Consider a system of two particles having masses `m_(1)` and `m_(2)`. If the particle of mass `m_(1)` is pushed towards the centre of mass of particles through a distance `d`, by what distance would the particle of mass `m_(2)` move so as to keep the mass centre of particles at the original position?

Options

A. `m_(1)/(m_(1)+m_(2)) d`
B. `m_(1)/m_(2) d`
C. `d`
D. `m_(2)/m_(1) d`

Answer

To solve the problem, we need to find the distance that mass \( m_2 \) moves when mass \( m_1 \) is pushed towards the center of mass by a distance \( d \), in order to keep the center of mass in its original position.

### Step-by-Step Solution:

1. **Understanding the System**:
- Let the mass of the first particle be \( m_1 \) and the mass of the second particle be \( m_2 \).
- Let the initial distances from the center of mass (CM) to \( m_1 \) and \( m_2 \) be \( X \) and \( Y \) respectively.

2. **Initial Condition**:
- The position of the center of mass (CM) is given by the formula:
\[ R_{CM} = \frac{m_1 \cdot X + m_2 \cdot Y}{m_1 + m_2} \]
- For the CM to remain at the original position, the following condition must hold:
\[ m_1 \cdot X = m_2 \cdot Y \quad \text{(Equation 1)} \]

3. **Final Condition After Moving \( m_1 \)**:
- When \( m_1 \) is pushed towards the CM by a distance \( d \), its new position becomes \( X - d \).
- Let \( m_2 \) move a distance \( D \) towards the CM, making its new position \( Y - D \).

4. **Setting Up the New Condition**:
- The new condition for the CM after moving the masses is:
\[ m_1 \cdot (X - d) = m_2 \cdot (Y - D) \quad \text{(Equation 2)} \]

5. **Expanding Equation 2**:
- Expanding Equation 2 gives:
\[ m_1 \cdot X - m_1 \cdot d = m_2 \cdot Y - m_2 \cdot D \]

6. **Substituting Equation 1 into Equation 2**:
- From Equation 1, we know that \( m_1 \cdot X = m_2 \cdot Y \). Substitute this into the expanded Equation 2:
\[ m_2 \cdot Y - m_1 \cdot d = m_2 \cdot Y - m_2 \cdot D \]
- This simplifies to:
\[ -m_1 \cdot d = -m_2 \cdot D \]

7. **Solving for \( D \)**:
- Rearranging gives:
\[ m_1 \cdot d = m_2 \cdot D \]
- Thus, we can solve for \( D \):
\[ D = \frac{m_1 \cdot d}{m_2} \]

### Final Answer:
The distance that the particle of mass \( m_2 \) moves to keep the center of mass at the original position is:
\[ D = \frac{m_1 \cdot d}{m_2} \]