Difference between revisions of "World Pendulum"

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=Description=
 
=Description=
The aim of this world experiment is to provide a constellation of pendulums across several latitudes to investigate the gravitational constant behaviour around the globe.
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[[File:Soyuz VS03 liftoff.jpg||thumb|Soyuz lift-off from French Guiana @ 5º north of the Equator .|right|border|236px]]
It is expected to have several countries among CPLP community contributing to this effort through their own high schools in complement to the e-lab on-line offer.
+
Rockets are launched to space from equatorial latitudes. This is due to the fact that the apparent weight of objects is gradually reduced from the poles to the equator. We will feel lighter at the equator than at the poles!
  
==Sponsors who contributed for this experiment==
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This small difference in apparent weight allows the same rocket to launch heavier payloads into orbit if launched nearer from the equator. For example, a Soyuz rocket launching into geostationary orbit from the French Guiana (5ºN) can carry 3 tons while it will only be capable of launching 1.7 tons of cargo when launched from Baikonur, Kazakhstan (46ºN).
  
 +
The goal of this experiment is to find the value of the gravity "constant" through a constellation of pendulums placed in various latitudes and remotely operated, through the internet, by anyone.
  
<!-- [[File:LogoAnoPortugalBrasil.png|border|204px]]-->
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It is expected that CPLP countries can contribute to this effort, bringing students, teachers and interested citizens closer together.
[[File:LogoSPF long.jpg|border|204px]]
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[[File:LogoPlanetarioRioJaneiro.png|border|204px]]
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There are two different activities occurring simultaneously: (i) access, through e-lab, of the pendulums located in different latitudes and (ii) the construction and local operation in schools or at home.
[[File:LogoPlanetarioGulbenkian.png|border|204px]]
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[[File:Logo_tap.png|border|204px]]
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Lisboa, Ilhéus, Faro e Rio de Janeiro were the first cities to contribute to the network in January 2013, making it possible for the first fits of experimental data to the theoretical equation within our project that describes how gravity changes with latitude to occur.
[[File:LogoPUC.PNG|border|65px]]
+
 
 +
If you want to be a part of the World Pendulum network, please contact us by sending us an [mailto:wwwelab@ist.utl.pt email].
 +
 
 +
<div class="toccolours mw-collapsible mw-collapsed" style="width:420px">
 +
'''Links'''
 +
<div class="mw-collapsible-content">
 +
 
 +
*Video Faro: rtsp://elabmc.ist.utl.pt/worldpendulum_ccvalg.sdp
 +
*Video Lisboa: rtsp://elabmc.ist.utl.pt/worldpendulum_planetarium.sdp
 +
*Video Ilhéus: rtsp://elabmc.ist.utl.pt/worldpendulum_ilheus.sdp
 +
*Video Rio Janeiro: rtsp://elabmc.ist.utl.pt/worldpendulum_puc.sdp
 +
*Video Maputo: rtsp://elabmc.ist.utl.pt/worldpendulum_maputo.sdp
 +
*Video São Tomé: rtsp://elabmc.ist.utl.pt/wp_saotome.sdp
 +
*Laboratory: World Pendulum in [http://elab.ist.utl.pt elab.ist.utl.pt]
 +
*Control room: Choose a location
 +
*Grade: *
 +
 
 +
</div>
 +
</div>
 +
 
 +
 
 +
==Who likes this idea==
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 +
[[File:PBA B1 1.png|border|180px|border|180px]]
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[[File:LogoSPF long.jpg|border|180px]]
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[[File:logo_EPS_blue.gif|border|80px]]
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[[File:Logo mar.png|border|80px]]
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[[File:LogoPlanetarioGulbenkian.png|border|180px]]
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[[File:LogoCCVALG.png|border|204px|border|180px]]
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[[File:LogoPlanetarioRioJaneiro.png|border|180px]]
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[[File:Logo info tech.png|border|180px]]
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[[File:Logo_tap.png|border|180px]]
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[[File:Cenfim Logo.jpg|border|180px]]
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[[File:LogoPUC.PNG|border|60px]]
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[[File:UESC BRASÃO ref.jpg|border|60px]]
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[[File:UFRPE.jpg|border|60px]]
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[[File:Logo_DGAE.png|border|380px]]
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[[File:LogosBeneficairesErasmus+RIGHT EN.jpg|border|280px]]
  
 
=Experimental apparatus=
 
=Experimental apparatus=
The pendulum used has been design and constructed by Dr. Jodl team in Germany <ref name="jodl">World pendulum—a distributed remotely controlled laboratory (RCL) to measure the Earth's gravitational acceleration depending on geographical latitude, Grober S, Vetter M, Eckert B and Jodl H J, European Journal of Physics - EUR J PHYS , vol. 28, no. 3, pp. 603-613, 2007</ref>. Minor changes have been introduced in the IST version. The following data have been used:
+
The pendulum design used was based in Dr. Jodl's design<ref name="jodl">World pendulum—a distributed remotely controlled laboratory (RCL) to measure the Earth's gravitational acceleration depending on geographical latitude, Grober S, Vetter M, Eckert B and Jodl H J, European Journal of Physics - EUR J PHYS , vol. 28, no. 3, pp. 603-613, 2007</ref>. Some minor changes were made to allow the same design to be easily replicated in high schools. The data concerning each pendulum follows:
  
 
[[File:WordlPendulum.JPG|thumb|Pendulum used for the world pendulum standard gravity experiment.]]
 
[[File:WordlPendulum.JPG|thumb|Pendulum used for the world pendulum standard gravity experiment.]]
 
[[File:Stringsuport.png|thumb|Pendulum string support to avoid elongation errors. The cable is fixed by soldering it into a brass M4 screw 40mm long.]]
 
