The Swinging Observatory · AI Generator #GEN-5138B063

Physics · mechanics

Problem

An astronomer has built a pendulum observatory on a mountaintop at latitude

\phi = 45°

to study the Earth's rotation. The observatory consists of a massive bob of mass

m = 50\;\text{kg}

suspended by a wire of length

L = 8.0\;\text{m}

from a fixed point. When the pendulum is displaced by a small angle

\theta_0 = 5.0°

from the vertical in the east-west direction and released from rest, it begins to oscillate. Due to the Coriolis effect caused by Earth's rotation, the plane of oscillation gradually rotates. The Earth rotates with angular velocity

\Omega = 7.27 \times 10^{-5}\;\text{rad/s}

. For small oscillations, the period of the pendulum (ignoring Earth's rotation) would be

T_0 = 2\pi\sqrt{L/g}

where

g = 9.81\;\text{m/s}^2

. (a) Calculate the natural period

T_0

of the pendulum in the absence of Earth's rotation. (b) Due to the Coriolis effect, the pendulum's motion can be decomposed into two perpendicular oscillations with slightly different frequencies. The effective angular frequency for east-west oscillations becomes

\omega_x = \sqrt{g/L} - \Omega \sin \phi

and for north-south oscillations becomes

\omega_y = \sqrt{g/L} + \Omega \sin \phi

. Calculate the beat period

T_{beat}

between these two modes, which represents how long it takes for the plane of oscillation to rotate through one complete cycle.

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