A Foucault pendulum can be used to demonstrate that earth is rotating. Explain how this is possible. What differences would you notice if you used the pendulum at the North Pole, at Earth's equator, and at altitudes between two points?
You must mean latitudes, not altitudes.
A concise derivation of the equation for the rotation of the plane of oscillation of a Foucault pendulum is hard to find. I once spend a half day in the college library looking for one, reviewing musty leather-bound journals over 150 years old. I finally found one.
It is not hard to convince yourself that the plane of oscillation (referred to ground-fixed coordinates) rotates once every 24 hours at the north and south poles, as the earth rotates below the pendulum. At the equator, the plane remains the same.
At an intermediate latitude A, the plane of oscillation rotates with a period (24 h)/|sin A|. You can think of the ground below as undergoing a combination of translation and rotation. The angle of rotation is determined by "unfolding" a conical surface that is tangent to the Earth as it makes one full rotation below the pendulum.
A Foucault pendulum consists of a long, heavy bob suspended from a high point so that it swings back and forth in a constant plane. The unique property of this pendulum is that over time, the plane of its swing appears to rotate. This rotation is not because of any external force, but rather due to the Earth's rotation.
The key to understanding how the Foucault pendulum demonstrates Earth's rotation lies in the principle of inertia. According to Newton's first law of motion, an object in motion tends to stay in motion along a straight line unless acted upon by an external force. In the case of the Foucault pendulum, as the pendulum swings back and forth, the Earth rotates underneath it. Due to its inertia, the pendulum's plane of swing remains fixed relative to space while the Earth rotates beneath it.
The rotation observed in the pendulum occurs because the Earth's surface rotates at different speeds depending on the latitude. Near the poles, the rotational speed is slower, while closer to the equator, the rotational speed is faster. As a result, the pendulum's plane of swing shifts over time, completing a full rotation in about 24 hours at the North Pole and not rotating at all at Earth's equator.
If you were to use a Foucault pendulum at different locations, you would notice the following differences:
1. North Pole: At the North Pole, the pendulum's plane of swing would make a complete rotation in approximately 24 hours. The rotation would be counterclockwise when viewed from above.
2. Equator: At the Earth's equator, the pendulum's plane of swing would not appear to rotate at all. This is because the equator experiences the Earth's fastest rotation speed, so the pendulum and the Earth move together.
3. Altitudes between two points: At intermediate latitudes, the rotation of the pendulum's plane of swing would be partial. The time taken for one full rotation would vary depending on the specific latitude. The farther away from the poles, the longer it would take for the pendulum's plane of swing to complete a rotation.
In conclusion, a Foucault pendulum demonstrates Earth's rotation by showing the apparent rotation of its plane of swing due to the Earth's rotation underneath it. The differences observed at different locations result from the varying rotational speeds of the Earth's surface at different latitudes.