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Q.

Two bodies of mass 1 kg and 3 kg have position vectors $\hat{i} + 2 \hat{j} + \hat{k}$ and $- 3 \hat{i} - 2 \hat{j} + \hat{k}$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector.

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a

$- 2 \hat{i} + 2 \hat{k}$

b

$- 3 \hat{i} - 2 \hat{j} + \hat{k}$

c

$\hat{i} - 2 \hat{j} + \hat{k}$

d

$- 2 \hat{i} - \hat{j} + 2 \hat{k}$

answer is A.

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Detailed Solution

$\begin{array}{l} \begin{array}{l} {\vec{r}}_{com} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2}}{m_{1} + m_{2}} = \frac{1 (\hat{i} + 2 \hat{j} + \hat{k}) + 3 (- 3 \hat{i} - 2 \hat{j} + \hat{k})}{1 + 3} \\ = - 2 \hat{i} - \hat{j} + \hat{k} \end{array} \\ | 2 \hat{i} - \hat{j} + \hat{k} | = \sqrt{(2)^{2} + (1)^{2} + (1)^{2}} = \sqrt{6} \end{array}$

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