Difference between revisions of "Péndulo mundial"

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= Descripción =
+
=Description=
 
[[File:Soyuz VS03 liftoff.jpg||thumb|Soyuz lift-off from French Guiana @ 5º north of the Equator .|right|border|236px]]
 
[[File:Soyuz VS03 liftoff.jpg||thumb|Soyuz lift-off from French Guiana @ 5º north of the Equator .|right|border|236px]]
Los cohetes se lanzan al espacio desde las latitudes ecuatoriales. Esto se debe al hecho de que el peso aparente de los objetos se reduce gradualmente desde los polos hasta el ecuador. ¡Nos sentiremos más ligeros en el ecuador que en los polos!
+
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!
  
Esta pequeña diferencia en el peso aparente permite que el mismo cohete lance cargas más pesadas en órbita si se lanza más cerca del ecuador. Por ejemplo, un cohete Soyuz que se lanza en órbita geoestacionaria desde la Guayana Francesa (5ºN) puede transportar 3 toneladas, mientras que solo será capaz de lanzar 1,7 toneladas de carga cuando se lance desde Baikonur, Kazajstán (46ºN).
+
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).
  
El objetivo de este experimento es encontrar el valor de la gravedad "constante" a través de una constelación de péndulos colocados en varias latitudes y operados remotamente, a través de Internet, por cualquier persona.
+
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.  
  
Se espera que los países de la CPLP puedan contribuir a este esfuerzo, acercando a estudiantes, maestros y ciudadanos interesados.
+
It is expected that CPLP countries can contribute to this effort, bringing students, teachers and interested citizens closer together.  
  
Hay dos actividades diferentes que ocurren simultáneamente: (i) acceso, a través del laboratorio elecrónico, de los péndulos ubicados en diferentes latitudes y (ii) la construcción y operación local en las escuelas o en tu casa.
+
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 y Río de Janeiro fueron las primeras ciudades en contribuir a la red en enero de 2013, haciendo posible que ocurran los primeros ajustes de datos experimentales a la ecuación teórica dentro de nuestro proyecto que describe cómo la gravedad cambia con la latitud.
+
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.
  
Si desea formar parte de la nueva red World Pendulum, contáctenos enviándonos un [mailto: wwwelab@ist.utl.pt email].
+
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">
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<div class="toccolours mw-collapsible mw-collapsed" style="width:420px">
'''Enlaces'''
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'''Links'''
<div class = "mw-collapsible-content">
+
<div class="mw-collapsible-content">
  
* Video Faro: rtsp: //elabmc.ist.utl.pt/worldpendulum_ccvalg.sdp
+
*Video Faro: rtsp://elabmc.ist.utl.pt/worldpendulum_ccvalg.sdp
* Video Lisboa: rtsp: //elabmc.ist.utl.pt/worldpendulum_planetarium.sdp
+
*Video Lisboa: rtsp://elabmc.ist.utl.pt/worldpendulum_planetarium.sdp
* Video Ilhéus: rtsp: //elabmc.ist.utl.pt/worldpendulum_ilheus.sdp
+
*Video Ilhéus: rtsp://elabmc.ist.utl.pt/worldpendulum_ilheus.sdp
* Video Río Janeiro: rtsp: //elabmc.ist.utl.pt/worldpendulum_puc.sdp
+
*Video Rio Janeiro: rtsp://elabmc.ist.utl.pt/worldpendulum_puc.sdp
* Video Maputo: rtsp: //elabmc.ist.utl.pt/worldpendulum_maputo.sdp
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*Video Maputo: rtsp://elabmc.ist.utl.pt/worldpendulum_maputo.sdp  
* Video Santo Tomé: rtsp: //elabmc.ist.utl.pt/wp_saotome.sdp
+
*Video São Tomé: rtsp://elabmc.ist.utl.pt/wp_saotome.sdp
* Laboratorio: Básico en [http://e-lab.ist.eu e-lab.ist.eu]
+
*Laboratory: Basic in [http://e-lab.ist.eu e-lab.ist.eu]
* Sala de control: Péndulo mundial
+
*Control room: World Pendulum
* Grado: *
+
*Grade: *
  
 
</div>
 
</div>
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== A quién le gusta esta idea ==
+
==Who likes this idea==
  
