The centre of mass of a system of two particle of masses `m_1 and m_2` is at a distance `d_1` from mass `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 regarding the center of mass of a system of two particles with masses \( m_1 \) and \( m_2 \), and their respective distances \( d_1 \) and \( d_2 \) from the center of mass, 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 certain point, which we will denote as the origin (0,0) for simplicity.

2. **Positioning the Masses**:
Let's place mass \( m_1 \) at position \( -d_1 \) on the x-axis and mass \( m_2 \) at position \( d_2 \) on the x-axis. The distances \( d_1 \) and \( d_2 \) are measured from the center of mass to each mass.

3. **Equation for Center of Mass**:
The formula for the center of mass of a two-particle system in one dimension is given by:
\[ x_{cm} = \frac{m_1 \cdot (-d_1) + m_2 \cdot d_2}{m_1 + m_2} \]
Since we have defined the center of mass to be at the origin, we set \( x_{cm} = 0 \).

4. **Setting Up the Equation**:
Substituting \( x_{cm} = 0 \) into the equation gives:
\[ 0 = \frac{m_1 \cdot (-d_1) + m_2 \cdot d_2}{m_1 + m_2} \]
This simplifies to:
\[ m_1 \cdot (-d_1) + m_2 \cdot d_2 = 0 \]

5. **Rearranging the Equation**:
Rearranging the above equation leads to:
\[ m_2 \cdot d_2 = m_1 \cdot d_1 \]

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

7. **Conclusion**:
Thus, the relationship between the masses and the distances from the center of mass is:
\[ \frac{m_2}{m_1} = \frac{d_1}{d_2} \]
This indicates that the ratio of the masses is equal to the ratio of their distances from the center of mass.