level: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<br />
=Who likes this idea=<br />
[[File:CienciaViva.gif|link=http://www.cienciaviva.pt/escolherciencia/|border|204px|border|204px]]<br />
[[File:ECB Logo.png|link=http://www.externatobenedita.net/|border|150px|border|150px]]<br />
[[File:Logo_alberto_cores.jpg|link=http://www.es-al-berto.com/|border|120px|border|120px]]<br />
[[File:LogoAEA_800x132.jpg|link=http://aealvalade.edu.pt/|border|280px|border|280px]]<br />
<br />
<br />
This idea is the result of the PEC26 contract with Ciência Viva - The Portuguese Agency for Cientific and Technological Culture - and its objectives are (i) sharing (through the internet) existing experimental setups available but not in use in some high-schools and (ii) promote teaching science through the use of experimental activities. The automation of the experiment was made by IST students as part of the e-lab project.<br />
<br />
The first schools involved in this experiment were Escola Secundária Padre António Vieira, Alvalade/Lisboa and Externato Cooperativo da Benedita, Leiria. Escola Secundária Poeta Al Berto in Sines also colaborated.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Colisao-fig1.png|thumb|Setup]]<br />
<br />
The setup is composed of a hollow air rail with tiny holes that, along with an air compressor, creates an air flow. This air flow is equal along the rail (just like air hockey tables) and it allows the cars to slide with minimum friction. This rail is marked in segments of 100mm long blue rectangles that allow a complete and thorough study of the experiment, through image processing.<br />
<br />
Two electromagnets (one at each end of the rail) made from linear solenoids and controlled by a microcontroller send a pre-configured pulse that launches the cars. The inicial speed of both cars are immediately measured by a '''photocell''', by measuring the time the car flags interrupt the sensors.<br />
<br />
Both cars A (215.4±1 gr.) and B (237.1±1 gr.) have two flags with a width of 10±0.2, separated by 20±0.2 mm, that interrupt the cell one after the other. This "double measure" of time allows an estimation of the experimental error.<br />
<!--<br />
{| border="1"<br />
! scope="col" | Quantity <br />
! scope="col" | Value<br />
|- <br />
|Car A's mass<br />
| 215.4±1 gr.<br />
|-<br />
|Car B's mass<br />
| 237.1±1 gr.<br />
|}<br />
--><br />
<br />
==The Cars==<br />
Despite its name, it doesn't look like a car at all. On the photo on the right we can see the 4 main parts:<br />
* A metal triangle that fits on the rail;<br />
* A rubber band in the front to make the collision as elastic as possible (it's impossible to create a perfectlly elastic collision, but this setup allows you to come close to it);<br />
* A U-shaped flag on top that interacts with the photocells;<br />
* A permanent magnet that will be repelled by the electromagnet, creating the pulse that starts the motion.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''What's a photocell?'''<br />
<div class="mw-collapsible-content"><br />
<br />
O photocell is a light sensor (photodiode) and a LED light. By measuring the time the flag interrupts the path between the two parts (emitter and receptor) and knowing the length of that part, we can determine the speed of the car.<br />
<br />
</div><br />
</div><br />
<br />
<br />
<br />
=Protocol=<br />
With this setup, the user can verify the conservation of linear momentum and kinetic energy in an elastic collision. To do this, the user must run the experiment with different starting pulses and observe the experiment in different reference frames.<br />
<br />
To run the experiment, the following parameters must be defined:<br />
# Pulse given to car A (as a percentage of maximum power);<br />
# Pulse given to car B (same as above);<br />
# Reference frame to observe the collision from (car A / car B / center of mass / laboratory).<br />
<br />
The results given are the time each car flags interrupt the photogate, which can be used to determine the speed. From there, the user can calculate both the kinetic energy and the linear momentum (initial and final).<br />
<br />
The video feed can be recorded using [http://www.videolan.org/vlc/ VLC] or a similar software. An image analysis tool (like [http://www.cabrillo.edu/~dbrown/tracker/ Tracker]) can be used to further verify conservation of the physical quantities mentioned.<br />
<br />
<br />
=Advanced Protocol=<br />
The motor that launches each car is a solenoid to which an electric pulse is applied. The duration of the pulse is chosen by the user (by choosing the pulse percentage). The minimum value is 30% (corresponding to a 45ms pulse) to ensure the cars return to the starting point.<br />
<br />
Based on the starting speeds and on the mass of the cars, the pulse created by each solenoid can be estimated. By running the experiment several times with different pulses the user can create a graphic relating the duration of the pulse with the average force created by the solenoid. Will the relation be linear?<br />
<br />
The following data is needed to estimate the motor's efficiency:<br />
* Applied voltage (to the solenoid): 8,5V <br />
* Average current: 3,5A<br />
* Pulse duration: [selected value (%)] \( \times \) 150ms <br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Linear | Portuguese Version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Linear_Momentum_Conservation&diff=1377Linear Momentum Conservation2013-10-22T21:25:37Z<p>Ist128595: </p>
<hr />
<div>=Decription of the Experiment=<br />
The pedagogic purpose of this experiment is to teach users about concepts like '''reference frame''', '''center of mass''' and '''conservation of linear momentum'''.<br />
<br />
This experiment implicates not only conservation of momentum, it also shows the collision in four different reference frames: those of the center of mass, of each vehicle and of Earth.<br />
<br />
To achieve this, two cars are launched and will collide with each other and, while that happens, the control system calculates the speed at which the camera should move to film the reference frame chosen by the user.<br />
<br />
By recording and reviewing the video, the user can see all the physical phenomena. The control room gives the inicial and final speeds of the cars.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [unavailable] <!-- rtsp://elabmc.ist.utl.pt/scuba.sdp --><br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: [unavailable]<br />
*Level: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<br />
=Who likes this idea=<br />
[[File:CienciaViva.gif|link=http://www.cienciaviva.pt/escolherciencia/|border|204px|border|204px]]<br />
[[File:ECB Logo.png|link=http://www.externatobenedita.net/|border|150px|border|150px]]<br />
[[File:Logo_alberto_cores.jpg|link=http://www.es-al-berto.com/|border|120px|border|120px]]<br />
[[File:LogoAEA_800x132.jpg|link=http://aealvalade.edu.pt/|border|280px|border|280px]]<br />
<br />
<br />
This idea is the result of the PEC26 contract with Ciência Viva - The Portuguese Agency for Cientific and Technological Culture - and its objectives are (i) sharing (through the internet) existing experimental setups that some high-schools have but are not in use and (ii) promote teaching science through the use of experimental activities. The automation of the experiment was made by IST students as part of the e-lab project.<br />
<br />
The first schools involved in this experiment were Escola Secundária Padre António Vieira, Alvalade/Lisboa and Externato Cooperativo da Benedita, Leiria. Escola Secundária Poeta Al Berto in Sines also colaborated.<br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Colisao-fig1.png|thumb|Setup]]<br />
<br />
The setup is composed of a hollow air rail with tiny holes that, along with an air compressor, creates an air flow. This air flow is equal along the rail (just like air hockey tables) and it allows the cars to slide with minimum friction. This rail is marked in segments of 100mm long blue rectangles that allow a complete and thorough study of the experiment, through image processing.<br />
<br />
Two electromagnets (one at each end of the rail) made from linear solenoids and controlled by a microcontroller send a pre-configured pulse that launches the cars. The inicial speed of both cars are immediately measured by a '''photocell''', by measuring the time the car flags interrupt the sensors.<br />
<br />
Both cars A (215.4±1 gr.) and B (237.1±1 gr.) have two flags with a width of 10±0.2, separeted by 20±0.2 mm, that interrupt the cell one after the other. This "double measure" of time allows an estimation of the experimental error.<br />
<!--<br />
{| border="1"<br />
! scope="col" | Quantity <br />
! scope="col" | Value<br />
|- <br />
|Car A's mass<br />
| 215.4±1 gr.<br />
|-<br />
|Car B's mass<br />
| 237.1±1 gr.<br />
|}<br />
--><br />
<br />
==The Cars==<br />
Despite its name, it doesn't look like a car at all. On the photo on the right we can see the 4 main parts:<br />
* A metal triangle that fits on the rail;<br />
* A rubber band in the front to make the collision as close to elastic as possible (it's impossible to create a perfectlly elastic collision, but this setup allows us to come close to it);<br />
* An U-shaped flag on top that interacts with the photocells;<br />
* A permanent magnet that will be repelled by the electromagnet, creating the pulse that starts the motion.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''What's a photocell?'''<br />
<div class="mw-collapsible-content"><br />
<br />
O photocell is a light sensor (photodiode) and a LED light. By measuring the time the flag interrupts the path between the two parts (emitter and receptor) and knowing the length of that part, we can determine the speed of the car.<br />
<br />
</div><br />
</div><br />
<br />
<br />
<br />
=Protocol=<br />
With this setup, the user can verify the conservation of linear momentum and kinetic energy in an elastic collision. To do this, the user must run the experiment with different starting pulses and observe the experiment in different reference frames.<br />
<br />
To run the experiment, the following parameters must be defined:<br />
# Pulse given to car A (as a percentage of maximum power);<br />
# Pulse given to car B (same as above);<br />
# Reference frame to observe the collision from (car A / car B / center of mass / laboratory).<br />
<br />
The results given are the time each car flags interrupt the photogate, which can be used to determine the speed. From there, the user can calculate both kinetic energy and linear momentum (inicial and final).<br />
<br />
The video feed can be recorded using [http://www.videolan.org/vlc/ VLC] or a similar software. An image analysis tool (like [http://www.cabrillo.edu/~dbrown/tracker/ Tracker]) can be used to further verify conservation of the physical quantities mentioned.<br />
<br />
<br />
=Advanced Protocol=<br />
The motor that launches each car is a solenoid to which an electric pulse is applied. The duration of the pulse is chosen by the user (by choosing the pulse percentage). The minimum value is 30% (corresponding to a 45ms pulse) to ensure the cars return to the starting point.<br />
<br />
Based on the starting speeds and on the mass of the cars, the pulse created by each solenoid can be estimated. By running the experiment several times with different pulses the user can create a graphic relating the duration of the pulse with the average force created by the solenoid. Will the relation be linear?<br />
<br />
The following data is needed to estimate the motor's efficiency:<br />
* Applied voltage (to the solenoid): 8,5V <br />
* Average current: 3,5A<br />
* Pulse duration: [selected value (%)] \( \times \) 150ms <br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Linear | Portuguese Version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Semi-cilinder_Optical_Behavior&diff=1376Semi-cilinder Optical Behavior2013-10-22T19:14:40Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment has multiple purposes:<br />
# To determine the refraction index of Plexiglas and verify Snell's law;<br />
# To measure the critical angle at which there is total internal reflection;<br />
# To study the power transmitted and reflected in an optical interface, proving conservation of energy;<br />
# To study the power transmitted and reflected as a function of laser polarization, determining the Brewster's angle.<br />
<br />
This analysis can be done qualitatively by processing the picture from the camera or quantitatively by using the light sensor that sweeps the semicilinder angle-wise. Due to experimental constraints, the avaliable angle range is only 0º through 240º.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/optica.sdp<br />
*Laboratory: Intermediate at e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: optica<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Protocol=<br />
<br />
==Protocol I==<br />
The ray tracing mode allows a qualitative analysis of the transmission and reflection of light in the interface between Plexiglas and air through the pictures captured by the camera.<br />
<br />
The user chooses the starting and the ending angle for the sweep. With some pictures and Snell's law it is possible to calculate the refraction index for the denser material (Plexiglas). We suggest using an image processing software like Corelpaint, Photoshop or Draw (openoffice.org) to analyse the pictures: simply find the light rays and measure the angle between them. <br />
<br />
Note: selecting angles between 0º and 180º means the light will shine upon the curved side of the semi-cylinder, whereas between 180º and 360º the light will shine on the straight side.<br />
<br />
==Protocol II==<br />
In "direct light" mode, the user selects the angle at which light shines upon the cylinder, and during the experiment itself the output will be the laser intensity. Two maxima can usually be found when sweeping the circle: one is the transmitted ray and the other the reflected one. <br />
At 0º, the user can determine the laser absolute power. We also suggest angles in [90º:180º] to study the Plexiglas-air interface and [270º-360º] for air-Plexiglas interface.<br />
This procedure can be done with; (i) non-polarized light, (ii) vertically polarized light, (iii) horizontally polarized light.<br />
<br />
==Protocol III==<br />
According to Snell's Law, there is an angle (it's called the Critical Angle and only exists in the case of light crossing from an optically denser material to a less dense one) where there is no transmitted beam, because the incident beam was completely reflected in the contact surface between the two materials.<br />
<br />
After determining the refraction index ([[#Protocol I|Protocol I]]), this angle can be calculated and the user can verify through experiment that there is no transmission beyond it. We recommend looking at the reflected and transmitted power in the neighborhood of this angle through small increases and find a mathematical function that best fits these points. The angles used should be between 90º and 180º (these correspond to a transition from the denser material to the less dense one). <br />
<br />
<br />
==Protocol IV==<br />
After polarizing the laser beam, we notice that the light behaves differently in the transitions ([[#Protocolo II |Protocolo II]]). Specifically, there is an angle (called Brewster's Angle) where a vertically polarized wave cannot be reflected, therefore creating a transmitted beam that is perpendicular to the incident one. The angle can be calculated using the refractive indexes. <br />
