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<!--
* @author Benjamin Thomas Schwertfeger (November 2022)
* @email development@b-schwertfeger.de
* @link https://github.com/btschwertfeger-AWI-Workspace/FoucaultsPendulumWebsite
-->
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<title>Foucaults Pendulum</title>
<meta name="description" content="">
<meta name="viewport" content="width=device-width, initial-scale=1">
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</head>
<body>
<!-- <img src="https://upload.wikimedia.org/wikipedia/commons/f/f7/AWI_Logo_2017.svg"
style="position:absolute; right: 10px; top: 10px; width: 15%;" \> -->
<div class="main">
<div class="placeholder" style="width:100%; height:42px"></div>
<div id="headline_content" style="text-align:center;">
<h1>Foucaults Pendulum</h1><br>
<p style="padding: 0px 15px 0px 15px;">
The Foucault pendulum, named after its inventor, the French
physicist <a href="https://en.wikipedia.org/wiki/L%C3%A9on_Foucault" target="_blank">Léon
Foucault</a>, is a very long spherical pendulum
with a large mass at the bottom. In an experiment with such
a pendulum in 1851, Foucault was able to prove for the first
time in a simple way that the earth rotates. Before that,
one always had to rely on long observations of the night sky
and calculations. With this experiment, it has been possible
to make the earth's rotation accessible in a relatively
simple way, even to people with little or no knowledge of
physics.
</p>
<p>The following equations are provided by Parmeet Singh Chani who
made his slides available <a
href="https://docs.google.com/presentation/d/1zGPZZ6i7xeJ5kZhfs0uXpLLQkGKN2nGrfFmFf72jizM/edit#slide=id.g126641e67d9_0_28"
target="_blank">here</a></a>. <br>
The basis of the foucaults pendulum is the equation of motion:
</p>
<p style="text-align: center;">
\(\ddot{x}-2\dot{y}sin(\lambda+\frac{gx}{l})=0\)<br>
\(\ddot{y}-2\dot{x}sin(\lambda+\frac{gy}{l})=0\) <br>
</p>
<p>which leads to the analytic solution:</p>
<p style="text-align: center">
\(p = x + iy = \biggl(C_{1}e^{i\sqrt{\frac{g}{l}}t}+C_{2}e^{-i\sqrt{\frac{g}{l}}t}\biggr)e^{-i\Omega
sin(\lambda)}\)
<br>
<br>
</p>
<p>
where:
<br><br>
<table style="
margin-left: auto;
margin-right: auto;
">
<tr>
<td>\(g=\)</td>
<td style="text-align: left;">gravity</td>
</tr>
<tr>
<td>\(l=\)</td>
<td style="text-align: left;">length of pendulum string</td>
</tr>
<tr>
<td>\(\Omega=\)</td>
<td style="text-align: left;">angular speed of earth</td>
</tr>
<tr>
<td>\(\lambda=\)</td>
<td style="text-align: left;">latitude</td>
</tr>
</table>
</p>
</div>
<!-- MAIN v1 ----- ---- ----- ---- ----- ---- ----- ---- ----- ---- ----- ---- -->
<div id="main_v1-wrapper">
<div class="wrapper">
<div class="content-left">
<div class="fp-plot-content-wrapper">
<div id="fp_line_plot_container" class="fp-line-plot-container">
<canvas id="fp_line_plot" width=500 height=500 class="canvas"></canvas>
</div>
<div id="fp_input_section" class="fp-input-section">
<div class="fp-input-wrapper input-wrapper">
<div>Gravity</div>
<div id="fp_g_value" name="fp_slide_value" class="fp-slide-value slide-value">9.81</div>
<input id="fp_g_slide" type="range" class="fp-slide slide" name="fp_slide" min="0"
max="20" step="0.1" value="9.8">
</div>
<div class="fp-input-wrapper input-wrapper">
<div>Angular speed of earth</div>
<div id="fp_R_value" name="fp_slide_value" class="fp-slide-value slide-value">0.1</div>
<input id="fp_R_slide" type="range" class="fp-slide slide" name="fp_slide" min="0"
max="10" step="0.1" value="0.1">
</div>
<div class="X-input-wrapper">
\(\lambda\):
<input id="fp_X_input_lambda" type="text" name="fp_input_field" class="fp_X_input_field"
value=3 />
</div>
<div class="X-input-wrapper">
\(l\):
<input id="fp_X_input_L" type="text" name="fp_input_field" class="fp_X_input_field"
value=6.7 />
</div>
<!-- <div class="X-input-wrapper">
\(k_2\):
<input id="fp_X_input_k_2" type="text"
name="fp_input_field"
class="fp_X_input_field" value=1 />
</div> -->
<button class="someBtn" id="fp_animate">Animate</button>
<button class="someBtn" id="fp_resetBtn">Reset</button>
</div>
</div>
</div>
<!-- INFORMATION -->
<div class="content-right">
<div class="description">
<h2>Proof that the earth is rotating</h2>
<h4>How is this experiment performed?</h4>
<p>
The pendulum is set up at a fixed point and
brought into an initial position, where the
release of a band with the help of fire sets the
pendulum in swinging motion. This is to prevent
unwanted movement and additional swing and spin.
</p>
<h4>What can be observed?</h4>
<p>
At the poles of our earth, after about 23.93
hours, it would look as if the pendulum had
turned 360°, but in fact it is the earth
that turns under the pendulum and this turn is
not transmitted to the swinging object. At the
equator this is different,
because there the path of the pendulum does not
seem to
change. This is due to the fact that a day on
earth, with its almost 24 hours, divided by the
sine of the latitude at the equator is 0
\(\bigl(0=\frac{23,93}{sin(0)}\bigr)\).
