A spherical shell of mass m and radius R is rolling up without slipping on a rough inclined plane as shown in the figure. The direction of static friction acting on the shell is

Options

A. Downwards along the inclined plane
B. Upward along the inclined plane
C. May be upwards or downward along the inclined plane
D. Static friction will not act

Answer

To determine the direction of static friction acting on a spherical shell rolling up an inclined plane, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Motion of the Shell:**
The spherical shell is rolling up the inclined plane. This means that it is moving in the upward direction along the incline.

2. **Understand the Forces Acting on the Shell:**
The forces acting on the shell include:
- The gravitational force (mg), which can be resolved into two components:
- Perpendicular to the incline: \( mg \cos(\theta) \)
- Parallel to the incline (down the slope): \( mg \sin(\theta) \)
- The normal force (N) acting perpendicular to the incline.
- The static friction force (f) acting along the incline.

3. **Determine the Direction of Acceleration:**
Since the shell is rolling up the incline, it is decelerating in the upward direction. This means that the net force acting on the shell must be directed down the incline.

4. **Apply Newton's Second Law:**
For the shell to roll up without slipping, the static friction must provide the necessary torque to prevent slipping. The direction of static friction must oppose the motion of the center of mass of the shell.

5. **Analyze the Torque:**
The static friction force must act in the direction that produces a clockwise torque about the center of mass. Since the shell is rolling up, the friction must act down the incline to create a clockwise rotation.

6. **Conclusion:**
Therefore, the direction of static friction acting on the shell is **downwards along the inclined plane**.

### Final Answer:
The direction of static friction acting on the shell is **downwards along the inclined plane**.

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