<br />
<br />
=Links=<br />
*[[Estudos de Óptica num Prisma Semi-cilíndrico | Portuguese Version (Versão em Português)]]<br />
*[http://en.wikipedia.org/wiki/Poly(methyl_methacrylate) Wikipedia article about Plexiglas]<br />
*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/efficacy.html Table with the conversion factors between Lumens and Watt]<br />
*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/lumpow.html#c1 Article on Hyperphysics about the relation between Lumens and Watt]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Semi-cilinder_Optical_Behavior&diff=1375Semi-cilinder Optical Behavior2013-10-22T19:07:24Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment has multiple purposes:<br />
# To determine the refraction index of Plexiglas and verify Snell's law;<br />
# To measure the critical angle at which there is total internal reflection;<br />
# To study the power transmitted and reflected in an optical interface, proving conservation of energy;<br />
# To study the power transmitted and reflected as a function of laser polarization, determining the Brewster's angle.<br />
<br />
This analysis can be done qualitatively by processing the picture from the camera or quantitatively by using the light sensor that sweeps the semicilinder angle-wise. Due to experimental constraints, the avaliable angle range is only 0º through 240º.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/optica.sdp<br />
*Laboratory: Intermediate at e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: optica<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Protocol=<br />
<br />
==Protocol I==<br />
The ray tracing mode allows a qualitative analysis of the transmission and reflection of light in the interface between Plexiglas and air through the pictures captured by the camera.<br />
<br />
The user chooses the starting and the ending angle for the sweep. With some pictures and Snell's law it is possible to calculate the refraction index for the denser material (Plexiglas). We suggest using an image processing software like Corelpaint, Photoshop or Draw (openoffice.org) to analyse the pictures: simply find the light rays and measure the angle between them. <br />
<br />
Note: selecting angles between 0º and 180º means the light will shine upon the curved side of the semi-cylinder, whereas between 180º and 360º the light will shine on the straight side.<br />
<br />
==Protocol II==<br />
In "direct light" mode, the user selects the angle at which light shines upon the cylinder, and during the experiment itself the output will be the laser intensity. Two maxima can usually be found when sweeping the circle: one is the transmitted ray and the other the reflected one. <br />
At 0º, the user can determine the laser absolute power. We also suggest angles in [90º:180º] to study the Plexiglas-air interface and [270º-360º] for air-Plexiglas interface.<br />
This procedure can be done with; (i) non-polarized light, (ii) vertically polarized light, (iii) horizontally polarized light.<br />
<br />
==Protocol III==<br />
According to Snell's Law, there is an angle (it's called the Critical Angle and only exists in the case of light crossing from an optically denser material to a lesser one) where there is no transmitted beam, because the incident beam was completely reflected in the contact surface between the two materials.<br />
<br />
After determining the refraction index ([[#Protocol I|Protocol I]]), this angle can be calculated and the user can verify through experiment that there is no transmission beyond it. We recommend looking at the reflected and transmitted power in the neighborhood of this angle, through small increases and find a mathematical function that best fits these points. The angles used should be between 90º and 180º (these correspond to a transition from the denser material to the less dense one). <br />
<br />
<br />
==Protocol IV==<br />
After polarizing the laser beam, we notice that the light behaves differently in the transitions ([[#Protocolo II |Protocolo II]]). Specifically, there is an angle (called Brewster's Angle) where a vertically polarized wave cannot be reflected, therefore creating a transmitted beam that is perpendicular to the incident one. The angle can be calculated using the refractive indexes. <br />
<br />
<br />
=Links=<br />
*[[Estudos de Óptica num Prisma Semi-cilíndrico | Portuguese Version (Versão em Português)]]<br />
*[http://en.wikipedia.org/wiki/Poly(methyl_methacrylate) Wikipedia article about Plexiglas]<br />
*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/efficacy.html Table with the conversion factors between Lumens and Watt]<br />
*[http://hyperphysics.phy-astr.gsu.edu/hbase/vision/lumpow.html#c1 Article on Hyperphysics about the relation between Lumens and Watt]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1374Angular Momentum Conservation2013-10-22T17:27:20Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Beyond that, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The motor of the apparatus can be used as a generator equipped with a switchable resistor, acting as an electromagnetic brake. The braking current and the voltage characteristic are measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figure 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this point, the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and the fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the rule of thumb of the friction deceleration related to angular velocity.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figure2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figure3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figure4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figure5: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance similar to the windings of the motor ('''Picture 3'''). These resistors will dissipate energy, acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results at the end of the session.<br />
<br />
'''Fugures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Figure 2''', before the braking, to do a linear regression, you can get the angular deceleration caused by friction, assumed constant all the time, from the slope of the rect. From this deceleration it will be possible to calculate the instantaneous loss of angular momentum differentially.<br />
<br />
Between each speed acquisition an energy balance is done . The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of one acquisition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used, the energy dissipates in 3 branches, so the power comes multiplied by 3. Besides, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
The ballance between each consecutive acquitition is summed in the end.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finally the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, we reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Figure 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all the energies that allows the verification of the conservation of energy throughout the experiment.<br />
<br />
The disc is in fact a ring having an interior radius of 13mm and an exterior radius of 47mm, so its theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9%, which gives a good approximation of the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum to the equations, you can infer an approximated value of the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1373Angular Momentum Conservation2013-10-22T17:24:40Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Beyond that, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The motor of the apparatus can be used as a generator equipped with a switchable resistor, acting as an electromagnetic brake. The braking current and the voltage characteristic are measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this point, the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and the fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the rule of thumb of the friction deceleration related to angular velocity.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figure2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figure3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figure4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figure5: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance similar to the windings of the motor ('''Picture 3'''). These resistors will dissipate energy, acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results at the end of the session.<br />
<br />
'''Fugures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Figure 2''', before the braking, to do a linear regression, you can get the angular deceleration caused by friction, assumed constant all the time, from the slope of the rect. From this deceleration it will be possible to calculate the instantaneous loss of angular momentum differentially.<br />
<br />
Between each speed acquisition an energy balance is done . The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of one acquisition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used, the energy dissipates in 3 branches, so the power comes multiplied by 3. Besides, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
The ballance between each consecutive acquitition is summed in the end.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finally the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, we reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Figure 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all the energies that allows the verification of the conservation of energy throughout the experiment.<br />
<br />
The disc is in fact a ring having an interior radius of 13mm and an exterior radius of 47mm, so its theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9%, which gives a good approximation of the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum to the equations, you can infer an approximated value of the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1372Angular Momentum Conservation2013-10-22T17:17:31Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Beyond that, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The motor of the apparatus can be used as a generator equipped with a switchable resistor, acting as an electromagnetic brake. The braking current and the voltage characteristic are measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this point, the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and the fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the rule of thumb of the friction deceleration related to angular velocity.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figura3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figura4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figura5: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance similar to the windings of the motor ('''Picture 3'''). These resistors will dissipate energy, acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results at the end of the session.<br />
<br />
'''Pictures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Picture 2''', before the braking, to do a linear regression, you can get the angular deceleration caused by friction, assumed constant all the time, from the slope of the rect. From this deceleration it will be possible to calculate the instantaneous loss of angular momentum differentially.<br />
<br />
Between each speed acquisition an energy balance is done . The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of one acquisition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used the energy dissipates in 3 branches, so the power comes multiplied by 3. Besides, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
The ballance between each consecutive acquitition is summed in the end.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finaly the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, we reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Picture 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all the energies that allows the verification of the conservation of energy throughout the experiment.<br />
<br />
The disc is in fact a ring having an interior radius of 13mm and an exterior radius of 47mm, so its theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9% which gives a good approximation of the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum to the equations, you can infer an approximated value of the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1368Angular Momentum Conservation2013-10-21T12:00:25Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Moreover, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The engine of the apparatus can be used as a generator equipped with a switchable resistor acting as an electromagnetic brake. The braking current and the voltage characteristic are measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this moment the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the friction deceleration related to angular velocity rule of thumb.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figura3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figura4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figura5: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance similar to the motor's windings ('''Figure 3'''). These resistors will dissipate energy, acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results in the end of the session.<br />
<br />
'''Pictures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Picture 2''', before the braking, to do a linear regression, you can get the angular deceleration caused by friction, assumed constant all the time, from the slope of the rect. From this deceleration it will be possible to calculate the instantaneous loss of angular momentum differentially.<br />
<br />
Between each speed acquisition an energy balance is done . The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of one acquisition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used the energy dissipates in 3 branches, so the power comes multiplied by 3. Besides, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
The ballance between each consecutive acquitition is summed in the end.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finaly the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, we reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Picture 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all energies that allows the verification of the conservation of energy throughout the experiment.<br />
<br />
The disc is in fact a ring having an interior radius of 13mm and an exterior radius of 47mm, so its theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9% which gives a good approximation of the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum on the equations, you can infer an approximated value of the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1367Angular Momentum Conservation2013-10-21T11:45:16Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Moreover, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The engine of the apparatus can be used as a generator equipped with a switchable resistor acting as an electromagnetic brake. The braking current and the voltage characteristic is measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this moment the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the friction deceleration related to angular velocity rule of thumb.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figura3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figura4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figura5: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance similar to the motor's windings ('''Figure 3'''). These resistors will dissipate energy, acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results in the end of the session.<br />