</p>
<p>
  ☞ <a target="_blank" href="FoucaultsPendulum.txt">View</a> R
implementation
</p>
<p>
  ☞ <a href="FoucaultsPendulum.html" target="_blank">View</a> or <a
href="FoucaultsPendulum.ipynb" target="_blank" download>download</a>
Jupyter
Notebook
</p>
<p>
  ☞ Other visualization: <a href="https://javalab.org/en/foucault_pendulum_en/"
target="_blank">javalab.org</a>
</p>
</div>
</div>
</div>
</div>
<div class="placeholder" style="width:100%; height: 70px;"></div>
<!-- MAIN v2 ----- ---- ----- ---- ----- ---- ----- ---- ----- ---- ----- ---- -->
<!-- <div id="main_v2-wrapper">
<div class="wrapper">
PLOT
<div class="content-left">
<div class="fp-plot-content-wrapper">
<div id="fp_v2_line_plot_container"
class="fp-line-plot-container">
<canvas id="fp_v2_line_plot" width=500
height=500
class="canvas"></canvas>
</div>
<div id="fp_v2_input_section"
class="fp-input-section">
<div class="fp-input-wrapper input-wrapper">
<div>\(g\)</div>
<div id="fp_v2_g_value"
name="fp_v2_slide_value"
class="fp-slide-value slide-value">9.81</div>
<input id="fp_v2_g_slide" type="range"
class="fp-slide slide"
name="fp_v2_slide"
min="0" max="20"
step="0.1" value="9.81">
</div>
<div class="X-input-wrapper">
\(lat\):
<input id="fp_v2_X_input_lat" type="text"
name="fp_v2_input_field"
class="fp_X_input_field" value=49 />
</div>
<div class="X-input-wrapper">
\(L\):
<input id="fp_v2_X_input_L" type="text"
name="fp_v2_input_field"
class="fp_X_input_field" value=6.7 />
</div>
<div class="fp-input-wrapper input-wrapper">
<button class="someBtn"
id="fp_v2_animateBtn">Animate</button>
<button class="someBtn" id="fp_v2_resetBtn">Reset</button>
</div>
</div>
</div>
</div>
INFORMATION
<div class="content-right">
<div class="description">
<h2>Proof that the earth is rotating</h2>
<h4>How is this experiment performed?</h4>
<p>
The pendulum is set up at a fixed point and
brought into an initial position, where the
release of a band with the help of fire sets the
pendulum in swinging motion. This is to prevent
unwanted movement and additional swing and spin.
</p>
<h4>What can be observed?</h4>
<p>
At the poles of our earth, after about 23.93
hours, it would look as if the pendulum had
turned 360°, but in fact it is the earth
that turns under the pendulum and this turn is
not transmitted to the swinging object. At the
equator this is different,
because there the path of the pendulum does not
seem to
change. This is due to the fact that a day on
earth, with its almost 24 hours, divided by the
sine of the latitude at the equator is 0 (
\(0=\frac{23,93}{sin(0)}\)).
</p>
<p>
  ☞ <a
target="_blank"
href="FoucaultsPendulum.txt">View</a> R
implementation
</p>
<p>
  ☞ <a href="FoucaultsPendulum.html"
target="_blank">View</a> or <a
href="FoucaultsPendulum.ipynb"
target="_blank" download>download</a>
Jupyter
Notebook
</p>
</div>
</div>
</div>
</div> -->
<div class="placeholder" style="height:62px; width:100%;"></div>
<!--<div id="imgs-container">
<div class="img-container">
<img
src="https://upload.wikimedia.org/wikipedia/commons/2/22/FoucaultGlobusAnima.gif">
<p style="font-size: 10px; text-align: center; margin:
0px;
padding: 0px;">
Source: <a
href="https://upload.wikimedia.org/wikipedia/commons/2/22/FoucaultGlobusAnima.gif">Wikipedia</a>
</p>
</div>
</div> -->
<h4 style="text-align: left;">References</h4>
<p style="padding: 0px 15px 0px 15px ;">
  ☞ <a target="_blank" href="https://www.kip.uni-heidelberg.de/image/f/oeffwiss/pendel/Foucault.pdf">
"The Foucault Pendulum"
</a> - a Simplified Trajectory
Analysis for a Pendulum on a Turntable and an
Outlook to a Pendulum on Earth, A. Reiser and J. Stiewe
(2018)
</p>
<p style="padding: 0px 15px 0px 15px ;">
  ☞ <a target="_blank"
href="https://github.com/singhmeet11/Foucalt-s-pendulum-a-simple-proof-that-earth-rotates">
"Foucalts pendulum a simple proof that earth rotates"
</a> - GitHub repository by
Parmeet Singh Chani (singhmeet11) (2022)
</p>
<!-- FOOTER -->
<div id="footer" style="text-align:center; margin:0 auto; margin-bottom: 2rem;
padding-top: 5rem; width: 50%; font-size: 16px;">
Site developed by <a href="https://www.b-schwertfeger.de" target="_blank">Benjamin T. Schwertfeger</a>
(2022)<br>
<a href="https://www.awi.de" target="_blank">Alfred Wegener Institute, Helmholtz Centre for Polar and Marine
Research, Bremerhaven, Germany</a>
</div>
</body>
</html>