A force is linearly increasing with time, on a block of mass m placed on the horizontal surface. The friction coefficient from the block and the ground is u. The graph of friction force and time is drawn. Which of the following graph is correct.

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Forces Acting on the Block
- We have a block of mass \( m \) placed on a horizontal surface.
- An external force \( F \) is applied to the block, which increases linearly with time. We can express this as \( F = k \cdot t \), where \( k \) is a constant.
- The friction force \( f \) acting on the block opposes the motion and is given by \( f = \mu \cdot N \), where \( N \) is the normal force. For a horizontal surface, \( N = mg \).

### Step 2: Calculate the Maximum Static Friction
- The maximum static friction force can be calculated as:
\[ f_{\text{max}} = \mu \cdot mg \]
- For example, if \( \mu = 0.1 \) and \( m = 10 \, \text{kg} \), then:
\[ f_{\text{max}} = 0.1 \cdot 10 \cdot 9.8 \approx 0.98 \, \text{N} \]

### Step 3: Determine When the Block Starts Moving
- The block will start moving when the applied force \( F \) equals the maximum static friction force \( f_{\text{max}} \).
- If \( F = k \cdot t \), we set \( k \cdot t = f_{\text{max}} \) to find the time \( t \) when the block starts moving:
\[ t = \frac{f_{\text{max}}}{k} \]

### Step 4: Analyze the Friction Force Over Time
- Before the block starts moving (for \( t < t_{\text{max}} \)), the friction force will equal the applied force, which increases linearly.
- Once the block starts moving (for \( t \geq t_{\text{max}} \)), the friction force becomes kinetic friction, which is constant and equal to \( f_{\text{max}} \).

### Step 5: Draw the Graph of Friction Force vs. Time
- For the time interval \( 0 \) to \( t_{\text{max}} \), the friction force increases linearly with time.
- After \( t_{\text{max}} \), the friction force remains constant at \( f_{\text{max}} \).

### Conclusion: Identify the Correct Graph
- The correct graph of friction force vs. time will show a linear increase until \( t_{\text{max}} \) and then a plateau at the maximum static friction force.