A block rests on a rough plane whose inclination `theta` to the horizontal can be varied. Which of the following graphs indicates how the friction force `F` between the block and the plane varies as `theta` is increased?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze how the friction force \( F \) between the block and the inclined plane varies as the angle \( \theta \) is increased.

### Step-by-Step Solution:

1. **Understanding the Forces**: When a block is resting on an inclined plane, the forces acting on it include gravitational force, normal force, and frictional force. The gravitational force can be resolved into two components: one parallel to the incline (which tries to slide the block down) and one perpendicular to the incline (which affects the normal force).

2. **Static Friction**: For small angles \( \theta \) (where \( \theta < \alpha \), with \( \alpha \) being the angle of repose), the block does not slide. The frictional force \( F \) will balance the component of the gravitational force acting down the slope. This can be expressed as:
\[ F = mg \sin \theta \]
Here, \( m \) is the mass of the block and \( g \) is the acceleration due to gravity.

3. **Behavior as \( \theta \) Increases**: As \( \theta \) increases, \( \sin \theta \) also increases, which means the frictional force \( F \) increases as long as the block remains stationary.

4. **Angle of Repose**: When \( \theta \) reaches the angle of repose \( \alpha \), the block is on the verge of sliding. At this point, the maximum static friction force is equal to:
\[ F_{\text{max}} = \mu_s mg \cos \theta \]
where \( \mu_s \) is the coefficient of static friction.

5. **Transition to Kinetic Friction**: Once \( \theta \) exceeds \( \alpha \), the block starts to slide down the plane. The frictional force now becomes kinetic friction, which is given by:
\[ F = \mu_k mg \cos \theta \]
where \( \mu_k \) is the coefficient of kinetic friction.

6. **Graphical Representation**:
- For \( \theta < \alpha \), \( F \) increases with \( \sin \theta \).
- For \( \theta \geq \alpha \), \( F \) decreases with \( \cos \theta \).

7. **Conclusion**: The graph that represents this behavior will show an increase in friction force \( F \) as \( \theta \) increases until it reaches a maximum at \( \theta = \alpha \), and then it will decrease as \( \theta \) continues to increase beyond \( \alpha \).

### Final Answer:
The correct graph indicates that the friction force \( F \) increases with \( \theta \) until it reaches the angle of repose and then decreases as \( \theta \) continues to increase.