A block of mass m is on an inclined plane of angle `theta`. The coefficient of friction between the block and the plane is `mu` and `tanthetagtmu`. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from `P_1=mg(sintheta-mucostheta)` to `P_2=mg(sintheta+mucostheta)`, the frictional force f versus P graph will look like

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Explanation

To solve the problem, we need to analyze the forces acting on the block on the inclined plane and how the frictional force varies with the applied force \( P \).

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Block**:
- The weight of the block \( mg \) acts downward.
- This weight can be resolved into two components:
- Parallel to the incline: \( mg \sin \theta \)
- Perpendicular to the incline: \( mg \cos \theta \)
- The normal force \( N \) acts perpendicular to the surface of the incline.
- The frictional force \( f \) acts opposite to the direction of motion (or impending motion).

2. **Establish the Conditions**:
- The block is held stationary by the applied force \( P \) parallel to the incline.
- The coefficient of friction is \( \mu \), and we know that \( \tan \theta > \mu \).

3. **Determine the Frictional Force**:
- When the applied force \( P \) is less than \( mg \sin \theta - \mu mg \cos \theta \) (denoted as \( P_1 \)), the frictional force \( f \) will act up the incline to prevent the block from sliding down.
- The equation for equilibrium in this case is:
\[ P + f = mg \sin \theta \]
Rearranging gives:
\[ f = mg \sin \theta - P \]

4. **Frictional Force at Maximum Static Friction**:
- When \( P \) is increased to \( P_1 \), the frictional force reaches its maximum static value:
\[ f = \mu mg \cos \theta \]

5. **Increasing the Applied Force \( P \)**:
- As \( P \) increases beyond \( P_1 \), the frictional force will decrease linearly until it reaches zero at \( P_2 = mg \sin \theta + \mu mg \cos \theta \).
- At this point, the block will start moving up the incline, and the frictional force will act downwards.

6. **Graph of Frictional Force \( f \) vs. Applied Force \( P \)**:
- The graph will start from \( f = mg \sin \theta - P \) when \( P < P_1 \) and will decrease linearly.
- When \( P \) exceeds \( P_1 \), the frictional force will decrease until it reaches zero at \( P_2 \).
- After \( P_2 \), the frictional force will act in the opposite direction (down the incline) and can be represented as \( f = P - mg \sin \theta \).

### Final Graph Description:
- The graph of frictional force \( f \) versus applied force \( P \) will be a linear decreasing line starting from the maximum static friction value at \( P_1 \) and reaching zero at \( P_2 \). After \( P_2 \), the graph will represent a constant downward frictional force as \( P \) continues to increase.