 
[[File:PBA B1 1.png|border|180px|border|180px]]
 
[[File:PBA B1 1.png|border|180px|border|180px]]
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[[File:LogosBeneficairesErasmus+RIGHT EN.jpg|border|280px]]
 
[[File:LogosBeneficairesErasmus+RIGHT EN.jpg|border|280px]]
  
= Aparato experimental =
+
=Experimental apparatus=
El diseño del péndulo utilizado se basó en el diseño del Dr. Jodl <ref name = "jodl"> Péndulo mundial: un laboratorio distribuido a control remoto (RCL) para medir la aceleración gravitacional de la Tierra según la latitud geográfica, Grober S, Vetter M, Eckert B y Jodl HJ, Revista Europea de Física - EUR J PHYS, vol. 28, no. 3, págs. 603-613, 2007 </ref>. Se hicieron algunos cambios menores para permitir que el mismo diseño se repita fácilmente en las escuelas secundarias. Los datos relativos a cada péndulo son los siguientes:
+
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 is as follows:
  
[[File: WordlPendulum.JPG | thumb | Pendulum utilizado para el experimento de gravedad estándar de péndulo mundial.]]
+
[[File:WordlPendulum.JPG|thumb|Pendulum used for the world pendulum standard gravity experiment.]]
[[File: Stringsuport.png | thumb | Soporte de cadena de péndulo para evitar errores de alargamiento. El cable se fija soldando a un tornillo M4 de latón de 40 mm de largo.]]
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[[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.]]
  
{| class = "wikitable"
+
{| class="wikitable"
! colspan = "6" | Tamaños físicos por lugar
+
!colspan="6"|Physical sizes by place
 
|-
 
|-
   | Sitio
+
   | Place
   | Latitud
+
   | Latitude
   | Longitud
+
   | Longitude
   | Altitud (m)
+
   | Altitude (m)
   | Longitud de la cuerda (mm)
+
   | String length (mm)
   | Diámetro de la esfera (mm)
+
   | Sphere diameter (mm)
 
|-
 
|-
   | CCV_Algarve / Faro
+
   | CCV_Algarve/Faro
 
   | 37º00'N
 
   | 37º00'N
 
   | 7º56'W
 
   | 7º56'W
 
   | 10 m
 
   | 10 m
   | 2677mm +/- 0.5mm @ 23ºC
+
   | 2677mm +/- 0.5mm @23ºC
 
   | 80.5 +/- 1.0 mm
 
   | 80.5 +/- 1.0 mm
 
|-
 
|-
   | UESC / Ilhéus
+
   | UESC/Ilhéus
 
   | 14º47'S
 
   | 14º47'S
 
   | 39º10'W
 
   | 39º10'W
 
   | 220m
 
   | 220m
   | 2705mm +/- 0.5mm @ 23ºC
+
   | 2705mm +/- 0.5mm @23ºC
 
   | 80.5 +/- 1.0 mm
 
   | 80.5 +/- 1.0 mm
 
|-
 
|-
   | Lisboa
+
   | Lisbon
 
   | 38º41'N
 
   | 38º41'N
 
   | 9º12'W
 
   | 9º12'W
 
   | 20m
 
   | 20m
   | 2677mm +/- 0.5mm @ 19ºC
+
   | 2677mm +/- 0.5mm @19ºC
 
   | 80.5 +/- 1.0 mm
 
   | 80.5 +/- 1.0 mm
 
|-
 
|-
Line 92: Line 92:
 
   | 32º36'E
 
   | 32º36'E
 
   | 80m
 
   | 80m
   | 2609.8mm +/- 0.5mm @ 27ºC
+
   | 2609.8mm +/- 0.5mm @27ºC
 
   | 80.5 +/- 1.0 mm
 
   | 80.5 +/- 1.0 mm
 
|-
 
|-
   | Santo Tomé
+
   | São Tomé
 
   | 0º21'N
 
   | 0º21'N
 
   | 6º43'E
 
   | 6º43'E
 
   | 50m
 
   | 50m
   | 2756.5mm +/- 0.5mm @ 29ºC
+
   | 2756.5mm +/- 0.5mm @29ºC
 
   | 81.8 +/- 0.5 mm
 
   | 81.8 +/- 0.5 mm
 
|-
 
|-
   | Praga - CTU
+
   | Prague - CTU
 
   | 50º5.47N
 
   | 50º5.47N
 
   | 14º24.97E
 
   | 14º24.97E
 
   | 150m
 
   | 150m
   | 2850mm +/- 0.5mm @ 25ºC
+
   | 2850mm +/- 0.5mm   @25ºC
 
   | 80.15 +/- 0.5 mm
 
   | 80.15 +/- 0.5 mm
 
|-
 
|-
Line 116: Line 116:
 
