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Rotation: Sphere over Sphere

Q- A solid sphere of radius R starts at rest on top of a semicircular track of radius S., as shown in the figure. The sphere rolls without slipping along the track until it falls off the track at point B. Point A is a vertical distance H above the ground; the height of point B is unknown. You may assume that R<<S.

(a) What is the angular speed ωA of the sphere at point A, expressed as a function of H, R, S, and the (downward) acceleration of gravity g?

(b) Find the speed of the sphere at point B, where it falls off the track.

Solution:

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