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, we need to analyze the motion of the block placed on the plank when the plank starts moving with a certain acceleration. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the Forces Acting on the Block
The block is subjected to two main forces:
1. The gravitational force acting downward, which is \( mg \).
2. The frictional force acting between the block and the plank, which can be calculated using the formula:
\[ f_{\text{friction}} = \mu mg \]
where \( \mu = 0.2 \) and \( g = 10 \, \text{m/s}^2 \).

### Step 2: Calculate the Maximum Static Friction
Using the values provided:
\[ f_{\text{friction}} = 0.2 \times m \times 10 = 2m \]

### Step 3: Analyze the Motion of the Plank
The plank accelerates to the right with an acceleration \( a_0 = 4 \, \text{m/s}^2 \). The block will tend to remain at rest due to inertia, but the frictional force will act to accelerate it.

### Step 4: Determine the Acceleration of the Block
The net force acting on the block can be expressed as:
\[ f_{\text{net}} = f_{\text{friction}} - m \cdot a \]
Where \( a \) is the acceleration of the block relative to the ground. The maximum frictional force is \( 2m \), and the force due to the plank's acceleration is \( 4m \).

Setting up the equation:
\[ 4m - 2m = ma \]
This simplifies to:
\[ 2m = ma \]
Thus, the acceleration \( a \) of the block relative to the ground is:
\[ a = 2 \, \text{m/s}^2 \]

### Step 5: Calculate the Displacement of the Block
Using the equation of motion for displacement:
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( u = 0 \) (initial velocity),
- \( a = 2 \, \text{m/s}^2 \),
- \( t = 1 \, \text{s} \).

Substituting the values:
\[ s = 0 \cdot 1 + \frac{1}{2} \cdot 2 \cdot (1)^2 = 1 \, \text{m} \]

### Step 6: Conclusion
The displacement of the block in 1 second, relative to the plank, is:
\[ \text{Displacement} = 1 \, \text{m} \]

### Final Answer
The correct option is that the displacement of the block is **1 meter relative to the plank**.

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