In this passage a brief idea is given of the motion of the rolling bodies on an inclined plane. We will consider three cases: objects are released on an incline plane
Case-A: which is smooth.
Case-B: where friction is insufficient to provide pure rolling.
Case-C: where friction is sufficient to provide pure rolling. Force diagram for three cases are as follows: (where symbols have their usual meanings)

If the four objects given in the above question are of same mass, same radius having the same friction coefficient and are released from the same height, then at the bottom the object which will have least kinetic energy for case `B` will be the:

Options

A. hollow sphere
B. solid sphere
C. ring
D. disc

Answer

C

Explanation

As given in the equation of case b
`muNR=Mk^(2)alpha` and `N=Mgcostheta`
As: `theta, M, R , mu` are same for all `alpha` will be least for the object for which `k` and hence `I` is maximum.
Therefore `alpha` for ring (`k=R`) and hence `omega` for ring at the bottom is minimum.
also, `Mgsintheta-muN=Ma`
Since `M, mu, theta, N` are same for all objects, they same linear acceleration and hene same linear velocity and hence same `1/2Mv_(cm)^(2)`
`:. KE=(1/2Mv_(cm)^(2)+1/2I_(cm)omega^(2))` is least for the ring,