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 well 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 forces at play during both motions.

### Step-by-step Solution:

1. **Understanding the Motion**:
- When the cylinder rolls up the incline, it is moving against gravity. The gravitational force component acting down the incline is given by \( mg \sin \theta \), where \( m \) is the mass of the cylinder and \( \theta \) is the angle of the incline.

2. **Forces Acting on the Cylinder While Rolling Up**:
- As the cylinder rolls up, the gravitational force component \( mg \sin \theta \) acts down the incline, trying to pull the cylinder back down.
- To counteract this and allow the cylinder to roll up, a frictional force must act up the incline. This frictional force helps provide the necessary torque for the cylinder to rotate in the direction of its motion.

3. **Analyzing the Rolling Condition**:
- For pure rolling motion, the point of contact (let’s call it point A) must have zero acceleration relative to the surface. This means that the acceleration due to translation and rotation must balance out.
- The frictional force acting up the incline provides the necessary torque to ensure that the cylinder rolls without slipping.

4. **Forces Acting on the Cylinder While Rolling Down**:
- When the cylinder reaches its maximum height and starts rolling back down, the same gravitational force component \( mg \sin \theta \) acts down the incline.
- In this case, the frictional force must also act up the incline to prevent slipping and to maintain pure rolling motion. This is because the cylinder is still rotating in the same direction as it rolls down.

5. **Conclusion**:
- In both scenarios (rolling up and rolling down), the frictional force acts **up the incline**. This is necessary to maintain pure rolling motion and to provide the required torque for rotation.

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

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