Question

The position vectors of three particles are (i + 2j + 3k), (5i - 2j + 2k) and (3i - 6j + 4k) and their masses are respectively 2 kg, 1 kg and 3 kg. The position vector of a fourth mass of 4 kg, so that the centre of mass of the system may be at the origin, is

• A. - 4 i + 4 j - 5 k
• B. 4 i - 4 j + 5 k
• C. - 4 i + 4 j + 5 k
• D. 4 i - 4 j - 5 k

Answer 1

The correct option is A

- 4 i + 4 j - 5 k

Given that ${\overrightarrow{r}}_{cm}=0$

$\frac{m_1{\overrightarrow{r}}_1+m_2{\overrightarrow{r}}_2+m_3{\overrightarrow{r}}_3+m_4{\overrightarrow{r}}_4}{m_1+m_2+m_3+m_4}={\overrightarrow{r}}_{cm}=0$

$\Rightarrow m_1{\overrightarrow{r}}_1+m_2{\overrightarrow{r}}_2+m_3{\overrightarrow{r}}_3+m_4{\overrightarrow{r}}_4=0$

$\Rightarrow 2(\hat{i}+2\hat{j}+3\hat{k})+1(5\hat{i}-2\hat{j}+2\hat{k})+3(3\hat{i}-6\hat{j}+4\hat{k})+4(x\hat{i}+y\hat{j}+z\hat{k}=0$

$\Rightarrow (16\hat{i}-16\hat{j}+20\hat{k})+4(x\hat{i}+y\hat{j}+z\hat{k}=0$

$\Rightarrow x\hat{i}+y\hat{j}+z\hat{k}=\frac{-(16\hat{i}-16\hat{j}+20\hat{k})}{4}$

$\Rightarrow {\overrightarrow{r}}_4=-4\hat{i}-4\hat{j}-5\hat{k}$