A smooth sphere of radius R is forced to translate along a straight line with a constant acceleration a . A particle placed on the top of the sphere is released from rest relative to the sphere. Determine the speed of the particle with respect to the sphere as a function of the angle through which it slides.

Step-by-step solution

Let the sphere accelerate horizontally with a constant acceleration a . We analyze the motion of the particle in the non-inertial frame of the sphere. In this frame, the particle experiences a pseudo force ma in the direction opposite to the sphere's acceleration, in addition to the downward gravitational force mg . As the particle slides through an angle heta from the top of the sphere, its displacement components relative to the sphere are: Horizontal displacement in the direction of the pseudo force: x = R Verti