Question

A smooth sphere of radius R is made to translate in a straight line with a constant acceleration a. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. Find the speed of the particle with respect to the sphere as a function of angle theta as it slides

Answer 1

Let the speed of particle is V

$$V=Rd\theta/dt---------(1)$$

Tangential force acting on the particle due to motion of sphere

$$mdv/dt=ma*cos\theta+mg*sin\theta$$

Thus,

$$dV/dt=a*Cos\theta+g*Sin\theta$$

$$VdV/dt=a*Cos\theta(Rd\theta/dt)+g*Sin\theta(Rd\theta/dt)$$ $$V^2/2=aRSin\theta-gRCos\theta+C-------(2)$$

Apply boundary conditions

$$\theta=0, V=0$$

From equation 2

$$C=gR$$

$$V^2/2=aRSin\theta-gRCos\theta+gR$$

Thus,

$$V=\sqrt{2R(aSin\theta-gCos\theta+g)}$$