Consider a system of two particles having masses 2kg and 5 kg the particle of mass 2 kg is pushed towards the centre of mass of particles through a distance 5 m,by what distance would particle of mass 5kg move so as to keep the centre of mass of particles at the original position?

Options

A. 2 m
B. 12.5m
C. 8m
D. 5m

Answer

To solve the problem, we need to determine how far the 5 kg particle moves when the 2 kg particle is pushed towards the center of mass by 5 m, in order to keep the center of mass of the system unchanged.

### Step-by-Step Solution:

1. **Identify the masses and movements**:
- Let \( m_1 = 2 \, \text{kg} \) (mass of the first particle).
- Let \( m_2 = 5 \, \text{kg} \) (mass of the second particle).
- The 2 kg particle moves a distance \( d_1 = 5 \, \text{m} \) towards the center of mass.

2. **Use the center of mass formula**:
The position of the center of mass (COM) for two particles is given by:
\[ R_{com} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2} \]
where \( r_1 \) and \( r_2 \) are the positions of the masses.

3. **Differentiate the COM position**:
Since the center of mass is to remain in the same position, we differentiate the COM equation with respect to time:
\[ \frac{dR_{com}}{dt} = 0 \]
This leads to:
\[ m_1 \frac{dr_1}{dt} + m_2 \frac{dr_2}{dt} = 0 \]
where \( dr_1 \) is the change in position of the 2 kg mass and \( dr_2 \) is the change in position of the 5 kg mass.

4. **Substituting known values**:
- The 2 kg mass moves towards the COM, so \( dr_1 = -5 \, \text{m} \) (negative because it's moving towards the center).
- We need to find \( dr_2 \).

Substituting the values into the differentiated equation:
\[ 2 \times (-5) + 5 \times dr_2 = 0 \]

5. **Solving for \( dr_2 \)**:
\[ -10 + 5 \times dr_2 = 0 \]
\[ 5 \times dr_2 = 10 \]
\[ dr_2 = \frac{10}{5} = 2 \, \text{m} \]

6. **Interpreting the result**:
The positive value of \( dr_2 \) indicates that the 5 kg mass moves 2 m in the opposite direction to the movement of the 2 kg mass.

### Final Answer:
The particle of mass 5 kg would move **2 meters** in the opposite direction to keep the center of mass at the original position.