Consider a sytem 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_2d`
B. `d`
C. `m_2/m_1`
D. `m_1/(m_1+m_2)d`

Explanation

To solve the problem, we need to determine how far the second particle (mass \( m_2 \)) moves when the first particle (mass \( m_1 \)) is pushed towards the center of mass by a distance \( d \), while keeping the center of mass of the system unchanged.

### Step-by-Step Solution:

1. **Understanding the Center of Mass (COM):**
The center of mass \( R \) of a two-particle system is given by the formula:
\[ R = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
where \( x_1 \) and \( x_2 \) are the positions of masses \( m_1 \) and \( m_2 \) respectively.

2. **Initial Position of the Center of Mass:**
Let the initial positions of the masses be \( x_1 \) and \( x_2 \). The initial position of the center of mass is:
\[ R_{initial} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

3. **Displacement of Mass \( m_1 \):**
When mass \( m_1 \) is pushed towards the center of mass by a distance \( d \), its new position becomes:
\[ x_1' = x_1 - d \]

4. **New Position of the Center of Mass:**
The new position of the center of mass after the displacement of \( m_1 \) becomes:
\[ R_{new} = \frac{m_1 (x_1 - d) + m_2 x_2}{m_1 + m_2} \]

5. **Condition for the Center of Mass to Remain at the Original Position:**
For the center of mass to remain at the original position, we set \( R_{new} = R_{initial} \):
\[ \frac{m_1 (x_1 - d) + m_2 x_2}{m_1 + m_2} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

6. **Simplifying the Equation:**
By multiplying both sides by \( m_1 + m_2 \) and simplifying, we get:
\[ m_1 (x_1 - d) + m_2 x_2 = m_1 x_1 + m_2 x_2 \]
This simplifies to:
\[ -m_1 d = 0 \]
which leads to:
\[ m_1 d = m_2 x_2' \]
where \( x_2' \) is the distance moved by mass \( m_2 \).

7. **Finding the Displacement of Mass \( m_2 \):**
Rearranging gives:
\[ x_2' = -\frac{m_1}{m_2} d \]
The negative sign indicates that \( m_2 \) moves towards the center of mass.

### Final Answer:
The distance that mass \( m_2 \) moves to keep the center of mass at the original position is:
\[ x_2' = \frac{m_1}{m_2} d \]