A marble in a spherical bowl oscillates with an apparently simple harmonic motion, however for large oscillations this motion becomes quite difficult to describe.

What is known:

marble radius r = 0.04 ft

marble mass m = 0.003 slugs

bowl radius R = 1 ft

Problem Diagram

Question

What is the frequency of the marble's motion in the bowl for small oscillations?

Approach

Determine the kinetic energy and potential energy for the marble.

Find the equation of motion by setting the time derivative of the total energy equal to zero.

Solve the equation of motion.

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