A heavy particle of mass is placed at the top of a semicircular block of radius . Find the height at which the particle falls off, assuming (i) the block is fixed to the ground, and (ii) the block has a mass and is free to move. Assume all surfaces are frictionless.

Related problem: Sliding on a block with a circular cut.

Solution:

(i) We first consider the case where the block is fixed to the ground. As the mass slides down the block, there are three forces acting on it: the weight , the centrifugal force , and …