[[File:Stringsuport.png|thumb|Pendulum string support to avoid elongation errors. The cable is fixed by soldering it into a brass M4 screw 40mm long.]]
<TABLE BORDER=1>
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[[File:Launcher_2.png|thumb|Standard launcher of the pendulum mass for the World Pendulum Alliance (WPA). This launcher uses a V-slot rail technology and it is characterized by a maximum horizontal launching distance of 250 mm.]]
<TR VALIGN=TOP>
+
 
<TD>
+
{| class="wikitable"
<P>String length (not counting the sphere)</P>
+
!colspan="6"|Physical sizes by place
</TD>
+
|-
<TD>
+
  | Place
<P>2705mm +/- 0.5mm</P>
+
  | Latitude
</TD>
+
  | Longitude
</TR>
+
  | Altitude (m)
<TR VALIGN=TOP>
+
  | Cable length (mm)
<TD>
+
  | Sphere diameter (mm)
<P>Sphere mass</P>
+
|-
</TD>
+
  | CCV_Algarve/Faro
<TD>
+
  | 37º00'N
<P>2kg +/- 75g</P>
+
  | 7º56'W
</TD>
+
  | 10
</TR>
+
  | 2677 +/- 0.5 @23ºC
<TR VALIGN=TOP>
+
  | 80.5 +/- 1.0
<TD>
+
|-
<P>Sphere radius</P>
+
  | UESC/Ilhéus
</TD>
+
  | 14º47'S
<TD>
+
  | 39º10'W
<P>81.5mm +/-0.5mm</P>
+
  | 220
</TD>
+
  | 2832.0 +/- 0.5 @23ºC
</TR>
+
  | 81.0 +/- 1.0
<TR VALIGN=TOP>
+
|-
<TD>
+
  | Lisbon
<P>String</P>
+
  | 38º41'N
</TD>
+
  | 9º12'W
<TD>
+
  | 20
<P>Remanium(r) - Stainless steel (Nickel chromium)
+
  | 2677 +/- 0.5 @19ºC
- 0,4mm</P>
+
  | 80.5 +/- 1.0
</TD>
+
|-
</TR>
+
  | Maputo
<TR VALIGN=TOP>
+
  | 25º56'S
<TD>
+
  | 32º36'E
<P>Modulus of elasticity of string</P>
+
  | 80
</TD>
+
  | 2609.8 +/- 0.5 @27ºC
<TD>
+
  | 80.5 +/- 1.0
<P>~200GPa</P>
+
|-
</TD>
+
  | São Tomé
</TR>
+
  | 0º21'N
<TR VALIGN=TOP>
+
  | 6º43'E
<TD>
+
  | 50
<P>Oscillation period measurement system</P>
+
  | 2756.5 +/- 0.5  @29ºC
</TD>
+
  | 81.8 +/- 0.5
<TD>
+
|-
<P>Micro processor with 7,3728MHz - 30ppm crystal
+
  | Prague - CTU
+ laser + PIN photodiode</P>
+
  | 50º5.5'N
</TD>
+
  | 14º25.0'E
</TR>
+
  | 150
<TR VALIGN=TOP>
+
  | 2803 +/- 0.5 @25ºC
<TD>
+
  | 80.1 +/- 0.5
<P>Wire CTE (25-500&ordm;C)</P>
+
|-
<P>(Coefficient of thermal expansion)</P>
+
  | Barcelona - UPC
</TD>
+
  | 41º24.6'N
<TD>
+
  | 2º13.1'E
<P>~14 x 10<SUP>-6</SUP> K<SUP>-1</SUP></P>
+
  | 55
</TD>
+
  | 2824 +/- 1
</TR>
+
  | 81.8 +/- 0.1
</TABLE>
+
|-
The experiment can be very easily replicated as for local gravity determination it can be human operated without the photo-gate and the microprocessor.
+
  | Rio de Janeiro - PUC
The stainless steel can be replace by brass structures.
+
  | 22º54.1'S
 +
  | 43º12'W
 +
  | 50
 +
  | 2826,0 +/- 0.5
 +
  | 81.6 +/- 0.1
 +
|-
 +
  | Praia - UniCV
 +
  | 14°56'N
 +
  | 23°31'W
 +
  | 40
 +
  | 2832,0 +/- 0.5
 +
  | 81.6 +/- 0.1
 +
|-
 +
  | Bogotá - UniAndes
 +
  | 4°36'N
 +
  | 74°3'W
 +
  | 2500
 +
  | 2824 +/- 0.5
 +
  | 82.0 +/- 0.1
 +
|-
 +
  | Bogotá - UNAD
 +
  | 4°35'N
 +
  | 74°5'W
 +
  | 2650
 +
  | 2835 +/- 0.5
 +
  | 82.0 +/- 0.1
 +
|-
 +
  | Panama city - UTP
 +
  | 9°1.3'N
 +
  | 79°31.9'W
 +
  | 82
 +
  | 2825 + /- 0.5 @28ºC
 +
  | 81.9 +/- 0.1
 +
|-
 +
  | Santiago - UChile
 +
  | 33°27.5'S
 +
  | 70°39.8'W
 +
  | 552
 +
  | 2825 +/- 0.5 @27ºC
 +
  | 81.9 +/- 0.1
 +
|-
 +
  | Valparaiso - UTFSM
 +
  | 33°1'S
 +
  | 71°37'W
 +
  | 30
 +
  | 2827.5 +/- 0.5 @28ºC
 +
  | 81.8 +/- 0.1
 +
|-
 +
  | Panama city - USMA
 +
  | 9°1'N
 +
  | 79°37'W
 +
  | 130
 +
  | 2800.0 +/- 0.5 @35ºC
 +
  | 81.8 +/- 0.1
 +
|-
 +
  | Brasilia - UnB
 +
  | 15° 46'S
 +
  | 47° 52'W
 +
  | 1034
 +
  | 2826.8 mm +/- 0.5 @26ºC
 +
  | 81.4 +/- 0.1
 +
|-
 +
  | Marseille - ECM
 +
  | 43°20.6'N
 +
  | 5°26.2'E
 +
  | 162
 +
  | 2828.0 mm +/- 0.5 @22ºC
 +
  | 82.0 +/- 0.1
 +
|-
 +
  | Punta Arenas - UMag
 +
  | 53°8'S
 +
  | 70°52'W
 +
  | 40
 +
  | 2823 +/- 0.5 @16.4ºC
 +
  | 81.7 +/- 0.1
 +
|}
 +
 
