A block of mass 8 kg is placed on a rough horizontal plane. A time dependent force `F=kt^2`acts on the block, where `k=3N//s^2`. Coefficient of friction `is 0.6`. Force of friction between the block and the plane at t=1s is

Options

A. 48N
B. 4N
C. 2N
D. 3N

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given values
- Mass of the block (m) = 8 kg
- Coefficient of friction (μ) = 0.6
- Time-dependent force (F) = kt², where k = 3 N/s²
- Time (t) = 1 s

### Step 2: Calculate the force acting on the block at t = 1 s
Using the formula for the time-dependent force:
\[ F = k \cdot t^2 \]
Substituting the values:
\[ F = 3 \cdot (1)^2 = 3 \, \text{N} \]

### Step 3: Calculate the maximum static friction (Fs_max)
The maximum static friction can be calculated using the formula:
\[ F_{s_{\text{max}}} = \mu \cdot m \cdot g \]
Where g (acceleration due to gravity) is approximately 10 m/s². Thus:
\[ F_{s_{\text{max}}} = 0.6 \cdot 8 \cdot 10 = 48 \, \text{N} \]

### Step 4: Compare the applied force and maximum static friction
Now we compare the applied force (3 N) with the maximum static friction (48 N):
- Since \( F < F_{s_{\text{max}}} \) (3 N < 48 N), the block will not move.

### Step 5: Determine the force of friction acting on the block
Since the block does not move, the force of friction will equal the applied force to prevent motion:
\[ F_{\text{friction}} = F = 3 \, \text{N} \]

### Final Answer:
The force of friction between the block and the plane at t = 1 s is **3 N**.

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