<br />
'''Figures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Figure 2''', before the braking, to do a linear regression, you can get the angular deceleration caused by friction, assumed constant all the time, from the slope of the rect. From this deceleration it will be possible to calculate differentially the instantaneous loss of angular momentum.<br />
<br />
Between each speed acquisition an energy balance is done . The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of the acquisition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used the energy dissipates in 3 branches so the power comes multiplied by 3. Also, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
In the end the ballance between each consecutive acquitition is summed.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finaly the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, it was reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Figure 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all energies that allows the verification of the conservation of energy through all the experiment.<br />
<br />
The disc is in fact a ring with interior radius 13mm and exterior 47mm so it's theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9% which gives a good approximation for the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum on the equations one can infer an approximated value for the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1352Angular Momentum Conservation2013-10-20T21:40:34Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by making collide a spinning disk with another one. Moreover, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The engine of the apparatus can be used as a generator equipped with a switchable resistor acting as an electromagnetic brake. The braking current and the voltage characteristic is measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this moment the engine is disconnected from power and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experiment in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the rule of thumb for the friction deceleration related to angular velocity.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figura3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figura4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figura5: Final energy balance showing electrical and mechanic component making it possible to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance is the same as the motor's windings ('''Figure 3'''). These resistors will dissipate energy acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results in the end of the session.<br />
<br />
'''Figures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Figure 2''', before the braking, to do a linear regression, one can get the angular deceleration caused by friction, assumed constant at all time, from the slope of the rect. From the deceleration it will be possible to calculate differentially the instantaneous loss of angular momentum.<br />
<br />
Between each speed acquisition it is done an energy balance. The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of the acquition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used the energy dissipates in 3 branches so the power comes multiplied by 3. Also, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
In the end the ballance between each consecutive acquitition is summed.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finaly the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, it was reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Figure 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all energies that allows the verification of the conservation of energy through all the experiment.<br />
<br />
The disc is in fact a ring with interior radius 13mm and exterior 47mm so it's theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9% which gives a good approximation for the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum on the equations one can infer an approximated value for the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&diff=1351Angular Momentum Conservation2013-10-20T21:33:13Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This control room allows the confirmation of angular momentum conservation by colliding a spinning disk with another one. Moreover, the disk inertia momentum can be extrapolated based on the principles of conservation of energy.<br />
<br />
<!-- Acho que este texto não está muito correcto. --><br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp<br />
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]<br />
*Control room: [unavaliable]<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|left}}<br />
<br />
<br />
=Experimental Apparatus=<br />
The experimental apparatus is based on a PC hard disk drive motor and its spinning disc with a mass of 115g, 12.5mm internal radius and 47.5mm external. A second disc with a mass of 69g and the same dimensions is held on top of it and can be dropped by a servo motor actuator.<br />
<br />
The engine of the apparatus can be used as a generator equipped with a switchable resistor acting as an electromagnetic brake. The braking current and the voltage characteristic is measured allowing an accurate calculation of energy dissipation.<br />
<br />
<br />
=Protocol - Angular Momentum Conservation=<br />
<br />
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figura 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]<br />
<br />
The bottom disc is accelerated by the motor until it reaches a selected angular velocity. At this instant the motor is disconnected from supply and the disc allowed to rotate freely. When a certain pre-determined velocity is reached, the servo lets the suspended disc initially at rest fall on top of the rotating disc.<br />
<br />
Data taken from the experiment is given and plotted with the disc velocity in function of time.<br />
<br />
'''Figure1''' is a plot of the results of an experient in which a servo lets the suspended disc fall when the disc below reaches 1000 rpm.<br />
<br />
Doing a linear regression between the deceleration and fall of the disc, it is possible to obtain the predicted velocity at any time. This gives us the rule of thumb for the friction deceleration related to angular velocity.<br />
<br />
<br />
=Advanced Protocol - Moment of Inertia Evaluation= <br />
<br />
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: Rotational velocity as function of time after the electromagnetic breaking]]<br />
[[File:Discos_tensao_2fases.png|thumb|alt=|Figura3: Circuit schematic for voltage measurement]]<br />
[[File:Discos_tensao.png|thumb|alt=|Figura4: Voltage between two phases during breaking]]<br />
[[File:Discos_balanco_energetico.png|thumb|alt=|Figura5: Final energy balance showing electrical and mechanic component making it possible to get the total moment of inertia]]<br />
<br />
The bottom disc is accelerated by the motor to a selected angular velocity. At that time the motor supply is disconnected and the disc is allowed to rotate freely. When a certain pre-determined velocity is reached, a relay puts each motor winding in parallel with a resistor resistance is the same as the motor's windings ('''Figure 3'''). These resistors will dissipate energy acting as an electromagnetic brake. Voltage and velocity as functions of time are given in a table of results in the end of the session.<br />
<br />
'''Figures 2''' and '''4''' are plots obtained through the table of results of an experiment in which the relay turns on when the rotating discs reach 1400 rpm.<br />
<br />
Using the first data from '''Figure 2''', before the braking, to do a linear regression, one can get the angular deceleration caused by friction, assumed constant at all time, from the slope of the rect. From the deceleration it will be possible to calculate differentially the instantaneous loss of angular momentum.<br />
<br />
Between each speed acquisition it is done an energy balance. The loss of total mechanical energy must be equal to the sum of losses by friction and electromagnetic breaking.<br />
<br />
$\Delta E_{mec} = \Delta E_{fr} + \Delta E_{ele}$<br />
<br />
The energy of a rotating body is $E_{rot}=\frac{I w^2}{2}$ I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:<br />
<br />
$\Delta E_{mec}=\frac{I(w_{2exp}^2-w_{1exp}^2)}{2}$<br />
<br />
$w_{2exp}$ and $w_{1exp}$ being the angular velocity in two consecutive acquisitions.<br />
<br />
The loss of energy by friction will be:<br />
<br />
$\Delta E_{fr}=Iw_{exp}\left(w_{2wo/fr}-w_{1wo/fr}\right)$<br />
<br />
$w_{exp}$ is the angular velocity of the disc in that acquisition and $w_{2wo/fr}$ $w_{1wo/fr}$ are the extrapolated velocity of the disc in a no friction situation at the time of the acquition and the previous one respectively.<br />
<br />
The dissipated power is:<br />
<br />
$P=VI=\frac{V^2}{R}$<br />
<br />
The rms voltage across one winding is:<br />
<br />
$V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}$<br />
<br />
In the setup used the energy dissipates in 3 branches so the power comes multiplied by 3. Also, each winding is in parallel with a resistor with the same resistence value $4,7\Omega$, which means the power equation will come multiplied by 2 and $R=4,7\Omega$ will be used.<br />
<br />
$P=3\times2\times\frac{V_{rms}^2}{R}=3\times2\times\left(\frac{V_{measured}}{\sqrt{3}\sqrt{2}}\right)^2\frac{1}{R}$<br />
<br />
$P=\frac{V^2}{R}$<br />
<br />
The energy dissipated will be:<br />
<br />
$\Delta E_{ele}=P*\Delta t$<br />
<br />
Where $\Delta t$ is the time between acquisitions.<br />
<br />
In the end the ballance between each consecutive acquitition is summed.<br />
<br />
$Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{ele}$<br />
<br />
Finaly the goal-seek function of Microsoft Excel is used to force the sum of balances to be 0 (zero) changing the value of I.<br />
<br />
Using this method, it was reached an experimental value of $1,274\times10^{-4}$ $kg$ $m^2$ for the moment of inertia.<br />
<br />
'''Figure 5''' shows the energy of the disc in function of time, the energy lost by friction and electromagnetic breaking and the sum of all energies that allows the verification of the conservation of energy through all the experiment.<br />
<br />
The disc is in fact a ring with interior radius 13mm and exterior 47mm so it's theoretical moment of inertia is:<br />
<br />
$I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%$<br />
<br />
This experiment gave results that differ by '''6,8%''' from the ones calculated theoreticaly.<br />
<br />
=Physics=<br />
Using the following quantities:<br />
<br />
L - angular momentum<br />
<br />
I - moment of inertia<br />
<br />
&omega; - angular velocity <br />
<br />
m - mass in rotation.<br />
<br />
For the angular momentum conservation:<br />
<br />
$L_i=L_f$<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}$<br />
<br />
$\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}$<br />
<br />
The experimental results give:<br />
<br />
$\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656$<br />
<br />
while the predicted mass ratio is<br />
<br />
$\frac{m_i}{m_f}=\frac{115}{115+69}=0,625$<br />
<br />
Evaluating the accuracy:<br />
<br />
$\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%$<br />
<br />
The speed ratio is different from the mass ratio by 4,9% which gives a good approximation for the angular momentum conservation.<br />
<br />
Knowing the exact dimensions of the disks ($r_1=12,5mm, r_2=47,5mm$) and adding an error momentum on the equations one can infer an approximated value for the motor rotor momentum of inertia (or its mass knowing its average radius).<br />
<br />
$I_i \omega_i=I_f \omega_f$<br />
<br />
$\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f$<br />
<br />
Solving in order to $I_m$<br />
<br />
$I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}$<br />
<br />
<br />
=Links=<br />
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Weather_Station&diff=1350Weather Station2013-10-20T21:18:36Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
Under construction.<br />
<br />
<br />
=Experimental Apparatus=<br />
This weather station is installed on top of [http://www.ist.utl.pt/ Instituto Superior Técnico]'s [http://maps.google.com/maps?daddr=38.736758,-9.139397&hl=en&ll=38.736904,-9.139182&spn=0.00213,0.005284&sll=38.736829,-9.139182&sspn=0.00228,0.005284&t=h&mra=mift&mrsp=1&sz=18&z=18 Central Pavilion], in Lisbon, Portugal. It is permanently recording the data it acquires in a database, averaging the values for each hour. The user can access the data from the last acquisition and the database itself, represented as graphs.<br />
<br />
The sensor is a thermistor, i.e. a resistor that varies with temperature. The temperature for a given value in electrical resistance can be found with the following formula (given by the manufacturer):<br />
<br />
This sensor was developed by the MERCATOR company, having been placed in a RHU217-AT model board. Operational values are between 0º and 60º (precision \( \pm 0.7 ºC\) at 25ºC).<br />
<br />
<br />
=Protocol=<br />
First, the user has to select the data to be observed (by default, the latest data is shown).<br />
<br />
[[File:Meteo-config1.jpg|thumb]]<br />
<br />
Sensor panel:<br />
* Select the type of data to be shown, by choosing which sensor to look at.<br />
<br />
Resolution panel:<br />
* Select the time scale for the graphical representation.<br />
<br />
Period panel:<br />
* Select the time interval between samples.<br />
<br />
Buttons:<br />
* OK - confirm the chosen selections, go to the experiment stage. <br />
* Cancel - close the config window.<br />
* Default config - default selection.<br />
<br />
[[File:Meteo-config2.jpg|thumb]]<br />
<br />
'''Data Visualization Window'''<br />
After the configuration, the chosen data can be observed. The latest acquisition is always shown and the data from the database is shown graphically.</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Weather_Station&diff=1349Weather Station2013-10-20T21:14:16Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
Under construction.<br />
<br />
<br />
=Experimental Apparatus=<br />
This weather station is installed on top of [http://www.ist.utl.pt/ Instituto Superior Técnico]'s [http://maps.google.com/maps?daddr=38.736758,-9.139397&hl=en&ll=38.736904,-9.139182&spn=0.00213,0.005284&sll=38.736829,-9.139182&sspn=0.00228,0.005284&t=h&mra=mift&mrsp=1&sz=18&z=18 Central Pavilion], in Lisbon, Portugal. It is permanently recording the data it acquires in a database, averaging the values for each hour. The user can access the data from the last acquisition and the database itself, represented as graphs.<br />
<br />
The sensor is a thermistor, i.e. a resistor that varies with temperature. The temperature for a given value in electrical resistance can be found with the following formula (given by the manufacturer):<br />
<br />
This sensor was developed by the MERCATOR company, having been placed in a RHU217-AT model board. Operational values are between 0º and 60º (precision \( \pm 0.7 ºC\) at 25ºC).<br />
<br />
<br />
=Protocol=<br />
First, the user has to select the data to be observed (by default, the latest data is shown).<br />
<br />
[[File:Meteo-config1.jpg|thumb]]<br />