   | 81.8mm
 
   | 81.8mm
 
|-
 
|-
   | Río de Janeiro - PUC
+
   | Rio de Janeiro - PUC
 
   | 22º54.13S
 
   | 22º54.13S
 
   | 43º12W
 
   | 43º12W
Line 124: Line 124:
 
|-
 
|-
 
   | Praia - UniCV
 
   | Praia - UniCV
   | 14 ° 56 'N
+
   | 14° 56 'N
   | 23 ° 31'W
+
   | 23° 31'W
 
   | 40 m
 
   | 40 m
 
   | 2826,0mm +/- 0.5mm
 
   | 2826,0mm +/- 0.5mm
Line 131: Line 131:
 
|-
 
|-
 
   | Bogotá - UniAndes
 
   | Bogotá - UniAndes
   | 4 ° 36 'N
+
   | 36 'N
   | 74 ° 3'W
+
   | 74° 3'W
 
   | 2650 m
 
   | 2650 m
   | 2815,3mm+/- 0.5mm
+
   | 2815,3mm +/- 0.5mm
 
   | 82.0mm
 
   | 82.0mm
 
|-
 
|-
   | Ciudad de Panamá - UTP
+
   | Panama city - UTP
   | 9 ° 1.34 'N
+
   | 1.34 'N
   | 79 ° 31.92'W
+
   | 79° 31.92'W
 
   | 82 m
 
   | 82 m
   | 2825mm +/- 0.5mm @ 28ºC
+
   | 2825mm +/- 0.5mm @28ºC
 
   | 81.9mm
 
   | 81.9mm
 
|-
 
|-
 
   | Santiago - UChile
 
   | Santiago - UChile
   | 33 ° 27.46 'S
+
   | 33° 27.46 'S
   | 70 ° 39.79'W
+
   | 70° 39.79'W
 
   | 552 m
 
   | 552 m
   | 2825mm +/- 0.5mm @ 27ºC
+
   | 2825mm +/- 0.5mm @27ºC
 
   | 81.9mm
 
   | 81.9mm
 
|-
 
|-
   | Valparaíso - UTFSM
+
   | Valparaiso - UTFSM
   | 9 ° 1.34 'N
+
   | 1.34 'N
   | 79 ° 31.92'W
+
   | 79° 31.92'W
 
   | 82 m
 
   | 82 m
   | 2825mm +/- 0.5mm @ 28ºC
+
   | 2825mm +/- 0.5mm @28ºC
 
   | 81.9mm
 
   | 81.9mm
 
|}
 
|}
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{| class = "wikitable"
+
{| class="wikitable"
! colspan = "2" | Cantidades típicas
+
!colspan="2"|Typical quantities
 
|-
 
|-
| Longitud de cadena (sin contar la esfera) || 2705mm +/- 0.5mm
+
| String length (not counting the sphere) || 2705mm +/- 0.5mm
 
|-
 
|-
| Masa de esfera || 2kg +/- 75g
+
| Sphere mass || 2kg +/- 75g
 
|-
 
|-
| Diámetro de esfera || 81.2mm +/- 1.5mm
+
| Sphere diameter || 81.2mm +/-1.5mm
 
|-
 
|-
| Cadena || Remanio (r) - Acero inoxidable (níquel cromo)- 0,4mm
+
| String || Remanium(r) - Stainless steel (Nickel chromium)
 +
- 0,4mm
 
|-
 
|-
| Módulo de elasticidad de la cuerda || ~ 200GPa
+
| Modulus of elasticity of string || ~200GPa
 
|-
 
|-
| Sistema de medición del período de oscilación || Microprocesador con 7,3728MHz - 30ppm de cristal+ láser + fotodiodo PIN
+
| Oscillation period measurement system || Microprocessor with 7,3728MHz - 30ppm crystal
 +
+ laser + PIN photodiode
 
