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Foucault's Pendulum

Foucault's Pendulum | Foucault's pendulum at Ellinogermaniki Agogi | Questions-Exercises

Jean Bernard L?on Foucault was born in Paris in 1819. His parents had planned to make a doctor out of him but he soon discovered that he couldn't even stand the sight of blood and thus he quit medical school. His main interests where optics and astronomy. Among other achievements he took the first picture of the sun and along with Fizeau they managed to measure the speed of light quiet accurately. However, his most recognized achievement is proving the Earth's rotation around its axis by using a pendulum.
It all began in 1848 when he made a remarkable observation. Even if someone turns the point from which a pendulum is hung the oscillation plane is not affected. Foucault realized that this bizarre behaviour could be very useful for proving the Earth's rotation. If the oscillation plane is not affected by Earth's movement, it wouldn't rotate along with it; it would remain fixed. Therefore, after some time the floor underneath the pendulum would have shifted, compared to the oscillation plane and thus proving the Earth's rotation.
In order to prove his theory he first had to overcome certain obstacles. The pendulum's oscillation would stop after some time due to air resistance. In order to perform the experiment he would have to make sure that the pendulum will swing long enough. His first attempt was made in his basement where he hung a 5 kgr ball from a 2 meters long cord. His pendulum didn't swing long, however it was enough for Foucault to notice a very slight clockwise drift. He repeated his experiment with a greater pendulum at Paris Observatory. This time the wire was 11 meters long and the oscillation lasted longer and thus confirming his first attempt's results. In 1851 he constructed an even greater pendulum, this time at the Panth?on in Paris. There, he had his first public demonstration. The 67 meters long pendulum made the expected clockwise drift causing the crowd's excitement.

How does it work?
Figure 1. Forces acting upon a pendulum. (Air resistance is neglected)A pendulum consists of a thin cord attached to a fixed point. A mass known as bob is hung at the other end of the cord. Once set to motion the pendulum's bob swings back and forth trying to get back to its initial equilibrium position. Due to air resistance the oscillation's amplitude diminishes as time passes and after a while it stops completely returning to its equilibrium position. The time it takes the bob to complete a full oscillation is called period (T). The number of oscillations per time interval is called frequency (f).
The forces working on the pendulum are air resistance, tension and gravity. Air resistance along with friction is the reason the pendulum will eventually stop oscillating. Tension is an external force that acts upon the pendulum and it is always perpendicular to the trajectory of the pendulum's bob. Thus, tension does not act upon the oscillation plane.
The last but not least of the forces acting upon the pendulum is gravity. Gravity always acts in a downward direction and does work on the bob but it cannot change the angle of the oscillation plane. Gravity acts like a restoring force that accelerates the pendulum back to its equilibrium position. Gravity pulls the bob downwards to its equilibrium position, but inertia prevents the bob from stopping, making it swing upwards. When the pendulum reaches its highest position, gravity once again pulls it downward.
The period of the swing depends on the length (L) of the pendulum and the acceleration of gravity (g). Contrary to common belief it does not depend on the mass of the bob. However, for a non - ideal pendulum the mass of the pendulum does affect the motion due to inertia. For small amplitudes and given that air resistance is neglected (which is an acceptable approximation especially for aero dynamical bobs) the period of the pendulum can be considered constant and equal to:

This is why in order to make the pendulum oscillate long enough to prove Earth's rotation; Foucault needed the pendulum to be as long as possible and have a quite heavy bob in order to increase the inertia.
As mentioned above there are no forces acting upon the pendulum that can change the oscillation plane. Thus, the drift the pendulum's oscillation plane seems to have can only be due to Earth rotating underneath it.

Does it always work?
Earth's angular velocity can always be analyzed into two components, one vertical and one parallel to the plane. These to components cause two different kinds of motion. The first makes the plane rotate around a perpendicular axis and the second component makes the plane move in a circle around an axis parallel to the plane.
The first out of the two motions is the one witnessed while observing Foucault's pendulum. Had this motion been observed at the North or at the South Pole, the plane would make a complete round within 24 hours. On the contrary, if the experiment was held at the equator it wouldn't work at all. This is because the plane at the equator is parallel to the angular velocity vector and thus there is no vertical component to cause the motion under discussion.
In any other case, the time it takes for a full twist depends on the latitude (θ). It can be easily proved that the plane's twist per day (φ) can be calculated by using the following formula:

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