<br />
Sensor panel:<br />
* Select the type of data to be shown, by choosing which sensor to look at.<br />
<br />
Resolution panel:<br />
* Select the time scale for the graphical representation.<br />
<br />
Period panel:<br />
* Select the time interval between samples.<br />
<br />
Buttons:<br />
* OK - confirm the chosen selections, go to the experiment stage. <br />
* Cancel - close the config window.<br />
* Default config - default selection.<br />
<br />
[[File:Meteo-config2.jpg|thumb]]<br />
<br />
'''Data Visualing Window'''<br />
After the configuration, the chosen data can be viewed. The latest acquisition is always shown and the data from the database is shown graphically.</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Radiation_Attenuation_over_Different_Materials&diff=1348Radiation Attenuation over Different Materials2013-10-20T20:52:28Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This laboratory uses a [[#Theoretical Principles|Geiger-Müller]] detector to measure the radiation from a radioactive source (\( ^{241}Am_{95} \), [http://www.speclab.com/elements/americium.htm Amerício]), and study the absorption of radiation of different materials by putting them between the source and the detector.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [indisponível]<br />
*Laboratory: Intermediate em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Radiare<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The user can set the distance between source and detector (between 6cm and 20cm), and choose which material to place between the two parts.<br />
<br />
The available materials are:<br />
<br />
{| border="1"<br />
! scope="col" | Position<br />
! scope="col" | Material<br />
! scope="col" | Thickness<br />
|-<br />
|1 <br />
|Wood <br />
|10 mm <br />
|-<br />
|2 <br />
|Corticite <br />
|10 mm <br />
|-<br />
|3 <br />
|Brick <br />
|10 mm <br />
|-<br />
|4 <br />
|Copper <br />
|0,2 mm <br />
|-<br />
|5 <br />
|Copper <br />
|0,4 mm <br />
|-<br />
|6 <br />
|Copper <br />
|0,8 mm <br />
|-<br />
|7 <br />
|Copper <br />
|1,6 mm <br />
|-<br />
|8 <br />
|Copper <br />
|3,2 mm <br />
|-<br />
|9 <br />
|Control window (air)<br />
|0,5 mm <br />
|-<br />
|10 <br />
|Lead <br />
|6 mm<br />
|}<br />
<br />
The tenth position is, actually, "closing" the source.<br />
<br />
<br />
=Protocol (Basic Laboratory)=<br />
The suggested protocol for this experiment in the Basic Laboratory is the following:<br />
# Study the variation of the radiation intensity (given by the detector's output) as the distance changes;<br />
# Verify the linear absorption rate of the different materials, trying to establish a theoretical law that relates the intensity with the surface mass of each element;<br />
# For copper, infer length of semi-reduction, i.e. the length it takes to halve the number of Geiger-Müller counts;<br />
# The user will observe that experiments conducted under the same conditions will give different values. This is caused by the randomness of nuclear processes. A way to prove this is to verify that the number of counts in the same conditions follow a Binomial or Poisson Distribution due to the extremely low probability of the event.<br />
Due to this last factor, the experimental error can be atenuated by adding several values of radioactive decay under the same conditions.<br />
<br />
<br />
=Advanced Protocol (Intermidiate Laboratory)=<br />
A slightly more advanced protocol is:<br />
# Looking at Americium's (\(^{241}Am_{95}\)) half-life and at the database, compare the number of decays under the same conditions (material in between and distance) in the past and in the present.<br />
# Verify the mathematical law for Radioactive Decay.<br />
# For the same material and distance, record a large number of decays.<br />
# Fit a Gaussian function to the data and see if the fitting improves by adding more experimental data.<br />
<br />
<br />
=Theoretical Principles=<br />
==Geiger-Müller Detector==<br />
This device detects radiation levels, providing a number corresponding to how many radioactive particles go through it. <br />
<br />
==Radioactive Decay==<br />
The mathematical law that describes radioactive decay tells us that the variation of the number of radioactive nuclei, i.e. the number of decays at a given moment is proportional to its quantity:<br />
<br />
\[<br />
\frac{dN}{dt} = - \lambda \times N<br />
\]<br />
<br />
where \( \lambda \) is the decay constant. The solution of this equation tells us that, for a given time, the number of remaining nuclei is<br />
\[<br />
N = N_0 \times e^{- \lambda t}<br />
\]<br />
<br />
Differentiating in relation to time, we get the sample's rate of decay (i.e. how many decays occur per second) a.k.a. radioactivity.<br />
\[<br />
R = \frac{dN}{dt} = N_0 \times \lambda \times e^{- \lambda t}<br />
\]<br />
<br />
The half-life, as the name implies, is the time it takes for an initial sample to decay into half:<br />
\[<br />
T_{^1/_2} = \frac{ln(2)}{\lambda}<br />
\]<br />
<br />
<br />
=Links=<br />
*[[Atenuação da Radiação em Diferentes Materiais | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Radiation_Attenuation_over_Different_Materials&diff=1347Radiation Attenuation over Different Materials2013-10-20T20:50:33Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This laboratory uses a [[#Theoretical Principles|Geiger-Müller]] detector to measure the radiation from a radioactive source (\( ^{241}Am_{95} \), [http://www.speclab.com/elements/americium.htm Amerício]), and study the absorption of radiation of different materials by putting them between the source and the detector.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [indisponível]<br />
*Laboratory: Intermediate em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Radiare<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The user can set the distance between source and detector (between 6cm and 20cm), and choose which material to place between the two parts.<br />
<br />
The available materials are:<br />
<br />
{| border="1"<br />
! scope="col" | Position<br />
! scope="col" | Material<br />
! scope="col" | Thickness<br />
|-<br />
|1 <br />
|Wood <br />
|10 mm <br />
|-<br />
|2 <br />
|Corticite <br />
|10 mm <br />
|-<br />
|3 <br />
|Brick <br />
|10 mm <br />
|-<br />
|4 <br />
|Copper <br />
|0,2 mm <br />
|-<br />
|5 <br />
|Copper <br />
|0,4 mm <br />
|-<br />
|6 <br />
|Copper <br />
|0,8 mm <br />
|-<br />
|7 <br />
|Copper <br />
|1,6 mm <br />
|-<br />
|8 <br />
|Copper <br />
|3,2 mm <br />
|-<br />
|9 <br />
|Control window (air)<br />
|0,5 mm <br />
|-<br />
|10 <br />
|Lead <br />
|6 mm<br />
|}<br />
<br />
The tenth position is, actually, "closing" the source.<br />
<br />
<br />
=Protocol (Basic Laboratory)=<br />
The suggested protocol for this experiment in the Basic Laboratory is the following:<br />
# Study the variation of the radiation intensity (given by the detector's output) as the distance changes;<br />
# Verify the linear absorption rate of the different materials, trying to establish a theoretical law that relates the intensity with the surface mass of each element;<br />
# For copper, infer length of semi-reduction, i.e. the length it takes to halve the number of Geiger-Müller counts;<br />
# The user will observe that experiments conducted under the same conditions will give different values. This is caused by the randomness of nuclear processes. A way to prove this is to verify that the number of counts in the same conditions follow a Binomial or Poisson Distribution due to the extremely low probability of the event.<br />
Because of this last factor, the experimental error can be atenuated by adding several values of radioactive decay to the same conditions.<br />
<br />
<br />
=Advanced Protocol (Intermidiate Laboratory)=<br />
A slightly more advanced protocol is:<br />
# Looking at Americium's (\(^{241}Am_{95}\)) half-life and at the database, compare the number of decays under the same conditions (material in between and distance) in the past and in the present.<br />
# Verify the mathematical law for Radioactive Decay.<br />
# For the same material and distance, record a large number of decays.<br />
# Fit a Gaussian function to the data and see if the fitting improves by adding more experimental data.<br />
<br />
<br />
=Theoretical Principles=<br />
==Geiger-Müller Detector==<br />
This device detects radiation levels, providing a number corresponding to how many radioactive particles go through it. <br />
<br />
==Radioactive Decay==<br />
The mathematical law that describes radioactive decay tells us that the variation of the number of radioactive nuclei, i.e. the number of decays at a given moment is proportional to its quantity:<br />
<br />
\[<br />
\frac{dN}{dt} = - \lambda \times N<br />
\]<br />
<br />
where \( \lambda \) is the decay constant. The solution of this equation tells us that, for a given time, the number of remaining nuclei is<br />
\[<br />
N = N_0 \times e^{- \lambda t}<br />
\]<br />
<br />
Differentiating in relation to time, we get the sample's rate of decay (i.e. how many decays occur per second) a.k.a. radioactivity.<br />
\[<br />
R = \frac{dN}{dt} = N_0 \times \lambda \times e^{- \lambda t}<br />
\]<br />
<br />
The half-life, as the name implies, is the time it takes for an initial sample to decay into half:<br />
\[<br />
T_{^1/_2} = \frac{ln(2)}{\lambda}<br />
\]<br />
<br />
<br />
=Links=<br />
*[[Atenuação da Radiação em Diferentes Materiais | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Radiation_Attenuation_over_Different_Materials&diff=1346Radiation Attenuation over Different Materials2013-10-20T20:48:00Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This laboratory uses a [[#Theoretical Principles|Geiger-Müller]] detector to measure the radiation from a radioactive source (\( ^{241}Am_{95} \), [http://www.speclab.com/elements/americium.htm Amerício]), and study the absorption of radiation of different materials by putting them between the source and the detector.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [indisponível]<br />
*Laboratory: Intermediate em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Radiare<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The user can set the distance between source and detector (between 6cm and 20cm), and choose which material to place between the two parts.<br />
<br />
The available materials are:<br />
<br />
{| border="1"<br />
! scope="col" | Position<br />
! scope="col" | Material<br />
! scope="col" | Thickness<br />
|-<br />
|1 <br />
|Wood <br />
|10 mm <br />
|-<br />
|2 <br />
|Corticite <br />
|10 mm <br />
|-<br />
|3 <br />
|Brick <br />
|10 mm <br />
|-<br />
|4 <br />
|Copper <br />
|0,2 mm <br />
|-<br />
|5 <br />
|Copper <br />
|0,4 mm <br />
|-<br />
|6 <br />
|Copper <br />
|0,8 mm <br />
|-<br />
|7 <br />
|Copper <br />
|1,6 mm <br />
|-<br />
|8 <br />
|Copper <br />
|3,2 mm <br />
|-<br />
|9 <br />
|Control window (air)<br />
|0,5 mm <br />
|-<br />
|10 <br />
|Lead <br />
|6 mm<br />
|}<br />
<br />
The tenth position is, actually, "closing" the source.<br />
<br />
<br />
=Protocol (Basic Laboratory)=<br />
The suggested protocol for this experiment in the Basic Laboratory is the following:<br />
# Study the variation of the radiation intensity (given by the detector's output) as the distance changes;<br />
# Verify the linear absorption rate of the different materials, trying to establish a theoretical law that relates the intensity with the surface mass of each element;<br />
# For copper, infer length of semi-reduction, i.e. the length it takes to halve the number of Geiger-Müller counts;<br />
# The user will observe that experiments conducted under the same conditions will give diferent values. This is caused by the randomness of nuclear processes. A way to prove this is to verify that the number of counts in the same conditions follow a Binomial or Poisson Distribution due to the extremelly low probability of the event.<br />
Because of this last factor, the experimental error can be atenuated by adding several values of radioactive decay under the same conditions.<br />
<br />
<br />
=Advanced Protocol (Intermidiate Laboratory)=<br />
A slightly more advanced protocol is:<br />
# Looking at Americium's (\(^{241}Am_{95}\)) half-life and at the database, compare the number of decays under the same conditions (material in between and distance) in the past and in the present.<br />
# Verify the mathematical law for Radioactive Decay.<br />
# For the same material and distance, record a large number of decays.<br />
# Fit a Gaussian function to the data and see if the fitting improves by adding more experimental data.<br />
<br />
<br />
=Theoretical Principles=<br />
==Geiger-Müller Detector==<br />
This device detects radiation levels, providing a number corresponding to how many radioactive particles go through it. <br />
<br />
==Radioactive Decay==<br />
The mathematical law that describes radioactive decay tells us that the variation of the number of radioactive nuclei, i.e. the number of decays at a given moment is proportional to its quantity:<br />
<br />
\[<br />
\frac{dN}{dt} = - \lambda \times N<br />
\]<br />
<br />
where \( \lambda \) is the decay constant. The solution of this equation tells us that, for a given time, the number of remaining nuclei is<br />
\[<br />
N = N_0 \times e^{- \lambda t}<br />
\]<br />
<br />
Differentiating in relation to time, we get the sample's rate of decay (i.e. how many decays occur per second) a.k.a. radioactivity.<br />
\[<br />
R = \frac{dN}{dt} = N_0 \times \lambda \times e^{- \lambda t}<br />
\]<br />
<br />
The half-life, as the name implies, is the time it takes for an initial sample to decay into half:<br />
\[<br />
T_{^1/_2} = \frac{ln(2)}{\lambda}<br />
\]<br />
<br />
<br />
=Links=<br />
*[[Atenuação da Radiação em Diferentes Materiais | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Radiation_Attenuation_over_Different_Materials&diff=1345Radiation Attenuation over Different Materials2013-10-20T20:45:26Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This laboratory uses a [[#Theoretical Principles|Geiger-Müller]] detector to measure the radiation from a radioactive source (\( ^{241}Am_{95} \), [http://www.speclab.com/elements/americium.htm Amerício]), and study the absorption of radiation of different materials by putting them between the source and the detector.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [indisponível]<br />
*Laboratory: Intermediate em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Radiare<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The user can set the distance between source and detector (between 6cm and 20cm), and choose which material to place between the two parts.<br />
<br />
The available materials are:<br />
<br />
{| border="1"<br />
! scope="col" | Position<br />
! scope="col" | Material<br />
! scope="col" | Thickness<br />
|-<br />
|1 <br />
|Wood <br />
|10 mm <br />
|-<br />
|2 <br />
|Corticite <br />
|10 mm <br />
|-<br />
|3 <br />
|Brick <br />
|10 mm <br />
|-<br />
|4 <br />
|Copper <br />
|0,2 mm <br />
|-<br />
|5 <br />
|Copper <br />
|0,4 mm <br />
|-<br />
|6 <br />
|Copper <br />
|0,8 mm <br />
|-<br />
|7 <br />
|Copper <br />
|1,6 mm <br />
|-<br />
|8 <br />
|Copper <br />
|3,2 mm <br />
|-<br />
|9 <br />
|Control window (air)<br />
|0,5 mm <br />
|-<br />
|10 <br />
|Lead <br />
|6 mm<br />
|}<br />
<br />
The tenth position is, actually, "closing" the source.<br />
<br />
<br />
=Protocol (Basic Laboratory)=<br />
The suggested protocol for this experiment in the Basic Laboratory is the following:<br />
