A uniform non – deformable cylinder of mass m and radius R is rolling without slipping on a horizontal rough surface. The force of friction is

Options

A. `mu mg`, where `mu` is the coefficient of sliding friction
B. Zero
C. Increases with time
D. Decreases with time

Answer

To solve the problem regarding the force of friction acting on a uniform non-deformable cylinder of mass \( m \) and radius \( R \) rolling without slipping on a horizontal rough surface, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Rolling Without Slipping**:
- In rolling motion, particularly "rolling without slipping," the point of contact between the cylinder and the surface does not slide. This means that the linear velocity of the center of mass of the cylinder is equal to the angular velocity multiplied by the radius.
- Mathematically, this is expressed as:
\[ v = R \omega \]
where \( v \) is the linear velocity of the center of mass, \( R \) is the radius, and \( \omega \) is the angular velocity.

2. **Analyzing the Forces**:
- When the cylinder rolls without slipping, the frictional force acts at the point of contact to prevent slipping. However, in pure rolling motion, the frictional force does not do any work since there is no relative motion at the contact point.

3. **Condition for Pure Rolling**:
- Since there is no relative motion at the point of contact, the static friction force is sufficient to provide the necessary torque for rolling. In an ideal case (without external forces causing acceleration), the friction force can be considered to be zero because it is not needed to prevent slipping.

4. **Conclusion**:
- Therefore, in the case of a uniform non-deformable cylinder rolling without slipping on a horizontal rough surface, the force of friction is effectively zero.

### Final Answer:
The force of friction acting on the cylinder is \( 0 \).

---