A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless surface. The man walks to the other end of the planck. If the mass of the planck is `M/2`, then the distance that the man moves relative to the ground is

Options

A. `3L/4`
B. `L/4`
C. `4L/5`
D. `L/3`

Answer

To solve the problem of a man walking on a plank that is on a frictionless surface, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Masses and Length**:
- Let the mass of the man be \( M \).
- The mass of the plank is \( \frac{M}{2} \).
- The length of the plank is \( L \).

2. **Understand the System**:
- The plank is on a frictionless surface, which means there are no external forces acting on the system.
- As the man walks from one end of the plank to the other, the plank will move in the opposite direction to conserve momentum.

3. **Define the Center of Mass**:
- The center of mass of the system (man + plank) must remain stationary because there are no external forces acting on it.
- Initially, we can assume the center of mass is at a position we can define as zero.

4. **Set Up the Equation for Center of Mass**:
- Let \( x \) be the distance the plank moves in the opposite direction while the man walks the length \( L \) of the plank.
- The position of the center of mass can be expressed as:
\[ \text{Center of Mass} = \frac{M \cdot (L - x) + \frac{M}{2} \cdot \left(-x\right)}{M + \frac{M}{2}} \]
- This simplifies to:
\[ \text{Center of Mass} = \frac{M(L - x) - \frac{Mx}{2}}{\frac{3M}{2}} = 0 \]

5. **Simplify the Equation**:
- Multiply both sides by \( \frac{3M}{2} \):
\[ M(L - x) - \frac{Mx}{2} = 0 \]
- Rearranging gives:
\[ ML - Mx + \frac{Mx}{2} = 0 \]
- Combine the terms involving \( x \):
\[ ML = Mx - \frac{Mx}{2} \]
\[ ML = \frac{Mx}{2} \]

6. **Solve for \( x \)**:
- Dividing both sides by \( M \):
\[ L = \frac{x}{2} \]
- Thus:
\[ x = 2L \]

7. **Calculate the Distance the Man Moves Relative to the Ground**:
- The distance the man moves relative to the ground is given by:
\[ \text{Distance}_{\text{man}} = L - x \]
- Substitute \( x = \frac{2L}{3} \):
\[ \text{Distance}_{\text{man}} = L - \frac{2L}{3} = \frac{L}{3} \]

### Final Answer:
The distance that the man moves relative to the ground is \( \frac{L}{3} \).