A sphere of mass m is rolling without sliding along positive x-axis on a rough horizontal surface of coefficient of friction `mu`. It elastically collides with a wall and then returns. The correct statement of frictional force (f) acting on the sphere is

Options

A. `f = mu mg hati` before collision and `f = mu mg hati` after collision
B. `f = 0` before collision and `f = +mu mg hati` after collision
C. `f lt mu mg hati` before collision and `f = mu mg hati` after collision
D. `f = mu mg hati` before collision and just after collision

Answer

To solve the problem regarding the frictional force acting on a sphere of mass \( m \) that is rolling without sliding and collides elastically with a wall, we can break down the situation into steps.

### Step-by-Step Solution:

1. **Understanding Initial Conditions**:
- The sphere is rolling without sliding along the positive x-axis. This means that the velocity of the center of mass \( v \) and the angular velocity \( \omega \) are related by the equation \( v = R \omega \), where \( R \) is the radius of the sphere.

2. **Collision with the Wall**:
- When the sphere collides elastically with the wall, it will reverse its linear velocity. If the initial velocity before collision is \( v \), after the collision, the velocity will become \( -v \) (assuming the wall is perfectly rigid and the collision is perfectly elastic).

3. **Friction Before Collision**:
- Before the collision, since the sphere is rolling without sliding, the frictional force acting on the sphere is zero. This is because there is no relative motion between the sphere and the surface at the point of contact.

4. **Friction After Collision**:
- After the collision, the sphere will try to continue moving in the negative x-direction. However, due to the nature of rolling without sliding, the frictional force will act in the direction opposite to the motion of the center of mass (which is now moving left).
- This means that the frictional force will act to the right (positive x-direction) to prevent slipping and to maintain rolling motion.

5. **Direction of Frictional Force**:
- The frictional force acts to the right after the collision because it needs to provide the necessary torque to keep the sphere rolling in the correct direction. The angular velocity \( \omega \) remains the same in magnitude but changes direction due to the collision.

### Conclusion:
- The correct statement regarding the frictional force acting on the sphere is that before the collision, the frictional force is zero, and after the collision, the frictional force acts in the positive x-direction.