A cylinder rolls up an inclined plane, reaches some height, and then rolls down (without slipping throughout these motions). The directions of the frictional force acting on the cylinder are.

Options

A. up the incline while ascending and down the incline while descending
B. up the incline while ascending as welll as descending
C. down the incline while ascending and up the incline while descending
D. down the incline while ascending as well as descending

Answer

To determine the direction of the frictional force acting on a cylinder that rolls up and down an inclined plane, we can analyze the motion in two parts: when the cylinder is rolling up the incline and when it is rolling down.

### Step-by-Step Solution:

1. **Understanding the Motion While Ascending:**
- When the cylinder rolls up the incline, it is moving against the force of gravity.
- The point of contact with the incline is moving upwards, while the center of mass of the cylinder is moving upwards as well.
- The tendency of the cylinder is to rotate in such a way that the bottom point (point of contact) moves backward relative to the incline.
- Therefore, to prevent slipping and to maintain rolling motion, the frictional force must act in the forward direction (up the incline) to oppose this backward tendency.

2. **Understanding the Motion While Descending:**
- When the cylinder rolls down the incline, the force of gravity pulls it downwards.
- The point of contact with the incline is moving downwards, while the center of mass is also moving downwards.
- In this case, if there were no friction, the cylinder would slide down without rolling.
- To ensure that the cylinder rolls without slipping, the frictional force must act upwards (against the direction of motion) to provide the necessary torque for rolling.
- Thus, the frictional force acts upwards along the incline.

3. **Conclusion:**
- In both cases (ascending and descending), the frictional force acts in the same direction, which is upwards along the incline.

### Final Answer:
The direction of the frictional force acting on the cylinder is upwards along the incline in both cases.