Question

4. 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 the angle it slides.
​

Answer 1

Answer:

force f acts on the sphere towards left, so the particle experience pseudo force f=ma towards right.

Initial kinetic energy of the particle is zero.

Let the speed of the particle at point C be v.

1. From work-energy theorem, Wg +Wf =∆K.E=K.Ef,
2. where Wg is work don be by gravity and Wf

is work done by pseudo force.

From figure, we get AB=R−RcosθR−Rcosθ=R(1−cosθ)

Also AC=Rsinθ

∴ mg(AB)+f(AC)= 1/2mvsquare Or mgR(1−cosθ)+(ma) (Rsinθ)=1/2mvsquare

Or mgR(1+sinθ−cosθ)=1/2mvsquare

(∵a=g)

⟹ v= 2gR(1+sinθ−cosθ)