|-
 
|-
| Alambre CTE (25-500ºC) (Coeficiente de expansión térmica) || ~ 14 x 10 <sup> -6 </sup> K <sup> -1 </sup>
+
| Wire CTE (25-500ºC) (Coefficient of thermal expansion) || ~14 x 10<sup>-6</sup> K<sup>-1</sup>
 
|}
 
|}
  
  
El aparato experimental se puede adaptar fácilmente a la operación humana, utilizando un buen cronómetro, para la ejecución local. Las estructuras de acero inoxidable pueden fabricarse en latón o bronce para facilitar el mecanizado. El cable utilizado puede ser reemplazado por un cable de acero para pesca deportiva y la masa puede ser reemplazada por un peso de entrenamiento de lanzamiento de pesas olímpico, que pesa 2 kg. Se debe usar una cinta de medición calibrada para medir la longitud del cable "," unos días después de ensamblar el aparato para permitir la expansión del cable ".
+
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'''.
  
= Socios locales =
+
=Local partners=
El péndulo <ref name = "serway"> Física para científicos e ingenieros, 5ª edición, Hardcourt College Publishers, R.Serway y R. Beichner, 2000 </ref>, aunque uno de los sistemas más simples comúnmente estudiados, es uno de los más rico en términos de física.
+
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.
  
Para construir un péndulo preciso, los factores más importantes son la medición precisa de la longitud del cable, su calidad y la de los soportes del péndulo. La selección de una masa entre 1 y 4 kg asegura que el error del período del péndulo sea lo suficientemente pequeño como para que se puedan detectar pequeños cambios de gravedad locales (menores del 0.1%), siempre que se use un cronómetro preciso para el cronometraje.
+
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.  
  
Se puede ensamblar un aparato local utilizando materiales fácilmente disponibles y los valores locales de <em> "g" </em> determinados usando dicho aparato se pueden comparar con los obtenidos a través de la constelación de péndulo remoto y el modelo teórico.
+
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.
  
La recopilación de estos datos a través de una red social permitirá una descripción más precisa de cómo <em> "g" </em> varía en todo el mundo. El "Péndulo mundial" puede ser una red de colaboración importante para la difusión de la física en las escuelas.
+
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.
  
Las instrucciones sobre cómo construir tal péndulo están disponibles en [[Precision Pendulum]].
+
Instructions on how to build such a pendulum are available in [[Precision Pendulum]].
La documentación del desarrollo y construcción de un péndulo está disponible en [[Precision Pendulum]] mientras que las instrucciones sobre cómo ensamblarlo están disponibles en [[Precision Pendulum Assembly]].
+
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]].
  
Si desea formar parte de la nueva red World Pendulum, contáctenos enviándonos un [mailto: wwwelab@ist.utl.pt email].
+
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].
  
= Física =
+
=Physics=
Determinar la aceleración de la gravedad en diferentes partes del mundo plantea preguntas sobre la importancia de los modelos en física. Es posible mostrar que la aceleración de la gravedad al nivel del mar cambia con la latitud y, por lo tanto, se necesita una corrección para cada ubicación individual. Este proceso nos permite desmitificar la ciencia y corregir el "mito urbano" existente en torno a algunas constantes físicas que solo son verdaderamente constantes cuando se realizan algunas aproximaciones. En este caso particular, mostraremos cómo la introducción de correcciones sucesivas a la "constante" de la gravedad conducirá a valores más cercanos a los obtenidos experimentalmente.
+
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.  
  
== Modelo geofísico ==
+
==Geophysical model==
El punto de partida es el valor constante de uso común de 9.81 ms <sup> -2 </sup>. Esto se obtiene al considerar que la Tierra es (i) una esfera (ii) que no está girando. Es trivial notar que este modelo, debido a la simetría de la forma esférica, no permite valores diferentes en diferentes ubicaciones. Esto cambia tan pronto como se tienen en cuenta la dinámica de rotación de la Tierra y la forma elipsoide (aplanamiento de los polos). Estos factores permiten que la gravedad cambie con la latitud, y de hecho, estos dos factores son los dos más importantes en este fenómeno, superando cualquier otro efecto, como (i) altitud, (ii) fuerzas de marea y (iii) composición del subsuelo .
+
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.
  
