Difference between revisions of "Angular Momentum Conservation"

From wwwelab
Jump to navigation Jump to search
m (Text replacement - "|left" to "|center")
 
(67 intermediate revisions by 4 users not shown)
Line 1: Line 1:
−
UNDER CONSTRUCTION
+
=Description of the Experiment=
 +
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.
 +
 
 +
<!-- Acho que este texto não está muito correcto. -->
 +
 
 +
 
 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px">
 +
'''Links'''
 +
<div class="mw-collapsible-content">
 +
 
 +
*Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp
 +
*Laboratory: Intermediate in [http://e-lab.ist.eu e-lab.ist.eu]
 +
*Control room: [unavaliable]
 +
*Level: ***
 +
 
 +
</div>
 +
</div>
 +
 
 +
{{#ev:youtube|X2OK3akY4GE|Slow motion video (12x) of the discs connecting.|center}}
  
−
=Description of the Experiment=
 
−
This control room allows the confirmation of angular momentum conservation by colliding a spinning disk with another. Moreover, the disk inertia momentum  can be extrapolate from energy conservation principles.
 
  
 
=Experimental Apparatus=
 
=Experimental Apparatus=
−
The experimental apparatus is based in a PC hard-drive motor and its spinning disk. A second disk is held on top of it and can be dropped by a servo motor actuator.
+
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.
 +
 
 +
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.
  
−
The apparatus´ motor can be used as a generator equipped with a switchable resistor acting as an electromagnetic brake. The braking current&voltage characteristic is measured allowing a rigorous energy dissipation calculation.
 
  
 
=Protocol - Angular Momentum Conservation=
 
=Protocol - Angular Momentum Conservation=
  
−
[[File:Discos_velocidade_protocolo1.png|thumb|alt=|Figura1: rotational speed]]
+
[[File:Discs_velocity_protocol1.png|thumb|alt=|Figure 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.]]
 +
 
 +
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.
 +
 
 +
Data taken from the experiment is given and plotted with the disc velocity in function of time.
 +
 
 +
'''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.
 +
 
 +
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.
 +
 
 +
 
 +
=Advanced Protocol - Moment of Inertia Evaluation=
 +
 
 +
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figure2: Rotational velocity as function of time after the electromagnetic breaking.|right|border|240px]]
 +
[[File:Discos_tensao_2fases.png|thumb|alt=|Figure3: Circuit schematic for voltage measurement.|right|border|240px]]
 +
[[File:Discos_tensao.png|thumb|alt=|Figure4: Voltage between two phases during breaking.|right|border|240px]]
 +
[[File:W_extrap.gif|thumb|alt=|Figure5: Extrapolation of w based on the initial slope of deceleration.|right|border|240px]]
 +
[[File:Discos_balanco_energetico.png|thumb|alt=|Figure6: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia.|right|border|240px]]
 +
 
 +
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.
 +
 
 +
'''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.
 +
 
 +
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 line. From this deceleration it will be possible to calculate  the instantaneous loss of angular momentum deferentially.
 +
 
 +
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.
 +
 
 +
<math>  \Delta E_{mec} = \Delta E_{friction} + \Delta E_{elec}</math>
 +
 
 +
The energy of a rotating body is <math> E_{rot}=\frac{I w^2}{2}</math> I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:
  
−
A disc with a total mass of 115g are accelerated by the hard-drive motor until they reach a selected angular velocity. In this instant the motor is disconnected from supply and the discs allowed to rotate freely being their speed monitored. When a certain pre-determined speed is reached the servo lets the suspended discs with a total mass of 69g initially at rest fall on top of the rotating disc.
+
<math>\Delta E_{mec}=\frac{I(w_{n+1}^2-w_{n}^2)}{2}</math>
  
−
Data taken from the experiment is given and plotted with the disc speed in function of time.
+
<math>w_{n+1}</math> and <math>w_{n}</math> being the angular velocity in two consecutive acquisitions.
  
−
'''Figure1''' is a plot created in Microsoft Excel using the table of results of an experience in which the servo let's the suspended discs fall when the discs below reach 1000 rpm.  
+
Using the initial slope of deceleration <math>a</math> due to mechanical friction, one can extrapolate <math>w_{n+1}</math> in absence of electromagnetic breaking.
  
