A block is placed over a plank. The coefficient of friction between the block and the plank is `mu= 0.2.` Initially both are at rest, suddenly the plank starts moving with acceleration `a_(0) = 4m//s^(2).` The displacement of the block in 1 is `(g=10m//s^(2))`

Options

A. 1 m relative to ground
B. 1 m relative to plank
C. zero relative to plank
D. 2 m relative to ground

Answer

To solve the problem step by step, we will analyze the situation involving the block and the plank, applying Newton's laws of motion and the concept of friction.

### Step 1: Identify the Given Information
- Coefficient of friction (μ) = 0.2
- Acceleration of the plank (a₀) = 4 m/s²
- Gravitational acceleration (g) = 10 m/s²
- Time (t) = 1 second

### Step 2: Determine the Maximum Static Friction
The maximum static friction force (f_max) that can act on the block is given by:
\[ f_{\text{max}} = \mu \cdot m \cdot g \]
Where:
- \( m \) is the mass of the block (we will see that it cancels out later).

Substituting the values:
\[ f_{\text{max}} = 0.2 \cdot m \cdot 10 = 2m \]

### Step 3: Write the Equation of Motion for the Block
When the plank starts moving, the block will experience a frictional force that tries to keep it stationary relative to the plank. The net force acting on the block can be expressed as:
\[ m \cdot a = m \cdot a_0 - f_{\text{friction}} \]

Where:
- \( a \) is the acceleration of the block,
- \( a_0 \) is the acceleration of the plank (4 m/s²),
- \( f_{\text{friction}} \) is the frictional force acting on the block.

Since the block can only accelerate up to the limit of static friction, we can set up the equation:
\[ m \cdot a = m \cdot 4 - f_{\text{friction}} \]

### Step 4: Determine the Frictional Force
The frictional force cannot exceed the maximum static friction. Therefore, we can write:
\[ f_{\text{friction}} = 2m \]

Substituting this into the equation gives:
\[ m \cdot a = m \cdot 4 - 2m \]

### Step 5: Simplify the Equation
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ a = 4 - 2 \]
\[ a = 2 \, \text{m/s}^2 \]

### Step 6: Calculate the Displacement of the Block
Now, we can use the equation of motion to find the displacement of the block in 1 second. The equation for displacement (s) when starting from rest is:
\[ s = ut + \frac{1}{2} a t^2 \]

Since the initial velocity (u) is 0:
\[ s = 0 + \frac{1}{2} \cdot 2 \cdot (1)^2 \]
\[ s = \frac{1}{2} \cdot 2 \cdot 1 = 1 \, \text{meter} \]

### Conclusion
The displacement of the block relative to the ground in 1 second is **1 meter**.

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