# Study the variation of the radiation intensity as the distance changes, given by the detector's output;<br />
# Verify the linear absorption rate of the different materials, trying to establish a theoretical law that relates the intensity with the surface mass of each element;<br />
# For copper, infer length of semi-reduction, i.e. the length it takes to halve the number of Geiger-Müller counts;<br />
# The user will observe that experiments conducted under the same conditions will give diferent values. This is caused by the randomness of nuclear processes. A way to prove this is to verify that the number of counts in the same conditions follow a Binomial or Poisson Distribution due to the extremelly low probability of the event.<br />
Because of this last factor, the experimental error can be atenuated by adding several values of radioactive decay under the same conditions.<br />
<br />
<br />
=Advanced Protocol (Intermidiate Laboratory)=<br />
A slightly more advanced protocol is:<br />
# Looking at Americium's (\(^{241}Am_{95}\)) half-life and at the database, compare the number of decays under the same conditions (material in between and distance) in the past and in the present.<br />
# Verify the mathematical law for Radioactive Decay.<br />
# For the same material and distance, record a large number of decays.<br />
# Fit a Gaussian function to the data and see if the fitting improves by adding more experimental data.<br />
<br />
<br />
=Theoretical Principles=<br />
==Geiger-Müller Detector==<br />
This device detects radiation levels, providing a number corresponding to how many radioactive particles go through it. <br />
<br />
==Radioactive Decay==<br />
The mathematical law that describes radioactive decay tells us that the variation of the number of radioactive nuclei, i.e. the number of decays at a given moment is proportional to its quantity:<br />
<br />
\[<br />
\frac{dN}{dt} = - \lambda \times N<br />
\]<br />
<br />
where \( \lambda \) is the decay constant. The solution of this equation tells us that, for a given time, the number of remaining nuclei is<br />
\[<br />
N = N_0 \times e^{- \lambda t}<br />
\]<br />
<br />
Differentiating in relation to time, we get the sample's rate of decay (i.e. how many decays occur per second) a.k.a. radioactivity.<br />
\[<br />
R = \frac{dN}{dt} = N_0 \times \lambda \times e^{- \lambda t}<br />
\]<br />
<br />
The half-life, as the name implies, is the time it takes for an initial sample to decay into half:<br />
\[<br />
T_{^1/_2} = \frac{ln(2)}{\lambda}<br />
\]<br />
<br />
<br />
=Links=<br />
*[[Atenuação da Radiação em Diferentes Materiais | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Radiation_Attenuation_over_Different_Materials&diff=1344Radiation Attenuation over Different Materials2013-10-20T20:40:06Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This laboratory uses a [[#Theoretical Principles|Geiger-Müller]] detector to measure the radiation from a radioactive source (\( ^{241}Am_{95} \), [http://www.speclab.com/elements/americium.htm Amerício]), and study the absorption of radiation of different materials by putting them between the source and the detector.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [indisponível]<br />
*Laboratory: Intermediate em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Radiare<br />
*Level: ***<br />
<br />
</div><br />
</div><br />
<br />
<br />
=Experimental Apparatus=<br />
The user can set the distance between source and detector (between 6cm and 20cm), and choose which material to place between the two parts.<br />
<br />
The available materials are:<br />
<br />
{| border="1"<br />
! scope="col" | Position<br />
! scope="col" | Material<br />
! scope="col" | Thickness<br />
|-<br />
|1 <br />
|Wood <br />
|10 mm <br />
|-<br />
|2 <br />
|Corticite <br />
|10 mm <br />
|-<br />
|3 <br />
|Brick <br />
|10 mm <br />
|-<br />
|4 <br />
|Copper <br />
|0,2 mm <br />
|-<br />
|5 <br />
|Copper <br />
|0,4 mm <br />
|-<br />
|6 <br />
|Copper <br />
|0,8 mm <br />
|-<br />
|7 <br />
|Copper <br />
|1,6 mm <br />
|-<br />
|8 <br />
|Copper <br />
|3,2 mm <br />
|-<br />
|9 <br />
|Control window (air)<br />
|0,5 mm <br />
|-<br />
|10 <br />
|Lead <br />
|6 mm<br />
|}<br />
<br />
The tenth position is, actually, "closing" the source.<br />
<br />
<br />
=Protocol (Basic Laboratory)=<br />
The suggested protocol for this experiment in the Basic Laboratory is the following:<br />
# Study the variation of the radiation's intensity as the distance changes, given by the detector's output;<br />
# Verify the linear absorption rate of the diferent materials, trying to establish a theoretical law that relates the intensity with the surface mass of each element;<br />
# For copper, infer lengh of semi-reduction, i.e. the lengh it takes to halve the number of Geiger-Müller counts;<br />
# The user will observe that experiments conducted under the same conditions will give diferent values. This is caused by the randomness of nuclear processes. A way to prove this is to verify that the number of counts in the same conditions follow a Binomial or Poisson Distribution due to the extremelly low probability of the event.<br />
Because of this last factor, the experimental error can be atenuated by adding several values of radioactive decay under the same conditions.<br />
<br />
<br />
=Advanced Protocol (Intermidiate Laboratory)=<br />
A slightly more advanced protocol is:<br />
# Looking at Americium's (\(^{241}Am_{95}\)) half-life and at the database, compare the number of decays under the same conditions (material in between and distance) in the past and in the present.<br />
# Verify the mathematical law for Radioactive Decay.<br />
# For the same material and distance, record a large number of decays.<br />
# Fit a Gaussian function to the data and see if the fitting improves by adding more experimental data, .<br />
<br />
<br />
=Theoretical Principles=<br />
==Geiger-Müller Detector==<br />
This device detects radiation levels, providing a number corresponding to how many radioactive particles go through it. <br />
<br />
==Radioactive Decay==<br />
The mathematical law that describes radioactive decay tells us that the variation of the number of radioactive nuclei, i.e. the number of decays at a given instant, is proportional to its quantity:<br />
<br />
\[<br />
\frac{dN}{dt} = - \lambda \times N<br />
\]<br />
<br />
where \( \lambda \) is the decay constant. The solution of this equation tells us that, for a given time, the number of remaining nuclei is<br />
\[<br />
N = N_0 \times e^{- \lambda t}<br />
\]<br />
<br />
Differentiating in relation to time, we get the sample's rate of decay (i.e. how many decays occur per second) a.k.a. radioactivity.<br />
\[<br />
R = \frac{dN}{dt} = N_0 \times \lambda \times e^{- \lambda t}<br />
\]<br />
<br />
The half-life, as the name implies, is the time it takes for an initial sample to decay into half:<br />
\[<br />
T_{^1/_2} = \frac{ln(2)}{\lambda}<br />
\]<br />
<br />
<br />
=Links=<br />
*[[Atenuação da Radiação em Diferentes Materiais | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Thermal_Conductivity_of_Metals&diff=1339Thermal Conductivity of Metals2013-10-19T16:27:10Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment revolves around the conduction of heat. The user can study and compare how fast heat propagates through three bars of different materials. <br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/radiare.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: [Unavaliable]<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="200" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/ConducaoCalorFinal.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
Two methods are used to study heat conduction through metals:<br />
<br />
# Single thermal pulse, applied to both ends, and subsequent evolution through the three monitored points; <br />
# Comparison of the output of each bar's first sensor. <br />
<br />
Each bar is 370mm long and has a 5mm diameter and each one is made of a different metal: iron, copper and brass.<br />
<br />
[[File:Conducao-montagem.jpg|thumb|Montagem: Mineral wool sandwich covering the metal cylinders.]]<br />
<br />
The first 70mm of each bar are wrapped in a heating resistor. 10mm away from the heat sink is the first batch of thermometers, the second being at 250mm and the third at 300mm. The heat sink keeps one of the ends at room temperature and is the position axis reference point. The bars are mounted in the middle of two 50mm thick layers of thermal insulation material (mineral wool) that prevent heat convection and minimize thermal losses.<br />
<br />
In a pulsed or periodic regime, an electric current is applied to the heating resistor that heats that end through Joule's effect. The heat generated will run through the bar and will be dissipated at the opposite end in the sink.<br />
<br />
By choosing the pulse width (it should be around one minute), heat propagation can be observed by measuring the three points described above.<br />
<br />
<br />
=Protocol= <br />
Run the experiment in pulse mode and observe how the pulse travels along the bars.<br />
<br />
'''Note:''' The heat sink keeps the ends of the bars at room temperature, which means that the readings of one of the thermometers will not vary, having been chosen as the distance reference point.<br />
<br />
<br />
=Advanced Protocol=<br />
In the periodic mode, the temperature reading has two components:<br />
<br />
# one comes from the oscillation of the heating itself, its period being the same as the heat source (on and off);<br />
# the other is the average heating of the bar as a whole, which is almost exponential.<br />
<br />
Through a graphical fitting of a function to the average temperature, we can extract the oscillating values by subtracting the average. By analyzing the oscillating data, we can determine the heat propagation constant by Fourier analysis or a simple sinusoidal fitting.<br />
<br />
For best results, please follow the procedure:<br />
<br />
# Execute two experiments in periodic mode, selecting a time interval of a few seconds between samples and a heating time of approximately one minute. For the first run, choose the maximum acquisition time possible, and for the second choose a reasonable value based on the data from the first one.<br />
# Copy the data to a spreadsheet or preferably to a science application, like Matlab or Origin;<br />
# Fit a math function to the thermal behavior of the bar; <br />
# Determine the period of oscillation for each experiment conducted and the phase shift between two consecutive thermometers, for each metal available;<br />
# Numerically adjust the function to the experimental data to get the highest precision possible. <br />
<br />
The thermal conductivity of each metal can be determined from the experimental data by using:<br />
<br />
\[<br />
k = \frac{C_p \rho n \omega (x_1 - x_2)^2}{2 (\phi_1 - \phi_2) ln(A_1/A_2)}<br />
\]<br />
<br />
where \( C_p \) is the metal's heat capacity, \( \rho \) the density, \( \omega \) the angular frequency, \( (x_2 - x_1) \) the distance between thermometers, \( (f_1 - f_2) \) phase between signals (measured in radians) and \( A_1/A_2 \) the ratio between amplitudes.<br />
<br />
We can also calculate the ''k'' experimental uncertainty by using propagation formulas.<br />
Note: the diameter of the bars was measured up to a ''0.01 mm'' and the sensor position up to ''0.5 mm''.<br />
<br />
Afterwards, compare it to the textbook value.<br />
<br />
A simple analysis can be done to check the experimental data, assuming the initial stimulus is sinusoidal. This way we can take ''n=1'' and the phase shift is determined by the distance between the maximum of the two experimental curves.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Experimental results for different metals<br />
|-<br />
! scope="col" | Metal <br />
! scope="col" | Specific heat<br /> \( C_p [J K^{-1} kg^{-1}] \)<br />
! scope="col" | Density<br /> \( \rho [kg m^{-3}] \)<br />
! scope="col" | Thermal conductivity<br /> \( k [W m^{-1} K^{-1}] \)<br />
|-<br />
|Brass <br />
|384 <br />
|8500 <br />
|113 <br />
|-<br />
|Copper <br />
|394 <br />
|8930 <br />
|390 <br />
|-<br />
|Iron <br />
|456 <br />
|7860 <br />
|81 <br />
|}<br />
<br />
<br />
=Links=<br />
*[[Determinação da Condutividade Térmica em Metais | Portuguese Version (Versão em português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Thermal_Conductivity_of_Metals&diff=1338Thermal Conductivity of Metals2013-10-19T16:14:31Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
This experiment revolves around the conduction of heat. The user can study and compare how fast heat propagates through three bars of different materials. <br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/radiare.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: [Unavaliable]<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="200" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/ConducaoCalorFinal.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
Two methods are used to study heat conduction through metals:<br />
<br />
# Single thermal pulse, applied to both ends, and subsequent evolution through the three monitored points; <br />
# Comparison of the output of each bar's first sensor. <br />
<br />
Each bar has a 5mm diameter and is 370mm long and each one is made of a different metal: iron, copper and brass.<br />
<br />
[[File:Conducao-montagem.jpg|thumb|Montagem: Mineral wool sandwich covering the metal cylinders.]]<br />
<br />
The first 70mm of each bar are wrapped in a heating resistor. 10mm away from the heat sink is the first batch of thermometers, the second being at 250mm and the third at 300mm. The heat sink keeps one of the ends at room temperature and is the reference point for the position axis. The bars are mounted in the middle of two 50mm thick layers of thermal insulation material (mineral wool) that prevent heat convection and minimize thermal losses.<br />
<br />
In a pulsed or periodic regime, an electric current is applied to the heating resistor that heats that end through Joule's effect. The heat generated will run through the bar and be dissipated at the opposite end in sink.<br />
<br />
By choosing the pulse width (it should be around one minute), heat propagation can be observed by measuring the three points described earlier.<br />
<br />
<br />
=Protocol= <br />
Run the experiment in pulse mode and observe how the pulse travels along the bars.<br />
<br />
'''Note:''' The heat sink keeps the ends of the bars at room temperature, which means that the readings of one of the thermometers will not vary, having been chosen as the reference point for distance.<br />
<br />
<br />
=Advanced Protocol=<br />
In the periodic mode, the temperature reading has two components:<br />
<br />
# one comes from the oscillation of the heating itself, its period being the same as the heat source (on and off);<br />
# the other is the average heating of the bar as a whole, which is almost exponential.<br />
<br />
Through a graphical fitting of a function to the average temperature, we can extract the oscillating values by subtracting the average. By analyzing the oscillating data, we can determine the heat propagation constant by Fourier analysis or a simple sinusoidal fitting.<br />
<br />
For best results, please follow the procedure:<br />
<br />
