The center of mass of a system of two particles divides the distance between them.

Options

A. in inverse ratio of square of masses of particles.
B. in direct ratio of square of masses of particles.
C. in inverse ratio of masses of particles.
D. in direct ratio of masses of particles.

Explanation

To solve the problem of how the center of mass of a system of two particles divides the distance between them, we can follow these steps:

### Step-by-Step Solution:

1. **Define the System**:
Let's consider two particles with masses \( m_1 \) and \( m_2 \) located on the x-axis. We denote the distance between them as \( r \), where \( r_1 \) is the distance from the center of mass to mass \( m_1 \) and \( r_2 \) is the distance from the center of mass to mass \( m_2 \).

2. **Positioning the Particles**:
We can place particle \( m_1 \) at position \( -r_1 \) and particle \( m_2 \) at position \( r_2 \) on the x-axis. The center of mass (CM) position can be denoted as \( x_{CM} \).

3. **Center of Mass Formula**:
The formula for the center of mass for two particles is given by:
\[ x_{CM} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \]
Substituting \( x_1 = -r_1 \) and \( x_2 = r_2 \), we have:
\[ x_{CM} = \frac{m_1 \cdot (-r_1) + m_2 \cdot r_2}{m_1 + m_2} \]

4. **Setting the Center of Mass to Zero**:
Since we want to find the ratio in which the center of mass divides the distance between the two particles, we can set \( x_{CM} = 0 \):
\[ 0 = \frac{-m_1 \cdot r_1 + m_2 \cdot r_2}{m_1 + m_2} \]

5. **Solving for the Distances**:
From the equation above, we can rearrange it to find:
\[ m_1 \cdot r_1 = m_2 \cdot r_2 \]

6. **Finding the Ratio**:
Dividing both sides by \( r_2 \) and \( m_1 \):
\[ \frac{r_1}{r_2} = \frac{m_2}{m_1} \]
This shows that the distances \( r_1 \) and \( r_2 \) are inversely proportional to the masses \( m_1 \) and \( m_2 \).

7. **Conclusion**:
Therefore, the center of mass divides the distance between the two particles in the ratio of the masses:
\[ r_1 : r_2 = m_2 : m_1 \]

### Final Answer:
The center of mass divides the distance between the two particles in the inverse ratio of their masses.