 +
 
 +
 
 +
{| class="wikitable"
 +
!colspan="2"|Typical quantities
 +
|-
 +
| String length (not counting the sphere) || min: 0.5 m nominal: 2.8m max: 12m
 +
|-
 +
| Sphere mass || 2kg +/- 75g
 +
|-
 +
| Sphere diameter || 81.2mm +/-1.5mm
 +
|-
 +
| String || Remanium(r) - Stainless steel (Nickel chromium)
 +
- 0,4mm
 +
|-
 +
| Modulus of elasticity of string || ~200GPa
 +
|-
 +
| Oscillation period measurement system || Microprocessor with 7,3728MHz - 30ppm crystal
 +
+ laser + PIN photodiode
 +
|-
 +
| Wire CTE (25-500ºC) (Coefficient of thermal expansion) || ~14 x 10<sup>-6</sup> K<sup>-1</sup>
 +
|}
 +
 
 +
 
 +
 
 +
{| class="wikitable"
 +
!colspan="2"|Penulum length limits*
 +
|-
 +
| Minimum || ~1.5 m
 +
|-
 +
| Maximum || virtually no limit (~63.5 m)
 +
|}
 +
<nowiki>*</nowiki>These limits were estimated for the standard World Pendulum Alliance launcher (WPA). A photo of a standard WPA launcher is shown on the figure on the right.
 +
Check [http://www.elab.tecnico.ulisboa.pt/wiki/index.php?title=Precision_Pendulum_Assembly:_Apparatus_description#Pendulum_length_limits Pendulum length limits] to understand how these limits were obtained.
 +
 
 +
 
 +
The experimental apparatus can be easily adapted to human operation, using a good chronometer, for local execution. The stainless steel structures can made in by brass or bronze for easier machining. The cable used can be replaced by a sport fishing steel cable and the mass can be replaced by a Olympic weight throw training weight, weighing 2Kg. A calibrated measuring tape should be used to measure the cable length, '''a few days after assembling the apparatus to allow for cable expansion'''.
  
 
=Local partners=
 
=Local partners=
The pendulum <ref name="serway">Physics for scientists and engineers, 5th edition, Hardcourt College Publishers, R.Serway and R. Beichner, 2000</ref> is one of the simplest apparatus with a lot of physics inside. As a matter of fact only the exact measurement of the hanging string and its material quality as to be precise and careful selected to have very accurate measurements. Choosing a heavy mass (e.g. 1-4 kg) the error regarding the period can be made relatively low, allowing an accurate measurement with the use of a simple although precise chronometer.
+
The pendulum<ref name="serway">Physics for scientists and engineers, 5th edition, Hardcourt College Publishers, R.Serway and R. Beichner, 2000</ref>, although one of the simplest systems commonly studied, is one of the richest in terms of physics.
  
A local apparatus manufacture is an easy task for those interested in their local "<em>g</em>" comparison with the remote experiment . The collection of data through a suitable social network could provide even more data for a complete description of “<em>g's</em> constants” around the globe.
+
In order to build a precise pendulum the most important factors are the precise measurement of the length of the cable, its quality, and of that of the pendulum supports. Selecting a mass between 1 to 4 Kg ensures that the pendulum's period error will be small enough for small local gravity changes (smaller than 0.1%) to be detectable, as long as a precise chronometer is used for timekeeping.  
This “world experiment” could be a strong link among various schools for a propitious physics dissemination and dialogue.
 
  
You can find a detail construction guide in the page [[Precision Pendulum]].
+
A local apparatus can be assembled using readily available materials and the local <em>"g"</em> values determined using such an apparatus can then be compared to the ones obtained through the remote pendulum constellation and the theoretical model.
 +
 
 +
Collecting this data through a social network will allow a more precise description of how <em>"g"</em> varies around the globe. The "World Pendulum" can be an important collaborative network for the dissemination of physics in schools.
 +
 
 +
Instructions on how to build such a pendulum are available in [[Precision Pendulum]].
 +
The documentation of the development and construction of a pendulum are available in [[Precision Pendulum]] while the instructions on how to assemble it are available in [[Precision Pendulum Assembly]].
 +
 
 +
If you want to be a part of the World Pendulum network, please contact us by sending us an [mailto:wwwelab@ist.utl.pt email].
  
 
=Physics=
 
=Physics=
The determination of earth gravitational force in different parts of the globe leads to interesting questions about how modelling in physics is so important. Starting from the basic premise that the gravitational accelerating force at sea level is constant, it can be showed very easily that this “constant” slightly changes and has to be corrected mostly due to latitude. So, a very interesting approach to demystify science and some “constants” can be to pursuit the addition of successive stages of approximation to this estimate.  
+
Determining gravity's acceleration in different parts of the globe raises questions about the importance of models in physics. It's possible to show that gravity's acceleration at sea level changes with latitude, and therefore a correction is needed for each individual location. This process allows us to demystify science and correct the existing "urban myth" around some physical constants that only are truly constant when some approximations are done. In this particular case, we will show how the introduction of successive corrections to gravity's "constant" will lead to values closer to those experimentally obtained.  
The starting point of this undertaking is of course the basic approach to the geophysical model of earth as a simple (i) non-rotating (ii) sphere which gives the very well known approximation of 9.82 m/s2. By symmetry reasons this model leads to a non-dependent latitude dependence on the gravitational acceleration and consequently can be measured in any place on earth disrespect its latitude <ref name="jodl"></ref>.
 
But as soon as we progress on the model we will find that a rigorous measurement depends on latitude mainly due to (i) the centrifugal force originating from earth’s rotation and (ii) the non-spherical mass distribution caused by the earth’s poles flattening. These two effects are the most contributing ones being almost of the same amplitude and far more important than (i) altitude, (ii) tides or (iii) sub-sole earth’s density.
 
To conduct a fine-tuning to these evidences, one has to determine the earth’s acceleration in several latitudes across the globe being a difficult task for any student. With this advance in the earth’s acceleration determination, students could easy question the gravitational “constant” and study superior methods for the correct interpretation of earth gravitational force.
 
  
A possible way to overcome this difficulty is to have a constellation of experiments across the globe easily reached by the internet.
+
==Geophysical model==
 +
The starting point is the commonly used, constant, value of 9.81 ms<sup>-2</sup>. This is obtained by considering the Earth as being (i) a sphere (ii) that is not rotating. It's trivial to note that this model, due to the symmetry of the spherical form, does not allow for different values in different locations. This changes as soon as Earth's rotation dynamics and ellipsoid shape (flattening of the poles) are taken in account. These factors allow for gravity to change with latitude, and in fact these two factors are the two most important ones in this phenomena, outweighing every other effect, such as (i) altitude, (ii) tidal forces, and (iii) subsoil composition.
  
