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Two bodies of mass $1 \mathrm{~kg}$ and $3 \mathrm{~kg}$ 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 this system has a position vector
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Verified Answer
The correct answer is:
$-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$
The position vector of centre of mass
$$
\begin{aligned}
\vec{r} & =\frac{m_1 \overrightarrow{r_1}+m_2 \vec{r}_2}{m_1+m_2} \\
& =\frac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3} \\
& =\frac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k}) \\
& =-2 \hat{i}-\hat{j}+\hat{k}
\end{aligned}
$$
The centre of mass changes its position only under the translatory motion. There is no effect of rotatory motion on centre of mass of the body.
$$
\begin{aligned}
\vec{r} & =\frac{m_1 \overrightarrow{r_1}+m_2 \vec{r}_2}{m_1+m_2} \\
& =\frac{1(\hat{i}+2 \hat{j}+\hat{k})+3(-3 \hat{i}-2 \hat{j}+\hat{k})}{1+3} \\
& =\frac{1}{4}(-8 \hat{i}-4 \hat{j}+4 \hat{k}) \\
& =-2 \hat{i}-\hat{j}+\hat{k}
\end{aligned}
$$
The centre of mass changes its position only under the translatory motion. There is no effect of rotatory motion on centre of mass of the body.
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