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, we need to analyze the motion of the sphere before and after it collides with the wall. Here are the steps to arrive at the solution:

### Step 1: Understand the Initial Motion of the Sphere
The sphere is rolling without sliding along the positive x-axis. This means that the center of mass of the sphere is moving in the positive x direction, while the point of contact with the ground is moving in the opposite direction (negative x direction).

**Hint:** Remember that for rolling without slipping, the velocity of the center of mass and the velocity of the point of contact have a specific relationship.

### Step 2: Identify the Direction of Friction Force Before Collision
Since the bottom point of the sphere is moving in the negative x direction, the frictional force must act in the positive x direction to prevent slipping. This frictional force helps maintain the rolling motion.

**Hint:** Friction acts in the direction that opposes the relative motion between the surfaces in contact.

### Step 3: Analyze the Collision with the Wall
When the sphere collides elastically with the wall, its velocity reverses direction. The center of mass will now move in the negative x direction immediately after the collision.

**Hint:** In an elastic collision, both momentum and kinetic energy are conserved.

### Step 4: Determine the Direction of Friction Force After Collision
After the collision, the bottom point of the sphere is still rotating in the clockwise direction. Therefore, the bottom point is again moving in the negative x direction. The frictional force will still act in the positive x direction to maintain rolling without slipping.

**Hint:** The direction of friction depends on the motion of the point of contact relative to the surface.

### Step 5: Calculate the Magnitude of the Frictional Force
The maximum static friction force can be calculated using the formula:
\[ f = \mu \cdot N \]
where \( N \) is the normal force. For a sphere of mass \( m \) on a horizontal surface, the normal force \( N \) is equal to the weight of the sphere, \( mg \). Therefore, the frictional force is:
\[ f = \mu \cdot mg \]

**Hint:** The normal force is influenced by the weight of the object, which is important for calculating friction.

### Step 6: Conclude the Result
Since the frictional force acts in the positive x direction both before and after the collision, we can summarize that:
- Before collision: \( f = \mu mg \hat{i} \)
- After collision: \( f = \mu mg \hat{i} \)

Thus, the correct statement is that the frictional force acting on the sphere is \( f = \mu mg \hat{i} \) both before and after the collision.

**Final Answer:** The correct statement is \( F = \mu mg \hat{i} \) before collision and \( F = \mu mg \hat{i} \) after collision.