−
Doing a linear regression between the deceleration and fall of the discs, it's possible to obtain the predicted rotational speed at the time that the falling discs stop sliding over the bottom discs.
+
<math>w_{n+1}= w_{n} + a \Delta t</math>
  
−
=Physics=
+
Substituting this extrapolated <math>w_{n+1}</math> in the equation of total energy variation it is possible to calculate the energy dissipation due to mechanical friction:
−
Using the following quantities:
+
 
 +
<math>\Delta E_{friction}=\frac{I(w_{n}^2+2w_{n}a\Delta t + a^2\Delta t^2-w_{n}^2)}{2}</math>
 +
 
 +
<math>\Delta E_{friction}=\frac{I(2w_{n}a\Delta t + a^2\Delta t^2)}{2}</math>
 +
 
 +
A set of extrapolations of <math>w_{n+1}</math> can be seen in '''Figure 5'''.
 +
 
 +
The dissipated power is:
 +
 
 +
<math>P=VI=\frac{V^2}{R}</math>
 +
 
 +
The rms voltage across one winding is:
 +
 
 +
<math>V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}</math>
 +
 
 +
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 resistance value <math>4,7\Omega</math>, which means the power equation will come multiplied by 2 and <math>R=4,7\Omega</math> will be used.
 +
 
 +
<math>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}</math>
 +
 
 +
<math>P=\frac{V^2}{R}</math>
 +
 
 +
The energy dissipated will be:
 +
 
 +
<math>\Delta E_{ele}=P*\Delta t</math>
 +
 
 +
Where <math>\Delta t</math> is the time between acquisitions.
  
−
L - angular momentum
+
The ballance between each consecutive acquitition is summed in the end.
  
−
I - moment of inertia
+
<math>Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{elec}</math>
  
−
&omega; - angular velocity
+
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.
  
−
m - mass in rotation.
+
Using this method, we reached an experimental value of <math>1,274\times10^{-4} kg m^2</math>  for the moment of inertia.
  
−
<div style="text-align: center;">We have for the angular momentum conservation:</div>
+
'''Figure 6''' 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.
−
\[L_i=L_f\]
 
−
\[I_i \omega_i=I_f \omega_f\]
 
−
\[\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}\]
 
−
\[\frac{\frac{m_i\left (r_1^2+r_2^2 \right )}{2}}{\frac{m_f\left (r_1^2+r_2^2 \right )}{2}}=\frac{\omega_f}{\omega_i}\]
 
−
\[\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}\]
 
  
−
<div style="text-align: center;">The experimental results give:</div>
+
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:
  
−
\[\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656\]
+
<math>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</math>
  
−
<div style="text-align: center;">while the predicted mass ratio is</div>
+
Evaluating the accuracy:
  
−
\[\frac{m_i}{m_f}=\frac{115}{115+69}=0,625\]
+
<math>\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%</math>
  
−
<div style="text-align: center;">Evaluating the accuracy</div>
+
This experiment gave results that differ by '''6,8%''' from the ones calculated theoretically.
  
−
\[\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%\]
+
=Physics=
 +
Using the following quantities:
  
−
The speed ratio is different from the mass ratio by 4,9% which confirms the angular momentum conservation.
+
L - angular momentum
  
−
=Protocol 2 - Moment of Inertia evaluation=
+
I - moment of inertia
  
−
[[File:Discos_velocidade_protocolo2.png|thumb|alt=|Figura2: rotational speed]]
+
&omega; - angular velocity
−
[[File:Discos_tensao.png|thumb|alt=|Figura3: voltage across one&nbsp;of&nbsp;the&nbsp;resistors]]
 
−
[[File:Discos_tensao_compensada.png|thumb|alt=|Figura4: compensated voltage]]
 
  
−
5 discs with a total mass of 115g are accelerated by the hard-drive motor until they reach 1500rpm. In this moment the motor is disconnected, the discs rotate freely and their speed is read. When a certain speed that the user defines previously is reached, the relay puts each motor winding in parallel with a resistor which resistance is the same as the motor windings'. These resistors will dissipate energy acting as an electromagnetic brake.
+
m - mass in rotation.
  