Para demostrar estos aspectos más finos, la aceleración de la gravedad debe determinarse en varias latitudes alrededor del globo distantes entre sí. Usando los datos recopilados, los estudiantes pueden preguntarse qué tan "constante" es realmente el valor y mejorar su intuición de la gravedad.
+
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.
  
=== Estudios experimentales ===
+
===Experimental studies===
==== Variación con latitud ====
+
====Variation with latitude====
Como se ve, el primer estudio posible consiste en usar los péndulos remotos para obtener una medición de la aceleración de la gravedad local para cada ubicación en la que se basan. Al considerar (o no) varios factores, es posible ajustar los datos a un experimento Descripción de la Tierra utilizando armónicos esféricos (ecuación \ eqref {harmonica-esferica}). Este trabajo experimental puedese llevará a cabo utilizando la constelación de péndulo de e-lab y [http://rcl-munich.informatik.unibw-muenchen.de/ los péndulos de nuestro socio].
+
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].
  
==== Determinación local ====
+
====Local determination====
Siguiendo las instrucciones disponibles en este wiki - [[Precision_Pendulum]] - o usando cualquier otro tipo de diseño que resulte en un aparato riguroso, se construye un péndulo local. Entonces es posible realizar mediciones de la gravedad local, siempre que se use un buen cronómetro. Además, también es posible contribuir al enriquecimiento de la red mundial.
+
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 | Estudio de mareas]] ====
+
====[[Tidal study]]====
Usando un almanaque apropiado para la ubicación, puede obtener los tiempos de alineaciones particulares de Luna / Sol (luna llena, luna nueva, creciente creciente y gibosa creciente). Al trazar un gráfico que abarque varios meses, se puede intentar verificar y cuantificar la influencia de las fuerzas de marea y las alineaciones Luna / Sol en el peso aparente. Es posible intentar y verificar la correlación entre las fases de la Luna y los cambios en la medición de la gravedad local, haciendo un estudio de un mes o un año.
+
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.
Los efectos de las mareas están en el límite de detección por los péndulos de la constelación de e-lab. Para que el experimento tenga éxito, es necesario ser muy riguroso en el momento en que se realizan las corridas experimentales y algunas técnicas numéricas avanzadas, como la transformada de Fourier, deben emplearse para que la señal se extraiga de los datos.
+
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.
  
==== Análisis de torsión de alambre ====
+
====Analysis of wire torsion ====
[[File: Torcao.jpg || thumb | Efecto de la torsión del alambre y la elipticidad de la esfera en la medición de la velocidad del péndulo. | Right | border | 240px]]
+
[[File:Torcao.jpg||thumb|Effect of wire torsion and sphere ellipticity in the measurement of pendulum speed.|right|border|240px]]
Quienes presten más atención notarán que la velocidad de la masa cambia debido a la torsión del alambre y debido a que la masa no es una esfera perfecta. Esto se ilustra en la imagen a la derecha. El péndulo puede estudiarse teniendo en cuenta el efecto de la torsión del alambre (para ello se recomienda el uso de las ecuaciones de Euler-Lagrange).
+
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).
  
== Movimiento circular uniformemente acelerado ==
+
==Uniformly accelerated circular movement==
La velocidad de la esfera en el punto más bajo de la trayectoria se determina midiendo cuánto tiempo se interrumpe el rayo láser. Conociendo el diámetro de la esfera, es trivial determinar la velocidad en el origen. A partir de esto, se puede calcular la energía cinética máxima y determinar la altura de lanzamiento del péndulo. El punto de lanzamiento calculado se puede comparar con el real.
+
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.
  
= CPLP como "proveedor de latitud" =
+
=CPLP as “latitude provider”=
  
[[file:G_latitude.png|link=https://docs.google.com/a/kic-innoenergy.com/spreadsheet/oimg?key=0AkxMmuJA92wgdHZnWHk5WHhaQldINGFqSTl6OGdpSlE&oid=1&zx=hfmrs4egtbuf|thumb|La constante gravitacional trazada contra la latitud con puntos de interés en todo el mundo resaltados. La isla Príncipe está por encima de la latitud cero. El valor de Lisboa se obtuvo con el experimento actual y ya se trazó en exceso en el gráfico.]]
+
[[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.]]
  