=CPLP as “latitude provider”=
+
To demonstrate these finer aspects, gravity's acceleration must be determined in various latitudes around the globe distant from each other. Using the data collected, students can then ask themselves about how "constant" the value truly is and improve their intuition of gravity.
 +
 
 +
===Experimental studies===
 +
====Variation with latitude====
 +
As seen, the first possible study consists of using the remote pendulums to obtain a measurement of the local gravity acceleration for each location they're based in. Through considering (or not) several factors, it is possible to fit the data to a experimental description of the Earth using spherical harmonics (equation \eqref{harmonica-esferica}). This experimental work can be conducted using e-lab's pendulum constellation and [http://rcl-munich.informatik.unibw-muenchen.de/ our partner's pendulums].
 +
 
 +
====Local determination====
 +
Following the instructions available in this wiki - [[Precision_Pendulum]] - or using any other kind of design that results in a rigorous apparatus, a local pendulum is built. It's then possible for measurements of local gravity to be made, as long as a good chronometer is used. Furthermore, it's also possible to contribute to the enrichment of the World Pendulum network's [https://docs.google.com/a/kic-innoenergy.com/spreadsheet/ccc?key=0AkxMmuJA92wgdHZnWHk5WHhaQldINGFqSTl6OGdpSlE#gid=0 spreadsheet].
 +
 
 +
====[[Tidal study]]====
 +
Using an almanac appropriate for the location, on can obtain the times of particular Moon/Sun alignments (full moon, new moon, waxing crescent and waxing gibbous). Plotting a graph spanning several months, one can try to verify and quantify the influence of tidal forces and Moon/Sun alignments in the apparent weight. It's possible to try and verify the correlation between Moon phases and changes in measurement of local gravity, by making a month or year-long study.
 +
Tidal effects are on the limit of detection by the pendulums of the e-lab constellation. For the experiment to be successful, it's necessary to be very rigorous on the time at which the experimental runs are made and some advanced numerical techniques, like the Fourier transform, need to by employed for the signal to be extracted from the data.
 +
 
 +
====Analysis of wire torsion ====
 +
[[File:Torcao.jpg||thumb|Effect of wire torsion and sphere ellipticity in the measurement of pendulum speed.|right|border|240px]]
 +
Those paying more attention will note that the speed of the mass changes due to wire torsion and due to the mass not being a perfect sphere. This is pictured in the image to the right. The pendulum can be studied taking into account the effect of the wire torsion (the use of Euler-Lagrange equations is recommended for this).
 +
 
 +
==Uniformly accelerated circular movement==
 +
The speed of the sphere in the lowest point of the trajectory is determined by measuring how much time the laser beam is interrupted. Knowing the sphere diameter, it's trivial to determine the speed at the origin. From this, the maximum kinetic energy can be calculated and the launching height of the pendulum determined. The calculated launching point can then be compared with the real one.
 +
 
 +
=Latitude provideres=
  
 
[[file:G_latitude.png|link=https://docs.google.com/a/kic-innoenergy.com/spreadsheet/oimg?key=0AkxMmuJA92wgdHZnWHk5WHhaQldINGFqSTl6OGdpSlE&oid=1&zx=hfmrs4egtbuf|thumb|The gravitational constant plotted against latitude with points of interest around the globe highlighted. Principe Island is just over zero latitude. Lisbon value was obtained with the current experiment and already over plotted on the graphic.]]
 
[[file:G_latitude.png|link=https://docs.google.com/a/kic-innoenergy.com/spreadsheet/oimg?key=0AkxMmuJA92wgdHZnWHk5WHhaQldINGFqSTl6OGdpSlE&oid=1&zx=hfmrs4egtbuf|thumb|The gravitational constant plotted against latitude with points of interest around the globe highlighted. Principe Island is just over zero latitude. Lisbon value was obtained with the current experiment and already over plotted on the graphic.]]
Mother language is one's nationality indicator factor and currently is the easier way to define what nations are designate as brother’s (“países irmãos”). Basically only four languages are spread around the globe being the Portuguese one of them. In particular the Portuguese community covers latitudes from ~30S to ~40N, almost a 75º span across the equator, giving an opportunity to operate as a latitude provider for the world pendulum constellation (see Figure).
 
  
Effectively to conduct this world experiment we need to consider at least four spaced points in order to have a proper fit. But due to the strong non-linearity of the equation more points are needed to provide a suitable adjustment in particular on the knee close to the earth’s equator. Brazil itself can provide almost four crucial points close to equator (e.g. Recife 8º, Salvador – 12º, Rio de Janeiro – 23º, Porto Alegre – 30º but lacks points with latitude where the characteristic varies more strongly, the almost linear region around 30º to 60º, where Portugal can give two good points, e.g. 37º and 41º. Mozambique can contribute for 27º (Maputo) and S. Tomé e Principe or Brazil are both good choices to cover the equator. Angola could give complementary points for those acquired in Brazil as the sensibility of the measurement is more pronounced close to equator and the poles.
+
Language is an important nationality factor ("My fatherland is the Portuguese language.", F. Pessoa) and a simple way to define what is called brother countries ("países irmãos"). Only four languages are disseminated around the world, Portuguese being one of them. The Portuguese speaking community covers latitudes from ~30S to ~40N, almost a 75º span across the equator. Therefore, CPLP countries can help by being "latitude providers" (see Figure).
 +
 
 +
To conduct this world experiment, at least four spaced points are needed in order to have a proper fit. But due to the strong non-linearity of the equation, more points are needed to provide a suitable adjustment, in particular on the "knee" close to the earth’s equator. Brazil itself can provide almost four crucial points close to the equator (e.g. Recife 8º, Salvador – 12º, Rio de Janeiro – 23º, Porto Alegre – 30º) but lacks points with a latitude where the characteristic varies more strongly, the almost linear region around 30º to 60º, where Portugal can give two good points (e.g. Porto - 37º and Faro - 41º). Mozambique can contribute with 27º (Maputo) and S. Tomé e Principe or Brazil are both good choices to cover the equator. Angola could give complementary points to those acquired in Brazil, as the sensibility of the measurement is more pronounced close to the equator and the poles.
  
