A block is placed on a rough horizontal plane. A time dependent horizontal force F=K t acts on the block K is a postive constant. Accelration-time graph of the block is

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Answer

To solve the problem of finding the acceleration-time graph of a block subjected to a time-dependent horizontal force \( F = Kt \) on a rough horizontal plane, we can follow these steps:

### Step 1: Understand the Forces Acting on the Block
The block is subjected to two forces:
1. The applied force \( F = Kt \) (which increases with time).
2. The frictional force \( f \), which opposes the motion. The maximum static friction can be expressed as \( f_{\text{max}} = \mu mg \), where \( \mu \) is the coefficient of friction and \( m \) is the mass of the block.

### Step 2: Determine the Condition for Motion
The block will not move until the applied force exceeds the maximum static friction. Therefore, we need to find the time \( t_0 \) when the applied force equals the maximum static friction:
\[ Kt_0 = \mu mg \]
From this, we can solve for \( t_0 \):
\[ t_0 = \frac{\mu mg}{K} \]

### Step 3: Analyze the Motion Before and After \( t_0 \)
- **For \( t < t_0 \)**: The applied force \( F \) is less than the maximum static friction, so the block does not move. Thus, the acceleration \( a = 0 \).

- **For \( t \geq t_0 \)**: The applied force exceeds the maximum static friction, and the block starts to move. The net force acting on the block can be expressed as:
\[ F_{\text{net}} = Kt - f_{\text{kinetic}} = Kt - \mu mg \]
Using Newton's second law, we can express the acceleration \( a \) as:
\[ ma = Kt - \mu mg \]
Thus, the acceleration can be expressed as:
\[ a = \frac{Kt - \mu mg}{m} \]
This shows that the acceleration is directly proportional to time \( t \) for \( t \geq t_0 \).

### Step 4: Sketch the Acceleration-Time Graph
- For \( t < t_0 \): The acceleration is \( 0 \).
- For \( t \geq t_0 \): The acceleration increases linearly with time.

The graph will be a horizontal line at \( a = 0 \) until \( t_0 \), and then it will rise linearly with a slope of \( \frac{K}{m} \).

### Final Result
The acceleration-time graph will have:
- A flat line at \( a = 0 \) for \( t < t_0 \).
- A straight line with a positive slope for \( t \geq t_0 \).