A man of mass `80 kg` stands on a plank of mass `40 kg`. The plank is lying on a smooth horizontal floor. Initianlly both are at rest. The man starts walking on the plank towards north and stops after moving a distance of `6 m` on the plank. Then

Options

A. The center of mas os plank man system remains stationary.
B. The plank will slide to the north by a distance 6m
C. The plank will slide to the south by a distance 6m
D. The plank will slids to the south by a distance 12m

Answer

To solve the problem, we need to analyze the situation using the principles of conservation of momentum and the concept of the center of mass. Here’s a step-by-step solution:

### Step 1: Understand the System
We have a man of mass \( m_1 = 80 \, \text{kg} \) standing on a plank of mass \( m_2 = 40 \, \text{kg} \). Initially, both the man and the plank are at rest.

### Step 2: Define the Displacements
When the man walks a distance of \( d_m = 6 \, \text{m} \) towards the north on the plank, the plank will move a distance \( d_p \) in the opposite direction (south) due to the conservation of momentum.

### Step 3: Apply Conservation of Momentum
Since there are no external horizontal forces acting on the system, the center of mass of the system must remain stationary. Therefore, the total momentum before and after the man starts walking must be equal.

Let \( d_p \) be the distance the plank moves south. The displacement of the man with respect to the ground can be expressed as:
\[ d_{man, ground} = d_m - d_p \]

### Step 4: Calculate the Center of Mass
The position of the center of mass \( x_{cm} \) of the system can be calculated using the formula:
\[ x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \]
where \( x_1 \) is the position of the man and \( x_2 \) is the position of the plank.

Since the initial center of mass is at rest, we can set up the equation:
\[ m_1 \cdot (d_m - d_p) + m_2 \cdot (-d_p) = 0 \]

### Step 5: Substitute Values
Substituting the values we have:
\[ 80 \cdot (6 - d_p) + 40 \cdot (-d_p) = 0 \]
Expanding this gives:
\[ 480 - 80d_p - 40d_p = 0 \]
Combining like terms:
\[ 480 - 120d_p = 0 \]

### Step 6: Solve for \( d_p \)
Now, solving for \( d_p \):
\[ 120d_p = 480 \]
\[ d_p = \frac{480}{120} = 4 \, \text{m} \]

### Conclusion
The plank moves south by a distance of \( 4 \, \text{m} \) while the man walks \( 6 \, \text{m} \) north on the plank. Thus, the center of mass of the system remains stationary.

### Final Answer
The plank moves south by \( 4 \, \text{m} \).