The centre of mass of a system of two particle of masses `m_1 and m_2` is at a distance `d_1` from `m_1` and at a distance `d_2` from mass `m_2` such that.

Options

A. `(d_1)/(d_2) = (m_2)/(m_1)`
B. `(d_1)/(d_2) = (m_1)/(m_2)`
C. `(d_1)/(d_2) = (m_1)/(m_1 + m_2)`
D. `(d_1)/(d_2) = (m_2)/(m_1 + m_2)`

Explanation

To solve the problem of finding the relationship between the distances \( d_1 \) and \( d_2 \) from the center of mass of two particles with masses \( m_1 \) and \( m_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two particles with masses \( m_1 \) and \( m_2 \).
- The center of mass (CM) of the system is located at a distance \( d_1 \) from mass \( m_1 \) and \( d_2 \) from mass \( m_2 \).

2. **Setting Up the Coordinate System**:
- We can place the center of mass at the origin of a coordinate system.
- Let’s assume the position of \( m_1 \) is at \( -d_1 \) and the position of \( m_2 \) is at \( d_2 \).

3. **Using the Formula for Center of Mass**:
- The formula for the center of mass \( x_{cm} \) for two particles is given by:
\[ x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \]
- Here, \( x_1 = -d_1 \) (for \( m_1 \)) and \( x_2 = d_2 \) (for \( m_2 \)).

4. **Substituting Values into the Formula**:
- Substituting the positions into the center of mass formula gives:
\[ 0 = \frac{m_1 \cdot (-d_1) + m_2 \cdot d_2}{m_1 + m_2} \]

5. **Simplifying the Equation**:
- Rearranging the equation, we get:
\[ m_1 \cdot (-d_1) + m_2 \cdot d_2 = 0 \]
- This simplifies to:
\[ m_1 \cdot d_1 = m_2 \cdot d_2 \]

6. **Finding the Relationship Between \( d_1 \) and \( d_2 \)**:
- Dividing both sides by \( m_1 \cdot m_2 \):
\[ \frac{d_1}{d_2} = \frac{m_2}{m_1} \]

### Final Answer:
The relationship between the distances \( d_1 \) and \( d_2 \) is given by:
\[ \frac{d_1}{d_2} = \frac{m_2}{m_1} \]