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

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining 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 experiences a horizontal force \( F = Kt \) and a frictional force opposing the motion. The frictional force can be static or kinetic depending on whether the block is moving or not.

### Step 2: Determine the Condition for Static Friction
Initially, at \( t = 0 \), the force \( F = Kt = 0 \). The static friction force \( F_s \) will act to prevent the block from moving. The maximum static friction force is given by:
\[ F_s = \mu_s mg \]
where \( \mu_s \) is the coefficient of static friction and \( m \) is the mass of the block.

### Step 3: Find the Time When the Block Starts Moving
The block will start moving when the applied force \( F \) exceeds the maximum static friction force. Setting \( F = F_s \):
\[ Kt = \mu_s mg \]
From this, we can solve for the time \( t \) at which the block starts to move:
\[ t = \frac{\mu_s mg}{K} \]

### Step 4: Analyze the Motion After the Block Starts Moving
Once the block starts moving, the frictional force becomes kinetic friction, which is given by:
\[ F_k = \mu_k mg \]
where \( \mu_k \) is the coefficient of kinetic friction.

The net force acting on the block after it starts moving is:
\[ F_{net} = F - F_k = Kt - \mu_k mg \]
Using Newton's second law \( F_{net} = ma \), we can write:
\[ ma = Kt - \mu_k mg \]
Dividing by \( m \):
\[ a = \frac{K}{m}t - \mu_k g \]

### Step 5: Determine the Acceleration-Time Relationship
The equation \( a = \frac{K}{m}t - \mu_k g \) indicates that the acceleration \( a \) is a linear function of time \( t \) after the block starts moving. The slope of this line is \( \frac{K}{m} \) and the y-intercept is \( -\mu_k g \).

### Step 6: Construct the Acceleration-Time Graph
1. For \( t < \frac{\mu_s mg}{K} \), the acceleration \( a = 0 \) (the block does not move).
2. For \( t \geq \frac{\mu_s mg}{K} \), the acceleration increases linearly with time.

### Conclusion
The acceleration-time graph will be:
- A horizontal line at \( a = 0 \) for \( t < \frac{\mu_s mg}{K} \).
- A line with a positive slope for \( t \geq \frac{\mu_s mg}{K} \).