Which of the following statements is `//`are true.

Options

A. work done by kinetic friction on an object may be positive.
B. A rigid body rolls up an inclined plane without sliding. The friction force on it will be up the incline. (only contact force and gravitational force is acting)
C. A rigid body rolls down an inclined plane without sliding. The friction force on it will be up the incline. (only contact force and gravitational force is acting)
D. A rigid body is left from rest and having no angular velocity from the top of a rough inclined plane. It moves down the plane with slipping. The friction force on it will be up the incline.

Explanation

To determine which statements are true regarding the work done by kinetic friction and the behavior of rigid bodies on inclined planes, we will analyze each statement step by step.

### Step-by-Step Solution:

1. **Understanding Kinetic Friction**:
- Kinetic friction acts between two surfaces in relative motion. It always opposes the relative motion but may do work that is positive under certain conditions.
- For example, if a block is sliding down a surface and a second block is on top of it, the kinetic friction acting on the second block can support its motion in the direction of the applied force.

**Hint**: Remember that kinetic friction opposes relative motion but can do positive work if it helps to minimize that motion.

2. **Analyzing the First Statement**:
- The statement claims that the work done by kinetic friction on an object may be positive.
- In the scenario where two blocks are in contact, the kinetic friction acting on the upper block can indeed do positive work if it helps to accelerate the block in the direction of the applied force.

**Conclusion**: This statement is **true**.

3. **Rigid Body Rolling Up an Incline**:
- When a rigid body rolls up an incline without sliding, the friction force acts up the incline. This is because friction must act to prevent slipping and ensure that the point of contact has zero velocity relative to the incline.
- The gravitational force component acting down the incline (mg sin θ) tries to pull the object down, while friction acts up to counteract this.

**Hint**: Consider the forces acting on the rolling body and how they relate to its motion.

4. **Rigid Body Rolling Down an Incline**:
- Similarly, when a rigid body rolls down an incline without sliding, the friction force again acts up the incline. This is to ensure that the point of contact does not slip and has the same velocity as the incline.
- The friction force provides the necessary torque to maintain rolling motion.

**Hint**: Analyze the direction of forces and the condition for rolling without slipping.

5. **Final Evaluation of Statements**:
- Both statements regarding the rolling of rigid bodies (up and down the incline) are true as they align with the principles of friction and motion.
- Therefore, all four statements presented in the question are correct.

### Final Conclusion:
All statements regarding the work done by kinetic friction and the behavior of rigid bodies on inclined planes are **true**.