# Execute two experiments in periodic mode, selecting a time interval of a few seconds between samples and a heating time of the order of one minute. For the first run, choose the maximum acquisition time possible, and for the second choose a reasonable value based on the data from the first.<br />
# Copy the data to a spreadsheet or preferably to a science application, like Matlab or Origin;<br />
# Fit a math function to the thermal behavior of the bar; <br />
# Determine the period of oscillation for each experiment conducted and the phase shift between two consecutive thermometers, for each metal available;<br />
# Numerically adjust the function to the experimental data to get the highest precision possible. <br />
<br />
The thermal conductivity of each metal can be determined from the experimental data by using:<br />
<br />
\[<br />
k = \frac{C_p \rho n \omega (x_1 - x_2)^2}{2 (\phi_1 - \phi_2) ln(A_1/A_2)}<br />
\]<br />
<br />
where \( C_p \) is the metal's heat capacity, \( \rho \) the density, \( \omega \) the angular frequency, \( (x_2 - x_1) \) the distance between thermometers, \( (f_1 - f_2) \) phase between signals (measured in radians) and \( A_1/A_2 \) the ratio between amplitudes.<br />
<br />
We can also calculate the experimental uncertainty that ''k'' has, by using propagation formulas.<br />
Note: the diameter of the bars was measured up to a ''0.01 mm'' and the sensor position up to ''0.5 mm''.<br />
<br />
Afterwards, compare it to the textbook value.<br />
<br />
A simple analysis can be done to check the experimental data, assuming the initial stimulus is sinusoidal. This way we can take ''n=1'' and the phase shift is determined by the distance between the maximum of the two experimental curves.<br />
<br />
{| border="1" style="text-align: center;"<br />
|+ Experimental results for different metals<br />
|-<br />
! scope="col" | Metal <br />
! scope="col" | Specific heat<br /> \( C_p [J K^{-1} kg^{-1}] \)<br />
! scope="col" | Density<br /> \( \rho [kg m^{-3}] \)<br />
! scope="col" | Thermal conductivity<br /> \( k [W m^{-1} K^{-1}] \)<br />
|-<br />
|Brass <br />
|384 <br />
|8500 <br />
|113 <br />
|-<br />
|Copper <br />
|394 <br />
|8930 <br />
|390 <br />
|-<br />
|Iron <br />
|456 <br />
|7860 <br />
|81 <br />
|}<br />
<br />
<br />
=Links=<br />
*[[Determinação da Condutividade Térmica em Metais | Portuguese Version (Versão em português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_the_Speed_of_Sound&diff=1337Determination of the Speed of Sound2013-10-18T22:57:14Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to determine the speed of sound.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/statsound.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: statsound<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="290" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/StatSound.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus (sometimes referred as "Kundt's tube") consists of a 1458 mm lon PVC tube. On one end there is a fixed speaker that can produce an audio sine, triangular or single pulse wave. On the opposite side there is a movable piston to change the effective tube length. Along the tube there are several microphones to register the sound intensity at some fixed points.<br />
<br />
The following table shows the positions of the microphones in relation to the source (speaker):<br />
<br />
{| border="1"<br />
|-<br />
! Designation<br />
! Distance to source (mm)<br />
|-<br />
| Mic 1 (reference) <br />
| 250<br />
|-<br />
| Mic 2 (center) <br />
| 750<br />
|-<br />
| Mic 3 (extreme) <br />
| 1250<br />
|-<br />
| Mic 4 (embolus surface) <br />
| Between 1260 e 1480<br />
|-<br />
| tube limit <br />
| 1450<br />
|+ Table 1 – Microphones distance to the sound source (speaker's membrane)<br />
|}<br />
<br />
<br />
The reference mic (Mic 1) should be used to verify that the sound is emitted as desired (i.e., there is no distortion caused by the speaker). On the piston surface there is another microphone (Mic 4) capable of moving between 1269mm and 1475mm.<br />
The sound is acquired through 2 channels of a sound card: the left channel (CH 1) is always bounded to the reference microphone (Mic 1); the other channel (CH 2) can be connected to one of the other three microphones.<br />
<br />
The experimental data is captured by the PC's sound-card and processed on-line (normalization) before being received by the user.<br />
<br />
<br />
=Protocol=<br />
This apparatus can also be used for the stationary wave experiment and thus has two modes of operation: in the "Speed of sound" mode the amplitude of the wave is registered over time.<br />
<br />
To determine \( v_{sound} \), the user must choose a "pulse" type of stimulus and measure the "time-of-flight" taken by the wave from Mic 1 and any other microphone. The speed can be determined with data from table 1 and the formula for speed determination:<br />
<br />
\[<br />
v_{sound} = \frac{\Delta s}{\Delta t}<br />
\]<br />
where \(s\) is the distance between the selected microphones.<br />
Of course, other waveforms could be used but this would require a closer look at the signals phase.<br />
<br />
<br />
=Advanced Protocol=<br />
Using the coherence function between the acquired signals, the phase determination can improve with higher accuracy among different microphones. Using an appropriate software package (like Matlab or Octave for instance) this phase is easily determined (mscohere). Pink or white noise are very suitable for this purpose as they won't show any phase indetermination.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Velocidade do Som | Portuguese Version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Determination_of_the_Speed_of_Sound&diff=1336Determination of the Speed of Sound2013-10-18T22:43:43Z<p>Ist128595: </p>
<hr />
<div>=Description of the Experiment=<br />
The purpose of this experiment is to determine the speed of sound.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt:554/statsound.sdp<br />
*Laboratory: Advanced in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: statsound<br />
*Level: ****<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="290" width="500">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/StatSound.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus (sometimes referred as "Kundt's tube") consists of a 1458 mm lon PVC tube. On one end there is a fixed speaker that can produce an audio sine, triangular or single pulse wave. On the opposite side there is a movable piston for changing the effective tube length. Along the tube there are several microphones to register the sound intensity at some fixed points.<br />
<br />
The following table shows the positions of the microphones in relation to the source (speaker):<br />
<br />
{| border="1"<br />
|-<br />
! Designation<br />
! Distance to source (mm)<br />
|-<br />
| Mic 1 (reference) <br />
| 250<br />
|-<br />
| Mic 2 (center) <br />
| 750<br />
|-<br />
| Mic 3 (extreme) <br />
| 1250<br />
|-<br />
| Mic 4 (embolus surface) <br />
| Between 1260 e 1480<br />
|-<br />
| tube limit <br />
| 1450<br />
|+ Table 1 – Microphones distance to the sound source (speaker's membrane)<br />
|}<br />
<br />
<br />
The reference mic (Mic 1) should be used to verify that the emitted sound is the required one (i.e., there is no distortion caused by the speaker). On the piston surface there is another microphone (Mic 4) capable of moving between 1269mm and 1475mm.<br />
The sound is acquired through 2 channels of a sound card: the left channel (CH 1) is always bounded to the reference microphone (Mic 1); the other channel (CH 2) can be connected to one of the other three microphones.<br />
<br />
The experimental data is captured by the PC's sound-card and processed on-line (normalization) prior to be received by the user.<br />
<br />
<br />
=Protocol=<br />
This apparatus can also be used for the stationary wave experiment and thus has two modes of operation: in the "Speed of sound" mode the amplitude of the wave is registered over time.<br />
<br />
To determine \( v_{sound} \), the user must choose a "pulse" type of stimulus and measure the "time-of-flight" taken by the wave from Mic 1 and any other microphone. The speed can be determined with data from table 1 and the formula for speed determination:<br />
<br />
\[<br />
v_{sound} = \frac{\Delta s}{\Delta t}<br />
\]<br />
where \(s\) is the distance between the selected microphones.<br />
Of course, other waveforms could be used but this would require a close look at the signals phase.<br />
<br />
<br />
=Advanced Protocol=<br />
Using the coherence function between the acquired signals, the phase determination can improve with higher accuracy among different microphones. Using an appropriate software package (like Matlab or Octave for instance) this phase is easily determined (mscohere). Pink or white noise are very suitable for this purpose as they won't show any phase indetermination.<br />
<br />
<br />
=Links=<br />
*[[Determinação da Velocidade do Som | Portuguese Version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Boyle-Mariotte_Law&diff=1335Boyle-Mariotte Law2013-10-18T22:11:16Z<p>Ist128595: </p>
<hr />
<div>=Experiment Description=<br />
The purpose of this experiment is to verify the relation \( p \propto \frac{1}{V} \) (i.e., pressure and volume are inversely proportional) of a gas during isothermal expansion or compression.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/pv.sdp<br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: pv<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="270" width="510">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/BoyleMariottePV.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is composed by a cylinder (5 ml syringe) filled with air, whose plunger is displaced by a servo motor.<br />
<br />
[[File:Boyle-mariotte-montagem.jpg|thumb|alt=Experimental setup|Setup]]<br />
<br />
It is possible to choose the compression/expansion limits of the chamber and measure the air pressure and volume inside it during the plunger's motion. The speed at which the transformation occurs can also be specified by choosing the time between samples.<br />
<br />
<br />
=Protocol=<br />
The user must specify the initial and final volume, the number of samples and time interval between them. This last option indirectly defines the speed at which the transformation occurs.<br />
<br />
If the experiment is executed with a long time interval between samples, the transformation is close to isothermal. That means the air temperature inside the syringe doesn't change and the Boyle-Mariotte Law is applicable.<br />
<br />
The example bellow is one such case where we can verify that the product \( pV \) is, in average, \( 49,3 \pm 0,3 l.kPa \), close to the theoretical value of \( nRT \) (where \( n = 2 \times 10 ^{-4} mol\), equivalent to the full syringe volume at rest \( 5 ml \), PTN).<br />
<br />
[[File:Boyle-mariotte-graf1.png|thumb|Graphical fit with experimental results with 1 sec. between samples.]]<br />
<br />
If the time between samples is decreased (i.e. the process is faster), the transformation is no longer isothermal and becomes slightly adiabatic, because there is not enough time for the heat exchange between the inside of the cylinder and the environment. This way, the compression no longer follows the Boyle-Mariotte Law, a fact that is clearly demonstrated by the deviation from the power function, which would be \( ^3/_5 \) in the ideal case (\( \gamma ^{-1} = \frac{c_v}{c_p} \)). <br />
<br />
It is difficult to obtain a fast enough compression to achieve this value, since thermalization occurs quickly (in the order of the sound speed). However this can be used to study how the transformation speed influences the deviation between the experimental data and the Boyle-Mariotte Law.<br />
<br />
<br />
=Advanced Protocol=<br />
With a high enough time between samples, it is also possible to determine the Ideal Gas Constant, if we consider room temperature to be 22ºC.<br />
<br />
Procedure:<br />
# Run the experiment with about 20 acquisitions, maximum extension between initial and final volume and maximum acquisition time allowed (we are trying to achieve an isothermal transformation).<br />
# Fit the experimental data to a \( a x ^b \) type function and find the proportion constant (which will equal \( nRT \)). <br />
# Assume the room temperature to be 22ºC. Determine the amount of substance (in mole) present at the inicial stage of the experiment ( \( 5 ml \) of air at NTP conditions). Find R.<br />
# Compare the R value that was found through this method with the one found through the average product of \( pV \) (using a math software, determine the product between p and V for each line and then find the average). <br />
# Find one last estimate for R setting the exponent at -1 (i.e., fitting the data to a \( a x ^{-1} \) function).<br />
<br />
<br />
=Who likes this idea=<br />
[[File:CienciaViva.gif|border|204px|border|204px]]<br />
<br />
<br />
=Links=<br />
*[[Lei de Boyle-Mariotte | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Boyle-Mariotte_Law&diff=1334Boyle-Mariotte Law2013-10-18T22:05:15Z<p>Ist128595: </p>
<hr />
<div>=Experiment Description=<br />
The purpose of this experiment is to verify the relation \( p \propto \frac{1}{V} \) (i.e., pressure and volume are inversely proportional) of a gas during isothermal expansion or compression.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/pv.sdp<br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: pv<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="270" width="510">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/BoyleMariottePV.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
The apparatus is composed by a cylinder (5 ml syringe) filled with air whose plunger is displaced by a servo motor.<br />
<br />
[[File:Boyle-mariotte-montagem.jpg|thumb|alt=Experimental setup|Setup]]<br />
<br />
It's possible to choose the compression/expansion limits of the chamber and measure the air pressure and volume inside it during the plunger's motion. The speed at which the transformation occurs can also be specified by choosing the time between samples.<br />
<br />
<br />
=Protocol=<br />
The user must specify the initial and final volume, the number of samples and time interval between them. This last option indirectly defines the speed at which the transformation occurs.<br />
<br />
If the experiment is executed with a long time interval between samples, the transformation is close to isothermal. That means the air temperature inside the syringe doesn't change and the Boyle-Mariotte Law is applicable.<br />
<br />
The example bellow is one such case where we can verify that the product \( pV \) is, in average, \( 49,3 \pm 0,3 l.kPa \), close to the theoretical value of \( nRT \) (where \( n = 2 \times 10 ^{-4} mol\), equivalent to the full syringe volume at rest \( 5 ml \), PTN).<br />
<br />
[[File:Boyle-mariotte-graf1.png|thumb|Graphical fit with experimental results with 1 sec. between samples.]]<br />
<br />
If the time between samples is decreased (i.e. the process is faster), the transformation is no longer isothermal and becomes slightly adiabatic, because there is not enough time for the heat exchange between the inside of the cylinder and the environment. This way, the compression no longer follows the Boyle-Mariotte, a fact that is clearly demonstrated by the deviation from the power function, which would be \( ^3/_5 \) in the ideal case (\( \gamma ^{-1} = \frac{c_v}{c_p} \)). <br />
<br />
It is difficult to obtain a fast enough compression to achieve this value, since thermalization occurs quickly (in the order of the sound speed). However this can be used to study how the transformation speed influences the deviation between the experimental data and the Boyle-Mariotte Law.<br />
