A block of mass `4 kg` is placed on a rough horizontal force plane. A time dependent horizontal force `F = k t` acts on the block. Here `k = 2 Ns^(-1)`. The frictional force between the block and plane at time `t = 2 s` is `(mu = 0.2)` a) 4 N b) 8 N c) 12 N d) 10 N

Options

A. `4 N`
B. `8 N`
C. `12 N`
D. `10 N`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- Mass of the block, \( m = 4 \, \text{kg} \)
- Coefficient of friction, \( \mu = 0.2 \)
- Time, \( t = 2 \, \text{s} \)
- Constant \( k = 2 \, \text{N/s} \)

### Step 2: Calculate the normal force
The normal force \( N \) acting on the block is equal to the weight of the block, which can be calculated using the formula:
\[ N = mg \]
Where \( g \) (acceleration due to gravity) is approximately \( 10 \, \text{m/s}^2 \).

Substituting the values:
\[ N = 4 \, \text{kg} \times 10 \, \text{m/s}^2 = 40 \, \text{N} \]

### Step 3: Calculate the maximum frictional force
The maximum frictional force \( f_{\text{max}} \) can be calculated using the formula:
\[ f_{\text{max}} = \mu N \]
Substituting the values:
\[ f_{\text{max}} = 0.2 \times 40 \, \text{N} = 8 \, \text{N} \]

### Step 4: Calculate the applied force at \( t = 2 \, \text{s} \)
The applied force \( F \) is given by the equation:
\[ F = kt \]
Substituting the values:
\[ F = 2 \, \text{N/s} \times 2 \, \text{s} = 4 \, \text{N} \]

### Step 5: Compare the applied force with the maximum frictional force
Now we compare the applied force with the maximum frictional force:
- Applied force \( F = 4 \, \text{N} \)
- Maximum frictional force \( f_{\text{max}} = 8 \, \text{N} \)

Since the applied force (4 N) is less than the maximum frictional force (8 N), the block will not move.

### Step 6: Determine the frictional force
In this case, the frictional force \( f \) will be equal to the applied force because the block does not move:
\[ f = F = 4 \, \text{N} \]

### Final Answer
The frictional force between the block and the plane at time \( t = 2 \, \text{s} \) is \( \boxed{4 \, \text{N}} \).