El idioma es un factor de nacionalidad importante ("Mi patria es el idioma portugués", F. Pessoa) y una forma sencilla de definir lo que se llama países hermanos ("países irmãos"). Solo cuatro idiomas se difunden en todo el mundo, siendo el portugués uno de ellos. La comunidad de habla portuguesa cubre latitudes de ~ 30S a ~ 40N, casi un tramo de 75º en el ecuador. Por lo tanto, los países CPLP pueden ayudar siendo "proveedores de latitud" (ver Figura).
+
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).
  
Para llevar a cabo este experimento mundial, se necesitan al menos cuatro puntos espaciados para tener un ajuste adecuado. Pero debido a la fuerte no linealidad de la ecuación, se necesitan más puntos para proporcionar un ajuste adecuado, en particular en la "rodilla" cerca del ecuador de la Tierra. El propio Brasil puede proporcionar casi cuatro puntos cruciales cerca del ecuador (por ejemplo, Recife 8º, Salvador - 12º, Río de Janeiro - 23º, Porto Alegre - 30º) pero carece de puntos con una latitud donde la característica varía más fuertemente, la región casi lineal alrededor 30º a 60º, donde Portugal puede dar dos buenos puntos (por ejemplo, Porto - 37º y Faro - 41º). Mozambique puede contribuir con 27º (Maputo) y S. Tomé e Principe o Brasil son buenas opciones para cubrir el ecuador. Angola podría dar puntos complementarios a los adquiridos en Brasil, ya que la sensibilidad de la medición es más pronunciada cerca del ecuador y los polos.
+
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.
  
= Ajuste de datos =
+
=Data fitting=
Referencias disponibles <ref name = "serway"> </ref> <ref name = "rcl"> http://rcl-munich.informatik.unibw-muenchen.de/ </ref> <ref name = "olsom"> Nelson, Robert; M. G. Olsson (febrero de 1987). "El péndulo - Física rica de un sistema simple". American Journal of Physics 54 (2):
+
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"> Péndulos en la literatura de educación física: una bibliografía, Gauld, Colin 2004 Science & Education, número 7, volumen 13, 811-832
+
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"> La ecuación exacta de movimiento de un péndulo simple de amplitud arbitraria: un enfoque hipergeométrico, MI 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 comprensi5 solución analítica del péndulo no lineal, Karlheinz Ochs 2011 Eur. J. Phys. 32 479 (http://dx.doi.org/10.1088/0143-0807/32/2/019) </ref> dan una muy buena descripción del modelo matemático necesario para ajustar los datos. Si se tienen en cuenta todos los factores principales, la gravedad en función de la latitud viene dada por:
+
(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:
  
<matemáticas>
+
<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>
 
</math>
  
donde \ (\ varphi \) es la latitud. Esta expresión es una de las mejores aproximaciones experimentales y los resultados del acuerdo de estandarización para ajustar la superficie de referencia del Sistema Geodésico Mundial (WSG84) a un elipsoide con radio r <sub> 1 </sub> = 6378137m en el ecuador y r <sub> 2 </sub> = 6356752m de radio semi-menor polar.
+
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.
Esta fórmula tiene en cuenta el hecho de que la Tierra es un elipsoide y que hay un aumento adicional en la aceleración de la gravedad cuando uno se acerca a los polos, debido a una fuerza centrífuga más débil. Sin embargo, los estudiantes podrían obtener una aproximación no armónica de primer orden teniendo en cuenta solo la fuerza centrífuga. Luego, como un segundo paso, podrían incluir los otros dos errores principales, la fuerza centrífuga y el formato de elipsoide de la Tierra.
+
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.
  
[[File: Period_over_time.png | thumb | La variabilidad del período con el tiempo transcurrido (ángulo de amplitud <7,5º), que muestra que este error es inferior al 0,05% independientemente de la amplitud inicial.]]
+
[[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.]]
  
