Question

Two bodies of masses $2\text{kg}$ and $4\text{kg}$ have position vector $(\hat{i}+2\hat{j})$ and $\left(2\hat{i}\right)$ respectively. Find the position vector for the centre of mass of the system.

• A. $2\hat{i}+5\hat{j}$
• B. $4\hat{i}+10\hat{j}$
• C. $\frac{(4\hat{i}+10\hat{j})}{6}$
• D. $\left(\frac{10\hat{i}+4\hat{j}}{6}\right)$

Answer 1

The correct option is D $\left(\frac{10\hat{i}+4\hat{j}}{6}\right)$

$m_1=2\text{kg},m_2=4\text{kg}$

Let the position vectors of masses $m_1$ and $m_2$ be $\overrightarrow{r_1}$, $\overrightarrow{r_2}$ respectively.
$\overrightarrow{r_1}=\hat{i}+2\hat{j}$ , $\overrightarrow{r_2}=2\hat{i}$

Applying formula for position vector of COM of system:

${\overrightarrow{r}}_{\text{COM}}=\frac{m_1\overrightarrow{r_1}+m_2\overrightarrow{r_2}}{m_1+m_2}=\frac{2(\hat{i}+2\hat{j})+4\left(2\hat{i}\right)}{2+4}$
$\therefore {\overrightarrow{r}}_{\text{COM}}=\left(\frac{10\hat{i}+4\hat{j}}{6}\right)$