Question

Two bodies of masses $2\text{kg}$ & $5\text{kg}$ have position vectors $\hat{i}-2\hat{j}+\hat{k}$ and $3\hat{i}+2\hat{j}-\hat{k}$ respectively. The centre of mass of the system has a position vector

• A. $2\hat{i}+4\hat{j}-3\hat{k}$
• B. $\left(\frac{4\hat{i}+5\hat{j}-3\hat{k}}{7}\right)$
• C. $17\hat{i}+6\hat{j}-3\hat{k}$
• D. $\left(\frac{17\hat{i}+6\hat{j}-3\hat{k}}{7}\right)$

Answer 1

The correct option is D $\left(\frac{17\hat{i}+6\hat{j}-3\hat{k}}{7}\right)$

Position vector of centre of mass for descrete particles, $\overrightarrow{r_{\text{com}}}=\frac{\sum m_i{\overrightarrow{r}}_i}{\sum m_i}=\frac{m_1\overrightarrow{r_1}+m_2\overrightarrow{r_2}}{m_1+m_2}$
(Given, $m_1=2\text{kg},\overrightarrow{r_1}=\overrightarrow{i}-2\hat{j}+\hat{k}$
$m_2=5\text{kg},\overrightarrow{r_2}=3\hat{i}+2\hat{j}-\hat{k}$)
$\because$$\overrightarrow{r_{\text{com}}}=\frac{m_1\overrightarrow{r_1}+m_2\overrightarrow{r_2}}{m_1+m_2}$
$=\frac{2(\hat{i}-2\hat{j}+\hat{k})+5(3\hat{i}+2\hat{j}-\hat{k})}{2+5}$
$=\left(\frac{17\hat{i}+6\hat{j}-3\hat{k}}{7}\right)$