−
In the end of the session a table is provided with the disc speed and tension across one of the resistors function of time, whose values will be used in plots created in some program the user chooses.
+
For the angular momentum conservation:
  
−
'''Figure2''' and '''3''' are plots created in Microsoft Excel using the table of results of an experience in which the relay turns on when the rotating discs reach 1000 rpm.
+
<math>L_i=L_f</math>
  
−
The difference in the deceleration slope is clear. At first the discs are slowed only by the kinetic friction, but when the relay turns on, the resistors and motor windings are put in parallel and energy starts being dissipated through the resistors. The contribution of the resistors to the deceleration and consequently energy of the discs is given by the difference between the slopes.
+
<math>I_i \omega_i=I_f \omega_f</math>
  
−
Quantifying:
+
<math>\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}</math>
  
−
\[\frac{d\omega_{res}}{dt}=\frac{d\omega_{atrito+res}}{dt}-\frac{d\omega_{atrito}}{dt}=-194,860-(-79,234)=-115,626rpm/s = -115,626\times\frac{2\pi}{60}=-12,11rad/s^2\]
+
<math>\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}</math>
  
−
The kinetic friction must also be taken into account in the voltages, so that the energy dissipated by the resistors corresponds to the energy loss given by the difference between slopes. This can be done, adding the function adjusted to the first data series to the second and adjusting a new function.
+
<math>\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}</math>
  
−
To clarify, a voltage of 0,0241(t-6,42)V will be added to each point in the table of results after the time at which the relay turns on and a new function is adjusted to that data series. The result of this process is shown in '''Figure4'''.
+
The experimental results give:
  
−
Using the voltage given by the adjust function in '''Figure4''' and knowing that the resistance is 5,3&Omega; in each of the 3 resistors, the dissipated power is calculated by:
+
<math>\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656</math>
  
−
\[P=VI=V\frac{V}{R}=\frac{V^2}{R}\]
+
while the predicted mass ratio is
  
−
The energy dissipated in the 3 resistors is obtained integrating the dissipated power over time:
+
<math>\frac{m_i}{m_f}=\frac{115}{115+69}=0,625</math>
  
−
\[\int_{6,67}^{9,62}\frac{dE}{dt}dt=3\int_{6,67}^{9,62}\frac{V^2}{R}dt=0,313J\]
+
Evaluating the accuracy:
  
−
The energy of a solid body in rotation and it's derivative is given by:
+
<math>\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%</math>
  
−
\[E=\frac{I\omega^2}{2}\;\;\;\;\;\;\;\;\;\;\frac{dE}{dt}=I\omega'\omega\]
+
The speed ratio is different from the mass ratio by 4,9%, which gives a good approximation of the angular momentum conservation.
  
−
The last equation is integrated knowing that \(\int_{6,67}^{9,62}\frac{dE}{dt}dt=-0,313J\) (the "-" is there because the energy is dissipated), &omega;' is constant = -12,11rad/s^2 (comes from the difference between slopes) and &omega; is given by the deceleration adjust function in rad/s,
+
Knowing the exact dimensions of the disks (<math>r_1=12,5mm, r_2=47,5mm</math>) 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).
  
−
\[\int_{6,67}^{9,62}\frac{dE}{dt}dt=\int_{6,67}^{9,62}I\omega'\omega \; dt\]
+
<math>I_i \omega_i=I_f \omega_f</math>
−
\[-0,313=I(-12,11)\int_{6,67}^{9,62}\omega \; dt\]
 
−
\[\frac{-0,313}{-12,11}=I\int_{6,67}^{9,62}-20,41t+230,93 dt\]
 
−
\[I=\frac{-0,313}{-12,11\times 190,837}\]
 
−
\[I=1,354\times 10^{-4}kg\, m^2\]
 
  
−
Now, the discs in this experience are in fact rings of interior and exterior radius 13mm and 47mm respectively. Their moment of inertia is then:
+
<math>\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f</math>
  
−
\[I=\frac{m\left(r_1^2+r_2^2\right)}{2}=\frac{0,115\left(0,013^2+0,047^2\right)}{2}=1,367\times 10^{-4}kg \; m^2\]
+
Solving in for <math>I_m</math>
  
−
Evaluating the accuracy,
+
<math>I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}</math>
  
−
\[\frac{\left|1,354\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=0,9\%\]
 
  
−
The experimental results of this experience differ only '''0,9%''' from the theoretical expectations!
+
=Links=
 +
*[[Conservação do Momento Angular | Portuguese version (Versão em Português)]]

Latest revision as of 15:12, 28 May 2015

Description of the Experiment

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.


Links

  • Video: rtsp://elabmc.ist.utl.pt/inertiadisks.sdp
  • Laboratory: Intermediate in e-lab.ist.eu
  • Control room: [unavaliable]
  • Level: ***


Experimental Apparatus

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.

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.


Protocol - Angular Momentum Conservation

Figure 1: Angular velocity (rpm) as a function of time for a collision at 1000 rpm.

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.

Data taken from the experiment is given and plotted with the disc velocity in function of time.

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.

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.


