Question

The position vectors of two identical particles with respect to the origin in three dimensional coordinate system are \( \vec{r}_1 \) and \( \vec{r}_2 \). The position vector of centre of mass of the system is given by

• A. $\vec{r}_1 + \vec{r}_2$
• B. $\frac{\vec{r}_1 - \vec{r}_2}{2}$
• C. $\vec{r}_1 - \vec{r}_2$
• D. $\frac{\vec{r}_1 + \vec{r}_2}{2}$
• E. $\frac{\vec{r}_1 + \vec{r}_2}{3}$

Hint:

For identical masses, COM is simply the average of their position vectors.

Answer 1

The Correct Option is D

Concept: The position vector of the centre of mass (COM) for a system of particles is given by: \[ \vec{R} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \] For two particles: \[ \vec{R} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2} \]

Step 1: Given particles are identical: \[ m_1 = m_2 = m \]

Step 2: Substitute: \[ \vec{R} = \frac{m\vec{r}_1 + m\vec{r}_2}{2m} \] \[ = \frac{\vec{r}_1 + \vec{r}_2}{2} \]

Step 3: Interpretation: Centre of mass lies at the midpoint of the line joining the two particles. Final Answer: \[ \vec{R} = \frac{\vec{r}_1 + \vec{r}_2}{2} \]