Question

Two bodies $A$ and $B$ of same mass are placed at position ${\overrightarrow{r}}_A=\hat{i}+2\hat{j}+\hat{k}$ and ${\overrightarrow{r}}_B=2\hat{i}+2\hat{k}$ respectively. If the bodies are displaced from their initial position to ${\overrightarrow{r}}_{A^{\prime }}=2\hat{i}+4\hat{j}+\hat{k}$ and ${\overrightarrow{r}}_{B^{\prime }}=3\hat{i}+4\hat{j}+2\hat{k}$, find the magnitude of displacement of COM of the system.

• A. $-\sqrt{18}$ units
• B. $-\sqrt{16}$ units
• C. $\sqrt{16}$ units
• D. $\sqrt{10}$ units

Answer 1

The correct option is D $\sqrt{10}$ units

Displacement of COM, $\Delta {\overrightarrow{r}}_{com}=\frac{m_A\Delta {\overrightarrow{r}}_A+m_B\Delta {\overrightarrow{r}}_B}{m_A+m_B}$
$\Delta {\overrightarrow{r}}_A={\overrightarrow{r}}_{A^{\prime }}-{\overrightarrow{r}}_A=(2\hat{i}+4\hat{j}+\hat{k})-(\hat{i}+2\hat{j}+\hat{k})$
$⟹\Delta {\overrightarrow{r}}_A=(\hat{i}+2\hat{j})$
and
$\Delta {\overrightarrow{r}}_B={\overrightarrow{r}}_{B^{\prime }}-{\overrightarrow{r}}_B=(3\hat{i}+4\hat{j}+2\hat{k})-(2\hat{i}+2\hat{k})$
$⟹\Delta {\overrightarrow{r}}_B=(\hat{i}+4\hat{j})$
Let $m_A=m_B=m$
$\Delta {\overrightarrow{r}}_{com}=\frac{m.(\hat{i}+2\hat{j})+m(\hat{i}+4\hat{j})}{m+m}$
$=\frac{m}{2m}(2\hat{i}+6\hat{j})$
$⟹\Delta {\overrightarrow{r}}_{com}=\hat{i}+3\hat{j}$
$|\Delta {\overrightarrow{r}}_{com}|=\sqrt{1^2+3^2}=\sqrt{10}$ units