Las imágenes muestran la desviación esperada de la "aceleración constante de la Tierra", la aceleración real para cada latitud. Hemos trazado el punto ya obtenido con este aparato en Lisboa y las marcas sobre las latitudes esperadas para futuros socios.
+
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.
Por supuesto, estas aproximaciones no incluyen una fuente importante de desviación de los datos reales al modelo matemático, el error experimental, ya que no incluimos la fuente experimental de error. Sin embargo, esos errores sistemáticos podrían estar bajo la precisión esperada necesaria (0,1%) para la aproximación anterior si se considera un diseño cuidadoso del aparato. Sin embargo, esos errores deben discutirse en cursos avanzados y su peso debe probarse al considerar el péndulo real.
+
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.
  
= Notas históricas =
+
=Historical notes=
La importancia del péndulo como base de los relojes y cronógrafos solo fue derrocada cuando la Royal Society convenció al parlamento inglés de crear un premio, que oscilaba entre 10k £ y 20k £ (equivalente hoy en día a más de 3.5M €), por la invención de un cronógrafo. eso no dependía de eso. La precisión temporal de los sistemas basados ​​en péndulo solo se ve mejorada por los sistemas electrónicos modernos.
+
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.
  
En la edad del descubrimiento, la longitud se determinó con un alto error, ya que los relojes y cronógrafos dependían de péndulos y estos eran muy sensibles a los barcos que se balanceaban, sufrían cambios en la frecuencia o incluso se detenían. La longitud local se calculó comparando la hora solar (u hora estelar) con la hora del reloj del barco.
+
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.
  
= Referencias =
+
=References=
  
<referencias />
+
<references />
  
= Enlaces =
+
=Links=
* [[Pêndulo Mundial | Versión portuguesa (Versão em Português)]]
+
*[[Pêndulo Mundial | Portuguese version (Versão em Português)]]
* [[Péndulo mundial | Versión en español (Versión en español)]]
+
*[[Péndulo mundial| Spanish version (Versión en español)]]
* [https://www.youtube.com/watch?v=ZOOFw_Nlee8&feature=youtu.be Construyendo tu propio péndulo]
+
*[https://www.youtube.com/watch?v=ZOOFw_Nlee8&feature=youtu.be Building your own pendulum]

Revision as of 10:11, 8 April 2020

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: Basic in e-lab.ist.eu
  • Control room: World Pendulum
  • 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 is as 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.
Physical sizes by place
Place Latitude Longitude Altitude (m) String length (mm) Sphere diameter (mm)
CCV_Algarve/Faro 37º00'N 7º56'W 10 m 2677mm +/- 0.5mm @23ºC 80.5 +/- 1.0 mm
UESC/Ilhéus 14º47'S 39º10'W 220m 2705mm +/- 0.5mm @23ºC 80.5 +/- 1.0 mm
Lisbon 38º41'N 9º12'W 20m 2677mm +/- 0.5mm @19ºC 80.5 +/- 1.0 mm
Maputo 25º56'S 32º36'E 80m 2609.8mm +/- 0.5mm @27ºC 80.5 +/- 1.0 mm
São Tomé 0º21'N 6º43'E 50m 2756.5mm +/- 0.5mm @29ºC 81.8 +/- 0.5 mm
Prague - CTU 50º5.47N 14º24.97E 150m 2850mm +/- 0.5mm @25ºC 80.15 +/- 0.5 mm
Barcelona - UPC 41º24.6N 2º13.12E 55 2756.5mm +/- 0.5mm 81.8mm
Rio de Janeiro - PUC 22º54.13S 43º12W 50 2826,0mm +/- 0.5mm 81.6mm
Praia - UniCV 14° 56 'N 23° 31'W 40 m 2826,0mm +/- 0.5mm 81.6mm
Bogotá - UniAndes 4° 36 'N 74° 3'W 2650 m 2815,3mm +/- 0.5mm 82.0mm
Panama city - UTP 9° 1.34 'N 79° 31.92'W 82 m 2825mm +/- 0.5mm @28ºC 81.9mm
Santiago - UChile 33° 27.46 'S 70° 39.79'W 552 m 2825mm +/- 0.5mm @27ºC 81.9mm
Valparaiso - UTFSM 9° 1.34 'N 79° 31.92'W 82 m 2825mm +/- 0.5mm @28ºC 81.9mm


Typical quantities
String length (not counting the sphere) 2705mm +/- 0.5mm
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


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.

CPLP as “latitude provider”

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.

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