<br />
<br />
=Advanced Protocol=<br />
With a high enough time between samples, it is also possible to determine the Ideal Gas Constant, if we consider room temperature to be 22ºC.<br />
<br />
Procedure:<br />
# Run the experiment with about 20 acquisitions, maximum extension between initial and final volume and maximum acquisition time allowed (we are trying to achieve an isothermal transformation).<br />
# Fit the experimental data to a \( a x ^b \) type function and find the proportion constant (which will equal \( nRT \)). <br />
# Assume the room temperature to be 22ºC. Determine the amount of substance (in mole) present at the inicial stage of the experiment ( \( 5 ml \) of air at NTP conditions). Find R.<br />
# Compare the value for R that was found through this method with the one found through the average product of \( pV \) (using a math software, determine the product between p and V for each line and then find the average). <br />
# Find one last estimate for R setting the exponent at -1 (i.e., fitting the data to a \( a x ^{-1} \) function).<br />
<br />
<br />
=Who likes this idea=<br />
[[File:CienciaViva.gif|border|204px|border|204px]]<br />
<br />
<br />
=Links=<br />
*[[Lei de Boyle-Mariotte | Portuguese version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Dice_Statistics&diff=1319Dice Statistics2013-10-16T22:59:05Z<p>Ist128595: </p>
<hr />
<div>=Description=<br />
<br />
This experiment consists of an apparatus that automatically shuffles a set of six-sided dice. To count the spots, it acquires and processes an image recognition pattern from the top side of each dice.<br />
<br />
By recording the number of times each side appears, you can study the law of probability and develop a statistical study of random phenomena. Using the images produced by this experiment, you can also develop your own algorithms using them in the study of computer recognition software.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [unavailable]<br />
*Laboratory: Básico em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Aleatorio<br />
*Grade: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|_qUFqXJQOpM|Dice in slow motion taken from the experiment (12x slow).|left}} <br />
<br />
<swf height="320" width="320">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/EstatisticaDados.swf</swf> <br />
<br />
<br />
<br />
<br />
<br />
=Experimental Apparatus=<br />
In this experiment there is a loudspeaker positioned horizontally with a platform on top of the cone made of k-line (structured light cardboard with polyurethane), where fourteen dice are spread. Above it, at 300mm from the platform, there is a video camera equipped with a white high brightness LED to illuminate it.<br />
<br />
[[File:EstatDados-montagem.jpg|thumb|alt=Apparatus for the Dice Statistics experiment|Apparatus]]<br />
<br />
The release (i.e. shuffling) of the dice is achieved through the speaker's stimulus with a sound wave, which makes the platform vibrate with the same frequency of the wave.<br />
<br />
This platform has a wall that prevents the dice from leaving the platform (and the web-cam field of view). The wall is quite high to block the entry of light from the outside of the lighting system.<br />
<br />
With this apparatus, pictures are obtained as we can see in picture 2.<br />
<br />
The image processing is made according to the flowchart below and the result of this process leads to an image like the one in picture 3.<br />
<br />
{|<br />
| [[File:EstatDados-foto1.jpg|thumb|Figure 2: Photograph of the dice]]<br />
| [[File:EstatDados-foto2.jpg|thumb|Figure 3: Figure 2 after software recognition]]<br />
| [[File:EstatDados-fluxograma.jpg|thumb|Fluxogram]]<br />
|}<br />
<br />
<br />
<br />
=Protocol=<br />
The experience execution protocol is simple because it consists in stimulating the platform conveniently, so that it can scramble the dice. The main features of the configurator control room are described below for a better understanding.<br />
<br />
<!-- [[File:EstatDados-interface.gif|thumb|Picture 4: The experiment "Control room" configuration ]] --><br />
<br />
==Dice shuffling==<br />
The dice are released (shuffled) by the oscillating movement of the platform where they are located. The user can select the starting and the ending frequencies of the sound wave that will be transmitted to the platform. This sound wave is synthesized on demand, there are no pre-recorded sound files.<br />
<br />
The frequency can be chosen between 20Hz and 150Hz. Bellow 20Hz there is no response from the hardware to vibrate the platform; above 150Hz, the inertia "forces" a low amplitude motion, so the dice don't move.<br />
<br />
The user can also choose the duration of the sound wave from 1.5 to 10 seconds. The lower value is enough to shuffle some of the dice at lower frequency but it will ultimately result in a small randomization. The upper limit allows very high randomization and is not even necessary in some cases.<br />
<br />
==Images==<br />
The user chooses how many frames should be analysed by choosing the number of samples between 1 and 20.<br />
<br />
One sample can be used to check how the recognition software works (what are the steps and how long they take). With 20 samples the results start to show a distribution that evolves towards the Gaussian distribution (even though, in theory, this only happens after 30 samples, minimum).<br />
<br />
==Video==<br />
The user can choose whether to watch or not to watch the shuffling process video.<br />
<br />
This video is composed of a series of .jpg pictures, which means that it is not an actual video, and the rate of display can change substantially with the connection quality. <br />
<br />
Since the video feed has a high demand on the internet connection, the user is advised to use it only once, as its purpose is merely to satisfy curiosity.<br />
<br />
=Advanced Protocol=<br />
After a number of samples a graph can be constructed with the number of times each number is recorded in each bin and a Gaussian distribution can be fitted:<br />
<br />
\[<br />
p(x) = y_0 + A e ^{- \frac{(x- \mu)^2}{\sigma ^2}}<br />
\]<br />
<br />
[[File:EstatDados-grafico.png|thumb|Figure 5: Example distribution]]<br />
<br />
Since there are 14 dice, the expected mean value is 49 (why?), which is confirmed by the build-up of values fitting.<br />
<br />
The best way to conduct this study is to merge the results of several users and see if the fit is improved by an increasing number of samples.<br />
<br />
The expected value for the average of N 6-sided dice is: <br />
<br />
\[<br />
\bar{\mu} = \frac{6N+N}{2}<br />
\]<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px"><br />
'''User Manual'''<br />
<div class="mw-collapsible-content"><br />
<br />
<br />
The user can change the parameters of the image recognition process.<br />
<br />
* ''Black&White Threshold'': the value, in grayscale, above which the pixels are considered white.<br />
<br />
* Hough Transform<br />
** ''Threshold1'': value above which a pixel from the Hough Transform image is taken to post processing.<br />
** ''Threshold2'': averages the pixels near the one that passed the previous test and, if that value is above Threshold2, this pixel is taken to post processing. <br />
** ''Threshold3'': in theory, any pixel in a white zone is ignored but, if the average (calculated in the previous test) is above Threshold3, the pixel is NOT ignored. This is used because some of the dice have white spots inside the black marks.<br />
<br />
* ''Convolution Threshold'': the value above which a pixel from the convolution image is taken to post processing.<br />
<br />
* ''Dice specifications'':<br />
** ''Mark radius'': the expected radius, given in pixels, of the marks (note that this algorithm expects the marks to be circular).<br />
** ''Dice width'': the maximum distance, in pixels, between marks. For example, in the 6-side of the dice, this value would be diagonal distance from one corner to the other.<br />
** ''Expected number of dice'': the number of dice in the experiment. If the algorithm finds more "dice" (i.e. sets of marks) than expected, and some of those sets are not considered "compatible" with dice, then these sets are eliminated ('''Note:''' The algorithm compares the relative position and distance between marks to create a set. These sets are then compared to what is expected in a dice; if the comparison is positive, the set is marked as "compatible", if not, some marks are switched or eliminated until a "compatible" set is achieved). <br />
</div><br />
</div><br />
<br />
<br />
=Links=<br />
*[[Estatística de Dados | Portuguese version (versão em português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Dice_Statistics&diff=1318Dice Statistics2013-10-16T22:44:10Z<p>Ist128595: </p>
<hr />
<div>=Description=<br />
<br />
This experiment consists on an apparatus that automatically shuffles a set of six-sided dice. To count the spots, it acquires and processes an image recognition pattern from the top side of each dice.<br />
<br />
By recording the number of times each side appears, you can study the law of probability and develop a statistical study of random phenomena. Using the images produced by this experiment, you can also develop your own algorithms using them in the study of computer recognition software.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [unavailable]<br />
*Laboratory: Básico em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Aleatorio<br />
*Grade: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|_qUFqXJQOpM|Dice in slow motion taken from the experiment (12x slow).|left}} <br />
<br />
<swf height="320" width="320">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/EstatisticaDados.swf</swf> <br />
<br />
<br />
<br />
<br />
<br />
=Experimental Apparatus=<br />
In this experiment there is a loudspeaker positioned horizontally with a platform on top of the cone made of k-line (structured light cardboard with polyurethane), where fourteen dice are spread. Above it, at 300mm from the platform, there is a video camera equipped with a white high brightness LED to illuminate it.<br />
<br />
[[File:EstatDados-montagem.jpg|thumb|alt=Apparatus for the Dice Statistics experiment|Apparatus]]<br />
<br />
The release (i.e. shuffling) of the dice is achieved through the speaker's stimulus with a sound wave, which makes the platform vibrate with the same frequency of the wave.<br />
<br />
This platform has a wall that prevents the dice from leaving the platform (and the web-cam field of view). The wall is quite high to prevent the entry of light from the outside of the lighting system.<br />
<br />
With this apparatus, pictures are obtained as we can see in picture 2.<br />
<br />
The image processing is made according to the flowchart below and the result of this process leads to an image like the one in picture 3.<br />
<br />
{|<br />
| [[File:EstatDados-foto1.jpg|thumb|Figure 2: Photograph of the dice]]<br />
| [[File:EstatDados-foto2.jpg|thumb|Figure 3: Figure 2 after software recognition]]<br />
| [[File:EstatDados-fluxograma.jpg|thumb|Fluxogram]]<br />
|}<br />
<br />
<br />
<br />
=Protocol=<br />
The experience execution protocol is simple because it consists in stimulating the platform conveniently, so that it can scramble the dice. And then, we describe the main features of the configurator control room for a better understanding.<br />
<br />
<!-- [[File:EstatDados-interface.gif|thumb|Picture 4: The experiment "Control room" configuration ]] --><br />
<br />
==Dice shuffling==<br />
The dice are released (shuffled) by the oscillating movement of the platform where they are located. The user can select the starting and the ending frequencies of the sound wave that will be transmitted to the platform. This sound wave is synthesized on demand, there are no pre-recorded sound files.<br />
<br />
The frequency can be chosen between 20Hz and 150Hz. Bellow 20Hz there is no response from the hardware to vibrate the platform; above 150Hz, the inertia "forces" a low amplitude motion, so the dice don't move.<br />
<br />
The user can also choose the duration of the sound wave from 1.5 to 10 seconds. The lower value is enough to shuffle some of the dice at lower frequency but it will ultimately result in a small randomization. The upper limit allows very high randomization and is not even necessary in some cases.<br />
<br />
==Images==<br />
The user chooses how many frames should be analysed by choosing the number of samples between 1 and 20.<br />
<br />
One sample can be used to check how the recognition software works (what are the steps and how long they take). With 20 samples the results start to show a distribution that evolves towards the Gaussian distribution (even though, in theory, this only happens after 30 samples, minimum).<br />
<br />
==Video==<br />
The user can choose whether to watch or not to watch the shuffling process video.<br />
<br />
This video is composed of a series of .jpg pictures, which means that it is not an actual video, and the rate of display can change substantially with the connection quality. <br />
<br />
Since the video feed has a high demand on the internet connection, the user is advised to use it only once, as its purpose is merely to satisfy curiosity.<br />
<br />
=Advanced Protocol=<br />
After enough samples a graph can be constructed with the number of times each number is recorded in each bin and a Gaussian distribution can be fitted:<br />
<br />
\[<br />
p(x) = y_0 + A e ^{- \frac{(x- \mu)^2}{\sigma ^2}}<br />
\]<br />
<br />
[[File:EstatDados-grafico.png|thumb|Figure 5: Example distribution]]<br />
<br />
Since there are 14 dice, the expected mean value is 49 (why?), which is confirmed by the build-up of values fitting.<br />
<br />
The best way to conduct this study is to merge the results of several users and see if the fit is improved with increasing number of samples.<br />
<br />
The expected value for the average of N 6-sided dice is: <br />
<br />
\[<br />
\bar{\mu} = \frac{6N+N}{2}<br />
\]<br />
<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px"><br />
'''User Manual'''<br />
<div class="mw-collapsible-content"><br />
<br />
<br />
The user can change the parameters of the image recognition process.<br />
<br />
* ''Black&White Threshold'': the value, in grayscale, above which the pixels are considered white.<br />
<br />
* Hough Transform<br />
** ''Threshold1'': value above which a pixel from the Hough Transform image is taken to post processing.<br />
** ''Threshold2'': averages the pixels near the one that passed the previous test and, if that value is above Threshold2, this pixel is taken to post processing. <br />
** ''Threshold3'': in theory, any pixel in a white zone is ignored but, if the average (calculated for the previous test) is above Threshold3, the pixel is NOT ignored. This is used because some of the dice have white spots inside the black marks.<br />
<br />
* ''Convolution Threshold'': the value above which a pixel from the convolution image is taken to post processing.<br />
<br />
* ''Dice specifications'':<br />
