Bug on a Bowling Ball

A bug sitting on top of a bowling ball begins to slide o? with negligible friction. Conclude the angle at which the bug leaves the surface.

This problem is easily solved with the aid of the diagram

The forces on the bug include the centripetal force due to gravity, the centrifugal force due to motion along the curve and the resulting normal force.

F_n = mv²/r - mg cos θ

The get done in kinetic energy as the bug slides is provided by the loss in potential energy. Since h = r - r cos θ we have

At the instant the bug leaves the ball the normal force F_n becomes zero. We are able to now substitute the value of the centrifugal force

F_n =(mv²/r)- mgcos θ

The get in kinetic energy as the bug slides is provided by the loss in potential energy. As h = r - r cos θ we have

mv²/2 = mgh

mv²/2 = mgr(1 - cos θ)

mv²/r = 2mgr(1 - cos θ)

At the moment the bug leaves the ball the normal force F_n becomes zero. We are able to now substitute the value of the centrifugal force