If `vec(R)_(CM)` is the position of the centre of mass of a system of two particles of masses `m_(1)` and `m_(2)` then `vec(R)_(CM)` is given by :

Options

A. `(m_(1)+m_(2))/(m_(1)vec(r)_(1)+m_(2)vecr_(2))`
B. `(m_(1)m_(2))/(m_(1)vec(r)_(1)+m_(2)vec(r)_(2))`
C. ` (m_(1)vec(r)_(1)+m_(2)vec(r)_(2))/(m_(1)+m_(2))`
D. `(m_(1)vec(r)_(1)+m_(2)vec(r)_(2))/(m_(1)m_(2))`

Answer

To find the position of the center of mass (\( \vec{R}_{CM} \)) of a system of two particles with masses \( m_1 \) and \( m_2 \), we can use the following formula:

### Step-by-Step Solution:

1. **Understanding the Position Vectors**:
- Let \( \vec{R}_1 \) be the position vector of the first particle (mass \( m_1 \)).
- Let \( \vec{R}_2 \) be the position vector of the second particle (mass \( m_2 \)).

2. **Formula for Center of Mass**:
- The position of the center of mass for a system of two particles is given by the formula:
\[ \vec{R}_{CM} = \frac{m_1 \vec{R}_1 + m_2 \vec{R}_2}{m_1 + m_2} \]

3. **Substituting Values**:
- If we substitute the values of \( m_1 \), \( m_2 \), \( \vec{R}_1 \), and \( \vec{R}_2 \) into the formula, we can calculate \( \vec{R}_{CM} \).

4. **Final Expression**:
- Thus, the final expression for the center of mass of the two-particle system is:
\[ \vec{R}_{CM} = \frac{m_1 \vec{R}_1 + m_2 \vec{R}_2}{m_1 + m_2} \]