 
=Data fitting=
 
=Data fitting=
Reference <ref name="serway"></ref> gives a very good description of the mathematical model needed to fit the data once established. The main harmonic equation to be fitted can be expressed generally:  
+
Available references <ref name="serway"></ref> <ref name="rcl">http://rcl-munich.informatik.unibw-muenchen.de/</ref> <ref name="olsom">Nelson, Robert; M. G. Olsson (February 1987). "The pendulum - Rich physics from a simple system". American Journal of Physics 54 (2):
\[
+
doi:10.1119/1.14703</ref> <ref name="gauld">Pendulums in the Physics Education Literature: A Bibliography, Gauld, Colin 2004 Science & Education, issue 7, volume 13, 811-832
 +
(http://dx.doi.org/10.1007/s11191-004-9508-7)</ref> <ref name="qureshi">The exact equation of motion of a simple pendulum of arbitrary amplitude: a hypergeometric approach, M I Qureshi et al 2010 Eur. J. Phys. 31 1485(http://dx.doi.org/10.1088/0143-0807/31/6/014)</ref> <ref name="ochs"> A comprehensive analytical solution of the nonlinear pendulum, Karlheinz Ochs 2011 Eur. J. Phys. 32 479 (http://dx.doi.org/10.1088/0143-0807/32/2/019)</ref> give a very good description of the mathematical model needed to fit the data. If all major factors are taken into account, gravity as a function of latitude is given by:
 +
 
 +
<math>
 
g_{n}(\varphi) = 9.780 326 772\times[1 + 0.005 302 33 \cdot  sin^{2}(\varphi) - 0.000 005 89 \cdot sin^{2}(2\varphi)]
 
g_{n}(\varphi) = 9.780 326 772\times[1 + 0.005 302 33 \cdot  sin^{2}(\varphi) - 0.000 005 89 \cdot sin^{2}(2\varphi)]
\]
+
</math>
where \(\varphi\) is the latitude. This expression is one of the best experimental approximation resulting from the standardization agreement to adjusting  the World Geodetic System datum surface (WSG84) to an ellipsoid  with radius r<sub>1</sub>=6378137m at the equator and r<sub>2</sub>=6356752m polar semi-minor radius.
 
This formula takes into account the fact that the Earth is an ellipsoid and that there is an additional increase in the acceleration of gravity when one moves nearer to the poles, due to a weaker centrifugal force. Nevertheless the students could derive a non-harmonic, first order approximation taking into account only the centrifugal force. As a second step they could include the two major errors, the centrifugal force and earth’s ellipsoid format.
 
  
[[File:Period_over_time.png|thumb|The variability of the period with elapsed time (angle amplitude < 7,5º), showing that this error is less than 0,15% regardless initial amplitude.]]
+
where \(\varphi\) is the latitude. This expression is one of the best experimental approximations and results from the standardization agreement to adjust the World Geodetic System datum surface (WSG84) to an ellipsoid with radius r<sub>1</sub>=6378137m at the equator and r<sub>2</sub>=6356752m polar semi-minor radius.
 +
This formula takes into account the fact that the Earth is an ellipsoid and that there is an additional increase in the acceleration of gravity when one moves nearer to the poles, due to a weaker centrifugal force. Nevertheless the students could derive a non-harmonic, first order approximation by taking into account only centrifugal force. Then, as a second step, they could include the two other major errors, the centrifugal force and earth’s ellipsoid format.
  
In the figure it is showed the expected deviation from “earth’s constant acceleration”, the real acceleration for each latitude. We have plotted the point already obtained with this apparatus in Lisbon and the marks over the expected latitudes for future partners.
+
[[File:Period_over_time.png|thumb|The variability of the period with elapsed time (angle amplitude < 7,5º), showing that this error is less than 0,05% regardless initial amplitude.]]
Of course these approximations do not include another important source of deviation from real data to the mathematical model as we do not include the experimental source of error. However those systematic errors could be under the expected precision needed (0,1%) for the former approximation if a careful design of the apparatus is considered. Nevertheless those errors must be discussed in advanced courses and proved their weight when considering the real pendulum.
 
  
=References=
+
The pictures shows the expected deviation from the “earth’s constant acceleration”, the real acceleration for each latitude. We have plotted the point already obtained with this apparatus in Lisbon and the marks over the expected latitudes for future partners.
 +
Of course these approximations do not include one important source of deviation from real data to the mathematical model, the experimental error, as we do not include the experimental source of error. However, those systematic errors could be under the expected precision needed (0,1%) for the former approximation if a careful design of the apparatus is considered. Nevertheless those errors must be discussed in advanced courses and their weight must be proved when considering the real pendulum.
 +
 
 +
=Historical notes=
 +
The pendulum importance as the basis of clocks and chronographs was only overthrown when the Royal Society convinced the English parliament to create an award, ranging from 10k£ to 20k£ (equivalent nowadays to more than 3.5M€), for the invention of a chronograph that didn't depend on it. The time precision of pendulum based systems is only bettered by modern electronic systems.
 +
 
 +
In the discovery age longitude was determined with a high error, since clocks and chronographs were reliant on pendulums and these were very sensitive to ships rocking, suffering changes in frequency or even stopping. Local longitude was calculated by comparing the solar hour (or stellar hour) with the ship's clock time.
 +
 
 +
=Frequently asked questions=
 +
 
 +
How long would it take to replace the cable that holds the sphere?
 +
 
 +
The remanium cable is a very strong material and only need to be replace in case of misuse.
 +
 
 +
Over time the cable can have some extension?
 +
 
 +
Cable length should undergone a rectified measure after ~3 months installation and thereafter once a year.
 +
 
 +
The temperature where the pendulum is installed interferes with the results of the experiments?
 +
 
 +
The variation induced by temperature is relevant when it comes to very precise measurements such as tide detection (local fluctuations), the reason why it is reported. NOTE that in some situations, due to incorrect housing of the temperature sensor, temperature increases during the reading as the sensor can be close to the power electronics devices. So, we advise a wise use of it.
  
<references />
+
Are the various pendulums subjected to regular calibrations?
 +
Regular calibrations shall take place at least once a year. Nevertheless we can not guarantee that every partner assure a regular calibration of the pendulums. Please monitor the course of each experience and use critical thinking in the face of any results obtained.
  
 +
What is the difference in measuring the cable with a laser or a tape measure, there is a noticeable difference between the two methods?
  
