A sphere rolling on a horizontal rough surface Collides elastically with a smooth vertical wall, as shown in Fig. State which of the following statements is true or false.
a. After collision, the velocity of the centre of mass gets reversed.
b. Angular momentum of the sphere about the point of contact with the wall is conserved.
c. Angular momentum of the sphere about a stationary point on the horizontal surface is conserved.
d. Just after collision the point of contact with the horizontal surface is moving towards the wall.
e. After collision the friction force acts on the sphere such that it decreases the linear speed and increases the angular speed.
f. Finally, when the sphere starts rolling, it is moving away from the wall.

Explanation

a. True: collision is elastic and the friction impulse is not generated.
b. True: The net impulse passes through the point of contact with the wall. Therefore,angular impulse about the point of contact with the wall is zero and hence the angular momentum is conserved.
c. False: Angular momentum is not conserved because the net impulse exerts torque about a stationary point on the floor.
d.False: The point of contact with the horizontal surface moves away from the wall.
e. False: Friction fore decreases both the speeds.
f. True: Wen the sphere starts rolling, it is moving away from the wall because the angular velocityy becomes zero before the linear velocity. The linear acceleration of the `CM` of the sphere is
`a=-mug`
The angular acceleration about the `CM` is
`alpha=(-5)/2 (mug)/R`
The negative sign shows that both the acceleration have the retarding effect. The time after which the linear velocity will become zero is
`t_(1)=v/(mug)`

and the time after which the angular velocity will become zero is
`t_(2)=2/5 (omegaR)/(mug)=2/5 v/(mugh)` (`:.v=omegaR)`
Note that `t_(2)ltt_(1)`