Foucault's Pendulum and Its Time Period

Foucault’s pendulum is a long, heavy simple pendulum designed to demonstrate that the Earth rotates. Léon Foucault first publicly demonstrated it in 1851 in the Panthéon in Paris (a 28 kg bob on a ~67 m wire). In an inertial frame the plane of oscillation stays fixed (by inertia). Because the Earth rotates beneath it, an observer on Earth sees the plane of swing slowly rotate (precess).

It consists of a heavy bob (mass) suspended from a long, flexible wire.

The key feature is that it is free to swing in any vertical plane (isotropic suspension).

As the pendulum swings, the plane of oscillation appears to rotate relative to the ground.

Hence, Foucault’s pendulum provides direct experimental evidence of the rotation of the Earth.

Angular Velocity of Foucault’s Pendulum

If the pendulum is situated at latitude \(\lambda\), the angular velocity of rotation of its plane of oscillation relative to the Earth is

\[ \boxed{\omega_F=\omega_E\sin\lambda} \]

Substituting \[ \omega_E=\frac{2\pi}{T_E}, \] we obtain

\[ \boxed{ \omega_F= \frac{2\pi}{T_E}\sin\lambda } \]

where \(\lambda\) represents the latitude of the place.

Time Period of Rotation of the Plane

The time period corresponding to the angular velocity \(\omega_F\) is

\[ T_F=\frac{2\pi}{\omega_F} \]

\[ T_F= \frac{2\pi} {\left(\frac{2\pi}{T_E}\sin\lambda\right)} \]

Therefore,

\[ \boxed{ T_F=\frac{T_E}{\sin\lambda} } \]

Since the period of Earth's rotation is approximately 24 hours,

\[ \boxed{ T_F=\frac{24}{\sin\lambda}\text{ hours} } \]

Case I: At the Pole \(\lambda=90^\circ\)

At the pole,

\[ \sin90^\circ=1 \]

Therefore,

\[ T_F=\frac{T_E}{\sin90^\circ} \] \[ T_F=\frac{T_E}{1}=T_E \]

Since \[ T_E=24\text{ h}, \] we obtain

\[ \boxed{T_F=24\text{ h}} \]

Result: At the poles, the plane of oscillation completes one complete rotation in approximately 24 hours.

Case II: At the Equator \(\lambda=0^\circ\)

At the equator,

\[ \sin0^\circ=0 \]

Therefore,

\[ T_F=\frac{T_E}{\sin0^\circ} \] \[ T_F=\frac{T_E}{0}\rightarrow\infty \]

Result: At the equator, the plane of oscillation does not rotate relative to the Earth.

\[ \boxed{T_F\rightarrow\infty} \]

Direction of Rotation

• In the Northern Hemisphere, the plane of oscillation appears to rotate clockwise when viewed from above.
• In the Southern Hemisphere, it appears to rotate anticlockwise.
• At the Equator, there is no apparent rotation of the plane of oscillation.