[2] http://rcl-munich.informatik.unibw-muenchen.de/
+
Usually both methods give the same incertitude because the limiting factor is the positioning of both devices by human usage which is barely the same, surpassing the intrinsic error of each device.
  
[4] Nelson, Robert; M. G. Olsson (February 1987). "The pendulum - Rich physics from a simple system". American Journal of Physics 54 (2):
+
The use of a frontal camera to analyze the motion of the pendulum, make a recording and then use some image analysis software to study the movements and oscillations will give good results?
doi:10.1119/1.14703
 
  
[5] Pendulums in the Physics Education Literature: A Bibliography, Gauld, Colin 2004 Science & Education, issue 7, volume 13, 811-832
+
This results can be obtained with the camera on top as well, a movie can be shot and analyses on Tracker software for instance.
(http://dx.doi.org/10.1007/s11191-004-9508-7)
 
  
[6] The exact equation of motion of a simple pendulum of arbitrary amplitude: a hypergeometric approach, M I Qureshi et al 2010 Eur. J. Phys. 31 1485(http://dx.doi.org/10.1088/0143-0807/31/6/014)
+
=References=
 +
 
 +
<references />
  
[7] A comprehensive analytical solution of the nonlinear pendulum, Karlheinz Ochs 2011 Eur. J. Phys. 32 479 (http://dx.doi.org/10.1088/0143-0807/32/2/019)
+
=Links=
 +
*[[Pêndulo Mundial | Portuguese version (Versão em Português)]]
 +
*[[Péndulo mundial| Spanish version (Versión en español)]]
 +
*[https://www.youtube.com/watch?v=ZOOFw_Nlee8&feature=youtu.be Building your own pendulum]

Latest revision as of 10:32, 13 October 2021

Description

Soyuz lift-off from French Guiana @ 5º north of the Equator .

Rockets are launched to space from equatorial latitudes. This is due to the fact that the apparent weight of objects is gradually reduced from the poles to the equator. We will feel lighter at the equator than at the poles!

This small difference in apparent weight allows the same rocket to launch heavier payloads into orbit if launched nearer from the equator. For example, a Soyuz rocket launching into geostationary orbit from the French Guiana (5ºN) can carry 3 tons while it will only be capable of launching 1.7 tons of cargo when launched from Baikonur, Kazakhstan (46ºN).

The goal of this experiment is to find the value of the gravity "constant" through a constellation of pendulums placed in various latitudes and remotely operated, through the internet, by anyone.

It is expected that CPLP countries can contribute to this effort, bringing students, teachers and interested citizens closer together.

There are two different activities occurring simultaneously: (i) access, through e-lab, of the pendulums located in different latitudes and (ii) the construction and local operation in schools or at home.

Lisboa, Ilhéus, Faro e Rio de Janeiro were the first cities to contribute to the network in January 2013, making it possible for the first fits of experimental data to the theoretical equation within our project that describes how gravity changes with latitude to occur.

If you want to be a part of the World Pendulum network, please contact us by sending us an email.

Links

  • Video Faro: rtsp://elabmc.ist.utl.pt/worldpendulum_ccvalg.sdp
  • Video Lisboa: rtsp://elabmc.ist.utl.pt/worldpendulum_planetarium.sdp
  • Video Ilhéus: rtsp://elabmc.ist.utl.pt/worldpendulum_ilheus.sdp
  • Video Rio Janeiro: rtsp://elabmc.ist.utl.pt/worldpendulum_puc.sdp
  • Video Maputo: rtsp://elabmc.ist.utl.pt/worldpendulum_maputo.sdp
  • Video São Tomé: rtsp://elabmc.ist.utl.pt/wp_saotome.sdp
  • Laboratory: World Pendulum in elab.ist.utl.pt
  • Control room: Choose a location
  • Grade: *


Who likes this idea

PBA B1 1.png LogoSPF long.jpg Logo EPS blue.gif Logo mar.png LogoPlanetarioGulbenkian.png LogoCCVALG.png LogoPlanetarioRioJaneiro.png Logo info tech.png Logo tap.png Cenfim Logo.jpg LogoPUC.PNG UESC BRASÃO ref.jpg UFRPE.jpg Logo DGAE.png LogosBeneficairesErasmus+RIGHT EN.jpg

Experimental apparatus

The pendulum design used was based in Dr. Jodl's design[1]. Some minor changes were made to allow the same design to be easily replicated in high schools. The data concerning each pendulum follows:

Pendulum used for the world pendulum standard gravity experiment.
Pendulum string support to avoid elongation errors. The cable is fixed by soldering it into a brass M4 screw 40mm long.
Standard launcher of the pendulum mass for the World Pendulum Alliance (WPA). This launcher uses a V-slot rail technology and it is characterized by a maximum horizontal launching distance of 250 mm.
Physical sizes by place
Place Latitude Longitude Altitude (m) Cable length (mm) Sphere diameter (mm)
CCV_Algarve/Faro 37º00'N 7º56'W 10 2677 +/- 0.5 @23ºC 80.5 +/- 1.0
UESC/Ilhéus 14º47'S 39º10'W 220 2832.0 +/- 0.5 @23ºC 81.0 +/- 1.0
Lisbon 38º41'N 9º12'W 20 2677 +/- 0.5 @19ºC 80.5 +/- 1.0
Maputo 25º56'S 32º36'E 80 2609.8 +/- 0.5 @27ºC 80.5 +/- 1.0
São Tomé 0º21'N 6º43'E 50 2756.5 +/- 0.5 @29ºC 81.8 +/- 0.5
Prague - CTU 50º5.5'N 14º25.0'E 150 2803 +/- 0.5 @25ºC 80.1 +/- 0.5
Barcelona - UPC 41º24.6'N 2º13.1'E 55 2824 +/- 1 81.8 +/- 0.1
Rio de Janeiro - PUC 22º54.1'S 43º12'W 50 2826,0 +/- 0.5 81.6 +/- 0.1
Praia - UniCV 14°56'N 23°31'W 40 2832,0 +/- 0.5 81.6 +/- 0.1
Bogotá - UniAndes 4°36'N 74°3'W 2500 2824 +/- 0.5 82.0 +/- 0.1
Bogotá - UNAD 4°35'N 74°5'W 2650 2835 +/- 0.5 82.0 +/- 0.1
Panama city - UTP 9°1.3'N 79°31.9'W 82 2825 + /- 0.5 @28ºC 81.9 +/- 0.1
Santiago - UChile 33°27.5'S 70°39.8'W 552 2825 +/- 0.5 @27ºC 81.9 +/- 0.1
Valparaiso - UTFSM 33°1'S 71°37'W 30 2827.5 +/- 0.5 @28ºC 81.8 +/- 0.1
Panama city - USMA 9°1'N 79°37'W 130 2800.0 +/- 0.5 @35ºC 81.8 +/- 0.1
Brasilia - UnB 15° 46'S 47° 52'W 1034 2826.8 mm +/- 0.5 @26ºC 81.4 +/- 0.1
Marseille - ECM 43°20.6'N 5°26.2'E 162 2828.0 mm +/- 0.5 @22ºC 82.0 +/- 0.1
Punta Arenas - UMag 53°8'S 70°52'W 40 2823 +/- 0.5 @16.4ºC 81.7 +/- 0.1


