A uniform wooden plank of mass 150 kg and length 8 m is floating on still water with a man of 50 kg at one end of it . The man walks to the other end of the plank and stops. Than the distance covered by the plank is

Options

A. 1 m
B. 4 m
C. 2 m
D. Zero

Answer

To solve the problem of how far the plank moves when the man walks from one end to the other, we can use the principle of conservation of the center of mass. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have a wooden plank of mass \( m_p = 150 \, \text{kg} \) and length \( L = 8 \, \text{m} \) floating on water. A man of mass \( m_m = 50 \, \text{kg} \) is initially at one end of the plank.

### Step 2: Define the Initial Position
Let’s define the initial position of the center of mass of the system. The center of mass of the plank is at its midpoint, which is at \( 4 \, \text{m} \) from either end. The man is at one end, which we can consider as position \( 0 \, \text{m} \).

### Step 3: Calculate the Initial Center of Mass
The initial position of the center of mass (\( x_{cm, initial} \)) can be calculated using the formula:
\[ x_{cm, initial} = \frac{m_m \cdot x_m + m_p \cdot x_p}{m_m + m_p} \]
Where:
- \( x_m = 0 \, \text{m} \) (position of the man)
- \( x_p = 4 \, \text{m} \) (position of the center of the plank)

Substituting the values:
\[ x_{cm, initial} = \frac{50 \cdot 0 + 150 \cdot 4}{50 + 150} = \frac{600}{200} = 3 \, \text{m} \]

### Step 4: Define the Final Position
When the man walks to the other end of the plank, he will be at position \( 8 \, \text{m} \). Let \( x \) be the distance the plank moves to the right.

The new position of the center of mass (\( x_{cm, final} \)) will be:
\[ x_{cm, final} = \frac{m_m \cdot (8 - x) + m_p \cdot (4 + x)}{m_m + m_p} \]

### Step 5: Set the Center of Mass Positions Equal
Since there are no external forces acting on the system, the center of mass does not move:
\[ x_{cm, initial} = x_{cm, final} \]
Substituting the values:
\[ 3 = \frac{50 \cdot (8 - x) + 150 \cdot (4 + x)}{200} \]

### Step 6: Simplify and Solve for \( x \)
Multiply both sides by \( 200 \):
\[ 600 = 50(8 - x) + 150(4 + x) \]
Expanding the equation:
\[ 600 = 400 - 50x + 600 + 150x \]
Combine like terms:
\[ 600 = 1000 + 100x \]
Rearranging gives:
\[ 100x = 600 - 1000 \]
\[ 100x = -400 \]
\[ x = -4 \, \text{m} \]

### Step 7: Calculate the Distance Covered by the Plank
Since \( x \) is negative, it indicates that the plank moves \( 2 \, \text{m} \) to the left (the direction opposite to the man's movement). Therefore, the distance covered by the plank is:
\[ \text{Distance covered by the plank} = 2 \, \text{m} \]

### Conclusion
The distance covered by the plank when the man walks from one end to the other is \( 2 \, \text{m} \).