A sphere and a block both are released from rest at the top of an inclined plane. After some time they reach the bottom. For all the options assume that angle made by the plane with the horizontal is more than angle of repose.

Options

A. If surface is frictionless then sphere and block reach the bottom simultaneously.
B. If surface is rough but not sufficient for pure rolling of sphere then sphere and block reach the bottom simultaneously.
C. If surface is sufficiently rough then sphere and block may not reach the bottom simultaneously.
D. If surface is sufficiently rough then at the bottom kinetic energy of sphere will be more than that of the block.

Answer

A, B, C, D

Explanation

In case of frictionless surface both the objects will have the same acceleration that is equal tog sin . So both will reach the bottom in same time. Hence option (a) is correct.
If surface is rough but not sufficiently rough to provide pure rolling to sphere then also acceleration of both the objects will be same that is equal to `(g sin theta - mu g cos theta)`. Hence in this case also both will reach the bottom simultaneously, so option (b) is correct.
When surface is sufficiently rough then acceleration of block will be given by (` g sin theta- mu g cos theta)`, but same for the sphere will be given by `(mg s in theta)/(m+(I)/(r^(2)))`. Due to different accelerations, time taken will be different to reach the bottom hence option (c) is correct.
In case of pure rolling total mechanical energy of the sphere remains conserved whereas there is loss of energy in case of block hence kinetic energy of the sphere at the bottom will be more than that of the block.