Typical quantities
String length (not counting the sphere) min: 0.5 m nominal: 2.8m max: 12m
Sphere mass 2kg +/- 75g
Sphere diameter 81.2mm +/-1.5mm
String Remanium(r) - Stainless steel (Nickel chromium)

- 0,4mm

Modulus of elasticity of string ~200GPa
Oscillation period measurement system Microprocessor with 7,3728MHz - 30ppm crystal

+ laser + PIN photodiode

Wire CTE (25-500ºC) (Coefficient of thermal expansion) ~14 x 10-6 K-1


Penulum length limits*
Minimum ~1.5 m
Maximum virtually no limit (~63.5 m)

*These limits were estimated for the standard World Pendulum Alliance launcher (WPA). A photo of a standard WPA launcher is shown on the figure on the right. Check Pendulum length limits to understand how these limits were obtained.


The experimental apparatus can be easily adapted to human operation, using a good chronometer, for local execution. The stainless steel structures can made in by brass or bronze for easier machining. The cable used can be replaced by a sport fishing steel cable and the mass can be replaced by a Olympic weight throw training weight, weighing 2Kg. A calibrated measuring tape should be used to measure the cable length, a few days after assembling the apparatus to allow for cable expansion.

Local partners

The pendulum[2], although one of the simplest systems commonly studied, is one of the richest in terms of physics.

In order to build a precise pendulum the most important factors are the precise measurement of the length of the cable, its quality, and of that of the pendulum supports. Selecting a mass between 1 to 4 Kg ensures that the pendulum's period error will be small enough for small local gravity changes (smaller than 0.1%) to be detectable, as long as a precise chronometer is used for timekeeping.

A local apparatus can be assembled using readily available materials and the local "g" values determined using such an apparatus can then be compared to the ones obtained through the remote pendulum constellation and the theoretical model.

Collecting this data through a social network will allow a more precise description of how "g" varies around the globe. The "World Pendulum" can be an important collaborative network for the dissemination of physics in schools.

Instructions on how to build such a pendulum are available in Precision Pendulum. The documentation of the development and construction of a pendulum are available in Precision Pendulum while the instructions on how to assemble it are available in Precision Pendulum Assembly.

If you want to be a part of the World Pendulum network, please contact us by sending us an email.

Physics

Determining gravity's acceleration in different parts of the globe raises questions about the importance of models in physics. It's possible to show that gravity's acceleration at sea level changes with latitude, and therefore a correction is needed for each individual location. This process allows us to demystify science and correct the existing "urban myth" around some physical constants that only are truly constant when some approximations are done. In this particular case, we will show how the introduction of successive corrections to gravity's "constant" will lead to values closer to those experimentally obtained.

Geophysical model

The starting point is the commonly used, constant, value of 9.81 ms-2. This is obtained by considering the Earth as being (i) a sphere (ii) that is not rotating. It's trivial to note that this model, due to the symmetry of the spherical form, does not allow for different values in different locations. This changes as soon as Earth's rotation dynamics and ellipsoid shape (flattening of the poles) are taken in account. These factors allow for gravity to change with latitude, and in fact these two factors are the two most important ones in this phenomena, outweighing every other effect, such as (i) altitude, (ii) tidal forces, and (iii) subsoil composition.

To demonstrate these finer aspects, gravity's acceleration must be determined in various latitudes around the globe distant from each other. Using the data collected, students can then ask themselves about how "constant" the value truly is and improve their intuition of gravity.

Experimental studies

Variation with latitude

As seen, the first possible study consists of using the remote pendulums to obtain a measurement of the local gravity acceleration for each location they're based in. Through considering (or not) several factors, it is possible to fit the data to a experimental description of the Earth using spherical harmonics (equation \eqref{harmonica-esferica}). This experimental work can be conducted using e-lab's pendulum constellation and our partner's pendulums.

Local determination

Following the instructions available in this wiki - Precision_Pendulum - or using any other kind of design that results in a rigorous apparatus, a local pendulum is built. It's then possible for measurements of local gravity to be made, as long as a good chronometer is used. Furthermore, it's also possible to contribute to the enrichment of the World Pendulum network's spreadsheet.

Tidal study

Using an almanac appropriate for the location, on can obtain the times of particular Moon/Sun alignments (full moon, new moon, waxing crescent and waxing gibbous). Plotting a graph spanning several months, one can try to verify and quantify the influence of tidal forces and Moon/Sun alignments in the apparent weight. It's possible to try and verify the correlation between Moon phases and changes in measurement of local gravity, by making a month or year-long study. Tidal effects are on the limit of detection by the pendulums of the e-lab constellation. For the experiment to be successful, it's necessary to be very rigorous on the time at which the experimental runs are made and some advanced numerical techniques, like the Fourier transform, need to by employed for the signal to be extracted from the data.

Analysis of wire torsion

Effect of wire torsion and sphere ellipticity in the measurement of pendulum speed.

Those paying more attention will note that the speed of the mass changes due to wire torsion and due to the mass not being a perfect sphere. This is pictured in the image to the right. The pendulum can be studied taking into account the effect of the wire torsion (the use of Euler-Lagrange equations is recommended for this).

Uniformly accelerated circular movement

The speed of the sphere in the lowest point of the trajectory is determined by measuring how much time the laser beam is interrupted. Knowing the sphere diameter, it's trivial to determine the speed at the origin. From this, the maximum kinetic energy can be calculated and the launching height of the pendulum determined. The calculated launching point can then be compared with the real one.

