A pendulum whose upper end is attached so as to allow the pendulum to swing freely in any direction can be used to repeat an experiment first shown publicly by Foucault in Paris in 1851. If the pendulum is set oscillating, the plane of oscillation slowly rotates with respect to a line drawn on the floor, even though the tension in the wire supporting the bob and the gravitational pull of the Earth on the bob lie in a vertical plane.

(a) Show that this is a result of the fact that the Earth is not an inertial reference frame.

(b) Show that for a Foucault endulum at a latitude e, the period of rotation of the plane, in hours, is 24 sin θ.

(c) Explain in simple terms the result at θ = 90° (the poles) and θ = 0° (the equator).

To understand the fascinating behavior of a Foucault pendulum, we need to delve into some fundamental concepts of physics, particularly regarding reference frames and the Earth's rotation. Let's break this down step by step.

Understanding Non-Inertial Reference Frames

First, let's tackle part (a) of your question. The Earth is not an inertial reference frame because it is rotating. An inertial frame is one where an object not subjected to any force moves at a constant velocity. However, because the Earth spins on its axis, any object on its surface, including our pendulum, experiences a fictitious force due to this rotation. This is often referred to as the Coriolis effect.

When the pendulum swings, it maintains its plane of oscillation due to inertia. However, as the Earth rotates beneath it, the pendulum appears to change direction relative to the surface of the Earth. This is why the plane of oscillation seems to rotate over time. The pendulum itself is not changing its motion; rather, it is the Earth that is moving beneath it.

Calculating the Period of Rotation

Now, moving on to part (b), we need to show that the period of rotation of the plane of oscillation for a Foucault pendulum at latitude θ is given by 24 sin θ hours. The key to this lies in understanding how the rotation of the Earth affects the pendulum's oscillation.

The Earth completes one full rotation (360 degrees) in 24 hours. At the poles (θ = 90°), the pendulum's plane of oscillation rotates once every 24 hours because the rotation axis of the Earth is vertical relative to the pendulum's motion. However, as you move towards the equator (θ = 0°), the effective rotation of the pendulum's plane decreases.

The relationship can be expressed mathematically as:

• At the poles (θ = 90°): The period of rotation is 24 hours.
• At the equator (θ = 0°): The pendulum does not rotate at all, so the period is infinite.
• At any latitude θ: The period of rotation is given by 24 hours multiplied by the sine of the latitude, or 24 sin θ hours.

This formula arises because the component of the Earth's rotation that affects the pendulum's plane of oscillation is proportional to the sine of the latitude. Thus, the further you are from the poles, the less effect the Earth's rotation has on the pendulum's plane.

Interpreting Extreme Cases

Finally, let's discuss part (c) regarding the special cases at the poles and the equator. At θ = 90° (the poles), the pendulum completes a full rotation in 24 hours. This means that if you were to observe the pendulum over a day, you would see it tracing a complete circle in the sky, demonstrating the Earth's rotation directly.

On the other hand, at θ = 0° (the equator), the pendulum does not appear to rotate at all. This is because the plane of oscillation remains fixed relative to the Earth's surface. The rotation of the Earth at the equator is horizontal to the pendulum's swing, meaning there is no vertical component to cause a change in the plane of oscillation.

In summary, the behavior of a Foucault pendulum beautifully illustrates the effects of Earth's rotation and the nature of non-inertial reference frames, revealing how our planet's movement influences even simple systems like a swinging pendulum.