** ''Mark radius'': the expected radius, given in pixels, of the marks (note that this algorithm expects the marks to be circular).<br />
** ''Dice width'': the maximum distance, in pixels, between marks. For example, in the 6-side of the dice, this value would be diagonal distance from one corner to the other.<br />
** ''Expected number of dice'': the number of dice in the experiment. If the algorithm finds more "dice" (i.e. sets of marks) than expected, and some of those sets are not considered "compatible" with dice, then these sets are eliminated ('''Note:''' The algorithm compares the relative position and distance between marks to create a set. These sets are then compared to what is expected in a dice; if the comparison is positive, the set is marked as "compatible", if not, some marks are switched or eliminated until a "compatible" set is achieved). <br />
</div><br />
</div><br />
<br />
<br />
=Links=<br />
*[[Estatística de Dados | Portuguese version (versão em português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Dice_Statistics&diff=1317Dice Statistics2013-10-16T22:38:55Z<p>Ist128595: </p>
<hr />
<div>=Description=<br />
<br />
This experiment consists on an apparatus that automatically shuffles a set of six-sided dice. To count the spots, it acquires and processes an image recognition pattern from the top side of each dice.<br />
<br />
By recording the number of times each side appears, you can study the law of probabilities and develop a statistical study of random phenomena. Using the images that this experiment outputs, you can also develop your own algorithms using them in the study of computer recognition software.<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: [unavailable]<br />
*Laboratory: Básico em e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: Aleatorio<br />
*Grade: ***<br />
<br />
</div><br />
</div><br />
<br />
{{#evp:youtube|_qUFqXJQOpM|Dice in slow motion taken from the experiment (12x slow).|left}} <br />
<br />
<swf height="320" width="320">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/EstatisticaDados.swf</swf> <br />
<br />
<br />
<br />
<br />
<br />
=Experimental Apparatus=<br />
In this experiment there is a loudspeaker positioned horizontally with a platform on top of the cone made of k-line (structured light cardboard with polyurethane), where fourteen dice are spread. Above it, at 300mm from the platform, there is a video camera equipped with a white high brightness LED to illuminate it.<br />
<br />
[[File:EstatDados-montagem.jpg|thumb|alt=Apparatus for the Dice Statistics experiment|Apparatus]]<br />
<br />
The release (i.e. shuffling) of the dice is achieved through the speaker's stimulus with a sound wave, which makes the platform vibrate with the same frequency of the wave.<br />
<br />
This platform has a wall that prevents the dice from leaving the platform (and the web-cam field of view). The wall is quite high to prevent the entry of light from the outside of the lighting system.<br />
<br />
With this apparatus, pictures are obtained as we can see in picture 2.<br />
<br />
The image processing is made according to the flowchart below and the result of this process leads to an image like the one in picture 3.<br />
<br />
{|<br />
| [[File:EstatDados-foto1.jpg|thumb|Figure 2: Photograph of the dice]]<br />
| [[File:EstatDados-foto2.jpg|thumb|Figure 3: Figure 2 after software recognition]]<br />
| [[File:EstatDados-fluxograma.jpg|thumb|Fluxogram]]<br />
|}<br />
<br />
<br />
<br />
=Protocol=<br />
The experience execution protocol is simple because it consists in stimulating the platform conveniently, so that it can scramble the dice. And then, we describe the main features of the configurator control room for a better understanding.<br />
<br />
<!-- [[File:EstatDados-interface.gif|thumb|Picture 4: The experiment "Control room" configuration ]] --><br />
<br />
==Dice shuffling==<br />
The dice are released (shuffled) by the oscillating movement of the platform where they are located. The user can select the starting and the ending frequencies of the sound wave that will be transmitted to the platform. This sound wave is synthesized on demand, there are no pre-recorded sound files.<br />
<br />
The frequency can be chosen between 20Hz and 150Hz. Bellow 20Hz there is no response from the hardware to vibrate the platform; above 150Hz, the inertia "forces" a low amplitude motion, so the dice don't move.<br />
<br />
The user can also choose the duration of the sound wave from 1.5 to 10 seconds. The lower value is enough to shuffle some of the dice at lower frequency but it will ultimately result in a small randomization. The upper limit allows very high randomization and is not even necessary in some cases.<br />
<br />
==Images==<br />
The user chooses how many frames should be analysed by choosing the number of samples between 1 and 20.<br />
<br />
One sample can be used to check how the recognition software works (what are the steps and how long they take). With 20 samples the results start to show a distribution that evolves towards the Gaussian distribution (even though, in theory, this only happens after 30 samples, minimum).<br />
<br />
==Video==<br />
The user can choose whether to watch or not to watch the shuffling process video.<br />
<br />
This video is composed of a series of .jpg pictures, which means that it is not an actual video, and the rate of display can change substantially with the connection quality. <br />
<br />
Since the video feed has a high demand on the internet connection, the user is advised to use it only once, as its purpose is merely to satisfy curiosity.<br />
<br />
=Advanced Protocol=<br />
After enough samples a graph can be constructed with the number of times each number is recorded in each bin and a Gaussian distribution can be fitted:<br />
<br />
\[<br />
p(x) = y_0 + A e ^{- \frac{(x- \mu)^2}{\sigma ^2}}<br />
\]<br />
<br />
[[File:EstatDados-grafico.png|thumb|Figure 5: Example distribution]]<br />
<br />
Since there are 14 dice, the expected mean value is 49 (why?), which is confirmed by the build-up of values fitting.<br />
<br />
The best way to conduct this study is to merge the results of several users and see if the fit is improved with increasing number of samples.<br />
<br />
The expected value for the average of N 6-sided dice is: <br />
<br />
\[<br />
\bar{\mu} = \frac{6N+N}{2}<br />
\]<br />
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<br />
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<div class="toccolours mw-collapsible mw-collapsed" style="width:600px"><br />
'''User Manual'''<br />
<div class="mw-collapsible-content"><br />
<br />
<br />
The user can change the parameters of the image recognition process.<br />
<br />
* ''Black&White Threshold'': the value, in grayscale, above which the pixels are considered white.<br />
<br />
* Hough Transform<br />
** ''Threshold1'': value above which a pixel from the Hough Transform image is taken to post processing.<br />
** ''Threshold2'': averages the pixels near the one that passed the previous test and, if that value is above Threshold2, this pixel is taken to post processing. <br />
** ''Threshold3'': in theory, any pixel in a white zone is ignored but, if the average (calculated for the previous test) is above Threshold3, the pixel is NOT ignored. This is used because some of the dice have white spots inside the black marks.<br />
<br />
* ''Convolution Threshold'': the value above which a pixel from the convolution image is taken to post processing.<br />
<br />
* ''Dice specifications'':<br />
** ''Mark radius'': the expected radius, given in pixels, of the marks (note that this algorithm expects the marks to be circular).<br />
** ''Dice width'': the maximum distance, in pixels, between marks. For example, in the 6-side of the dice, this value would be diagonal distance from one corner to the other.<br />
** ''Expected number of dice'': the number of dice in the experiment. If the algorithm finds more "dice" (i.e. sets of marks) than expected, and some of those sets are not considered "compatible" with dice, then these sets are eliminated ('''Note:''' The algorithm compares the relative position and distance between marks to create a set. These sets are then compared to what is expected in a dice; if the comparison is positive, the set is marked as "compatible", if not, some marks are switched or eliminated until a "compatible" set is achieved). <br />
</div><br />
</div><br />
<br />
<br />
=Links=<br />
*[[Estatística de Dados | Portuguese version (versão em português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Liquid_Pressure_Variation_with_Depth&diff=1316Liquid Pressure Variation with Depth2013-10-16T22:03:24Z<p>Ist128595: </p>
<hr />
<div>===Description of the Experiment===<br />
In this experiment, we study the density of four different liquids by taking into account that pressure variation with depth depends on it.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/scuba.sdp<br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: scuba<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="550" width="480">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/Scuba.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Hidroestatica-montagem.jpg|thumb|Photo of the four tubes used in this experiment.]]<br />
<br />
In this experiment, there are four acrylic tubes with a diameter of thirty millimiters and one meter long. Each tube is filled with a different liquid: distilled water, salt water, glycerin and vegetable oil. Inside each tube there is a bell with an air bubble that allows pressure to be measured through a flexible tube, which is attached to a pressure sensor located outside the liquid.<br />
<br />
The change in volume can be estimated considering that each bell has a volume of approximately \( 2 cm^3 \). The hose has a cross-section of \(1 mm\) and a length of \(1 m\), but it can easily be ignored. <br />
<!-- (why?)(because the hose's volume doesn't change significantly with the change in pressure?). --><br />
<br />
The tubes are mounted vertically, and the four probes move simultaneously as established by the configuration chosen. The latter pauses for a second at each measuring point to allow the pressure to stabilize before measuring. The experiment will take longer if the user requests many points.<br />
<br />
<br />
=Protocol=<br />
The user must define the maximum and minimum height, as well as the number of samples to take across the path. This means that he can choose the initial and final depth of the probe's motion and obtain the data (each liquid's) on the variation of the pressure as depth changes.<br />
Afterwards, the data can be fitted to the following equation, and from that, the density of the various liquids can be determined.<br />
<br />
\[<br />
p(h) = p_0 + \rho g h<br />
\]<br />
<br />
If multiple runs are made (with different starting and ending points), the experimental error will be lower.<br />
<br />
The following table shows the four liquid's density accepted values.<br />
<br />
{| border="1"<br />
|-<br />
! Material<br />
! Accepted density (\( \frac{kg}{m^3} \))<br />
|-<br />
| Water <br />
| \( 1,00 \times 10 ^3 \)<br />
|-<br />
| Glycerine<br />
| \( 1,26 \times 10 ^3 \)<br />
|-<br />
| Salty water<br />
| \( 1,03 \times 10 ^3 \)<br />
|-<br />
| Vegetable oil <br />
| \( 0,92 \times 10 ^3 \)<br />
|}<br />
<br />
<br />
=Theoretical Principles=<br />
The pressure exerted by a liquid is proportional to the weight of the fluid column, meaning that it depends not only on depth but also on density. This can be determined through the relation between pressure and depth. This relation can be expressed mathematically by: <br />
<br />
\[ <br />
p = p_0 + \rho g h <br />
\] <br />
<br />
where \( p_0 \) represents the pressure at the liquid's surface and \( \rho = m/V\) it's density, being <i>g</i> the local gravity acceleration and <i>h</i> the depth.<br />
Recalling Pascal's principle note that \( p_0 \) is evenly distributed through the whole liquid.<br />
<br />
<br />
=Links=<br />
*[[Variação da Pressão num Líquido com a Profundidade |Portuguese Version (Versão em Português)]]</div>Ist128595http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Liquid_Pressure_Variation_with_Depth&diff=1315Liquid Pressure Variation with Depth2013-10-16T22:00:29Z<p>Ist128595: </p>
<hr />
<div>===Description of the Experiment===<br />
In this experiment, we study the density of four different liquids by taking into account that pressure variation with depth depends on it.<br />
<br />
<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px"><br />
'''Links'''<br />
<div class="mw-collapsible-content"><br />
<br />
*Video: rtsp://elabmc.ist.utl.pt/scuba.sdp<br />
*Laboratory: Basic in e-lab.ist.eu[http://e-lab.ist.eu]<br />
*Control room: scuba<br />
*Grade: **<br />
<br />
</div><br />
</div><br />
<br />
<br />
<swf height="550" width="480">http://www.elab.tecnico.ulisboa.pt/anexos/descricoes-flash/Scuba.swf</swf><br />
<br />
<br />
=Experimental Apparatus=<br />
[[File:Hidroestatica-montagem.jpg|thumb|Photo of the four tubes used in this experiment.]]<br />
<br />
In this experiment, there are four acrylic tubes with a diameter of thirty millimiters and one meter long. Each tube is filled with a different liquid: distilled water, salt water, glycerin and vegetable oil. Inside each tube there is a bell with an air bubble that allows pressure to be measured through a flexible tube, which is attached to a pressure sensor located outside the liquid.<br />
<br />
The change in volume can be estimated considering that each bell has a volume of approximately \( 2 cm^3 \). The hose has a cross-section of \(1 mm\) and a length of \(1 m\), but it can easily be ignored. <br />
<!-- (why?)(because the hose's volume doesn't change significantly with the change in pressure?). --><br />
<br />
The tubes are mounted vertically, and the four probes move simultaneously as established by the configuration chosen. The latter pause for a second at each measuring point to allow the pressure to stabilize before measuring. The experiment will take longer if the user requests many points.<br />
<br />
<br />
=Protocol=<br />
The user must define the maximum and minimum height, as well as the number of samples to take across the path. This means that he can choose the initial and final depth of the probe's motion and obtain the data (for each liquid) on the variation of the pressure as depth changes.<br />
Afterwards, the data can be fitted to the following equation, and from that, the density of the various liquids can be determined.<br />
<br />
\[<br />
p(h) = p_0 + \rho g h<br />
\]<br />
<br />
If multiple runs are made (with different starting and ending points), the experimental error will be lower.<br />
<br />
The following table shows the four liquid's density accepted values.<br />
<br />
{| border="1"<br />
|-<br />
! Material<br />
! Accepted density (\( \frac{kg}{m^3} \))<br />
|-<br />
| Water <br />
| \( 1,00 \times 10 ^3 \)<br />
|-<br />
| Glycerine<br />
| \( 1,26 \times 10 ^3 \)<br />
|-<br />
| Salty water<br />
| \( 1,03 \times 10 ^3 \)<br />
|-<br />
| Vegetable oil <br />
| \( 0,92 \times 10 ^3 \)<br />
|}<br />
<br />
<br />
=Theoretical Principles=<br />
The pressure exerted by a liquid is proportional to the weight of the fluid column, meaning that it depends not only on depth but also on density. This can be determined through the relation between pressure and depth. This relation can be expressed mathematically by: <br />
<br />
\[ <br />
p = p_0 + \rho g h <br />
\] <br />
<br />
where \( p_0 \) represents the pressure at the liquid's surface and \( \rho = m/V\) it's density, being <i>g</i> the local gravity acceleration and <i>h</i> the depth.<br />
Recalling Pascal's principle note that \( p_0 \) is evenly distributed through the whole liquid.<br />
<br />
<br />
=Links=<br />
*[[Variação da Pressão num Líquido com a Profundidade |Portuguese Version (Versão em Português)]]</div>Ist128595