Latitude provideres

The gravitational constant plotted against latitude with points of interest around the globe highlighted. Principe Island is just over zero latitude. Lisbon value was obtained with the current experiment and already over plotted on the graphic.

Language is an important nationality factor ("My fatherland is the Portuguese language.", F. Pessoa) and a simple way to define what is called brother countries ("países irmãos"). Only four languages are disseminated around the world, Portuguese being one of them. The Portuguese speaking community covers latitudes from ~30S to ~40N, almost a 75º span across the equator. Therefore, CPLP countries can help by being "latitude providers" (see Figure).

To conduct this world experiment, at least four spaced points are needed in order to have a proper fit. But due to the strong non-linearity of the equation, more points are needed to provide a suitable adjustment, in particular on the "knee" close to the earth’s equator. Brazil itself can provide almost four crucial points close to the equator (e.g. Recife 8º, Salvador – 12º, Rio de Janeiro – 23º, Porto Alegre – 30º) but lacks points with a latitude where the characteristic varies more strongly, the almost linear region around 30º to 60º, where Portugal can give two good points (e.g. Porto - 37º and Faro - 41º). Mozambique can contribute with 27º (Maputo) and S. Tomé e Principe or Brazil are both good choices to cover the equator. Angola could give complementary points to those acquired in Brazil, as the sensibility of the measurement is more pronounced close to the equator and the poles.

Data fitting

Available references [2] [3] [4] [5] [6] [7] give a very good description of the mathematical model needed to fit the data. If all major factors are taken into account, gravity as a function of latitude is given by:

[math] g_{n}(\varphi) = 9.780 326 772\times[1 + 0.005 302 33 \cdot sin^{2}(\varphi) - 0.000 005 89 \cdot sin^{2}(2\varphi)] [/math]

where \(\varphi\) is the latitude. This expression is one of the best experimental approximations and results from the standardization agreement to adjust the World Geodetic System datum surface (WSG84) to an ellipsoid with radius r1=6378137m at the equator and r2=6356752m polar semi-minor radius. This formula takes into account the fact that the Earth is an ellipsoid and that there is an additional increase in the acceleration of gravity when one moves nearer to the poles, due to a weaker centrifugal force. Nevertheless the students could derive a non-harmonic, first order approximation by taking into account only centrifugal force. Then, as a second step, they could include the two other major errors, the centrifugal force and earth’s ellipsoid format.

The variability of the period with elapsed time (angle amplitude < 7,5º), showing that this error is less than 0,05% regardless initial amplitude.

The pictures shows the expected deviation from the “earth’s constant acceleration”, the real acceleration for each latitude. We have plotted the point already obtained with this apparatus in Lisbon and the marks over the expected latitudes for future partners. Of course these approximations do not include one important source of deviation from real data to the mathematical model, the experimental error, as we do not include the experimental source of error. However, those systematic errors could be under the expected precision needed (0,1%) for the former approximation if a careful design of the apparatus is considered. Nevertheless those errors must be discussed in advanced courses and their weight must be proved when considering the real pendulum.

Historical notes

The pendulum importance as the basis of clocks and chronographs was only overthrown when the Royal Society convinced the English parliament to create an award, ranging from 10k£ to 20k£ (equivalent nowadays to more than 3.5M€), for the invention of a chronograph that didn't depend on it. The time precision of pendulum based systems is only bettered by modern electronic systems.

In the discovery age longitude was determined with a high error, since clocks and chronographs were reliant on pendulums and these were very sensitive to ships rocking, suffering changes in frequency or even stopping. Local longitude was calculated by comparing the solar hour (or stellar hour) with the ship's clock time.

Frequently asked questions

How long would it take to replace the cable that holds the sphere?

The remanium cable is a very strong material and only need to be replace in case of misuse.

Over time the cable can have some extension?

Cable length should undergone a rectified measure after ~3 months installation and thereafter once a year.

The temperature where the pendulum is installed interferes with the results of the experiments?

The variation induced by temperature is relevant when it comes to very precise measurements such as tide detection (local fluctuations), the reason why it is reported. NOTE that in some situations, due to incorrect housing of the temperature sensor, temperature increases during the reading as the sensor can be close to the power electronics devices. So, we advise a wise use of it.

Are the various pendulums subjected to regular calibrations?

Regular calibrations shall take place at least once a year. Nevertheless we can not guarantee that every partner assure a regular calibration of the pendulums. Please monitor the course of each experience and use critical thinking in the face of any results obtained.

What is the difference in measuring the cable with a laser or a tape measure, there is a noticeable difference between the two methods?

Usually both methods give the same incertitude because the limiting factor is the positioning of both devices by human usage which is barely the same, surpassing the intrinsic error of each device.

The use of a frontal camera to analyze the motion of the pendulum, make a recording and then use some image analysis software to study the movements and oscillations will give good results?

This results can be obtained with the camera on top as well, a movie can be shot and analyses on Tracker software for instance.

References

  1. World pendulum—a distributed remotely controlled laboratory (RCL) to measure the Earth's gravitational acceleration depending on geographical latitude, Grober S, Vetter M, Eckert B and Jodl H J, European Journal of Physics - EUR J PHYS , vol. 28, no. 3, pp. 603-613, 2007
  2. 2.0 2.1 Physics for scientists and engineers, 5th edition, Hardcourt College Publishers, R.Serway and R. Beichner, 2000
  3. http://rcl-munich.informatik.unibw-muenchen.de/
  4. Nelson, Robert; M. G. Olsson (February 1987). "The pendulum - Rich physics from a simple system". American Journal of Physics 54 (2): doi:10.1119/1.14703
  5. Pendulums in the Physics Education Literature: A Bibliography, Gauld, Colin 2004 Science & Education, issue 7, volume 13, 811-832 (http://dx.doi.org/10.1007/s11191-004-9508-7)
  6. The exact equation of motion of a simple pendulum of arbitrary amplitude: a hypergeometric approach, M I Qureshi et al 2010 Eur. J. Phys. 31 1485(http://dx.doi.org/10.1088/0143-0807/31/6/014)
  7. A comprehensive analytical solution of the nonlinear pendulum, Karlheinz Ochs 2011 Eur. J. Phys. 32 479 (http://dx.doi.org/10.1088/0143-0807/32/2/019)

Links