Advanced Protocol - Moment of Inertia Evaluation

Figure2: Rotational velocity as function of time after the electromagnetic breaking.
Figure3: Circuit schematic for voltage measurement.
Figure4: Voltage between two phases during breaking.
Figure5: Extrapolation of w based on the initial slope of deceleration.
Figure6: Final energy balance showing electrical and mechanical component allowing you to get the total moment of inertia.

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.

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.

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 line. From this deceleration it will be possible to calculate the instantaneous loss of angular momentum deferentially.

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.

[math] \Delta E_{mec} = \Delta E_{friction} + \Delta E_{elec}[/math]

The energy of a rotating body is [math] E_{rot}=\frac{I w^2}{2}[/math] I being the moment of inertia. Then, the variation of mechanical energy between each acquisition will be:

[math]\Delta E_{mec}=\frac{I(w_{n+1}^2-w_{n}^2)}{2}[/math]

[math]w_{n+1}[/math] and [math]w_{n}[/math] being the angular velocity in two consecutive acquisitions.

Using the initial slope of deceleration [math]a[/math] due to mechanical friction, one can extrapolate [math]w_{n+1}[/math] in absence of electromagnetic breaking.

[math]w_{n+1}= w_{n} + a \Delta t[/math]

Substituting this extrapolated [math]w_{n+1}[/math] in the equation of total energy variation it is possible to calculate the energy dissipation due to mechanical friction:

[math]\Delta E_{friction}=\frac{I(w_{n}^2+2w_{n}a\Delta t + a^2\Delta t^2-w_{n}^2)}{2}[/math]

[math]\Delta E_{friction}=\frac{I(2w_{n}a\Delta t + a^2\Delta t^2)}{2}[/math]

A set of extrapolations of [math]w_{n+1}[/math] can be seen in Figure 5.

The dissipated power is:

[math]P=VI=\frac{V^2}{R}[/math]

The rms voltage across one winding is:

[math]V_{rms}=\frac{V_{measured}}{\sqrt{3}\sqrt{2}}[/math]

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 resistance value [math]4,7\Omega[/math], which means the power equation will come multiplied by 2 and [math]R=4,7\Omega[/math] will be used.

[math]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}[/math]

[math]P=\frac{V^2}{R}[/math]

The energy dissipated will be:

[math]\Delta E_{ele}=P*\Delta t[/math]

Where [math]\Delta t[/math] is the time between acquisitions.

The ballance between each consecutive acquitition is summed in the end.

[math]Balance = \Delta E_{mec} - \Delta E_{friction} - \Delta E_{elec}[/math]

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.

Using this method, we reached an experimental value of [math]1,274\times10^{-4} kg m^2[/math] for the moment of inertia.

Figure 6 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.

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:

[math]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[/math]

Evaluating the accuracy:

[math]\frac{\left|1,274\times 10^{-4}-1,367\times 10^{-4}\right|}{\left|1,367\times 10^{-4}\right|}\times 100=6,8\%[/math]

This experiment gave results that differ by 6,8% from the ones calculated theoretically.

Physics

Using the following quantities:

L - angular momentum

I - moment of inertia

ω - angular velocity

m - mass in rotation.

For the angular momentum conservation:

[math]L_i=L_f[/math]

[math]I_i \omega_i=I_f \omega_f[/math]

[math]\frac{I_i}{I_f}=\frac{\omega_f}{\omega_i}[/math]

[math]\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}[/math]

[math]\frac{m_i}{m_f}=\frac{\omega_f}{\omega_i}[/math]

The experimental results give:

[math]\frac{\omega_f}{\omega_i}=\frac{623}{950}=0,656[/math]

while the predicted mass ratio is

[math]\frac{m_i}{m_f}=\frac{115}{115+69}=0,625[/math]

Evaluating the accuracy:

[math]\frac{\left|0,656-0,625\right|}{\left|0,625\right|}\times 100=4,9\%[/math]

The speed ratio is different from the mass ratio by 4,9%, which gives a good approximation of the angular momentum conservation.

Knowing the exact dimensions of the disks ([math]r_1=12,5mm, r_2=47,5mm[/math]) 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).

[math]I_i \omega_i=I_f \omega_f[/math]

[math]\left (I_m + I_{Di}\right ) \omega_i=\left (I_m + I_{Df}\right ) \omega_f[/math]

Solving in for [math]I_m[/math]

[math]I_m = \frac{I_{Df} \omega_f - I_{Di} \omega_i}{\omega_i-\omega_f}[/math]


Links

Retrieved from "http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Angular_Momentum_Conservation&oldid=1715"