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

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

• 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.