Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3   m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be x…

Question

Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually perpendicular sides equal to 3 &#160; m each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be x &#160; m . The value of x is

Aicharya · Accepted Answer

The diagram represents the locations of the masses as mentioned in the question.
For point $1$ , $\vec{r_{1}} = 0 \hat{i} + 0 \hat{j}$ for point $2$ , $\vec{r_{2}} = 3 \hat{i} + 0 \hat{j}$ and for point $3$ , $\vec{r_{3}} = 0 \hat{i} + 3 \hat{j}$ .
The formula for centre of mass is
${\vec{r}}_{c o m} = \frac{m_{1} \vec{r_{1}} + m_{2} \vec{r_{2}} + m_{3} \vec{r_{3}}}{m_{1} + m_{2} + m_{3}}$
$\Rightarrow {\vec{r}}_{com} = \frac{M (0 \hat{i} + 0 \hat{j}) + M (3 \hat{i}) + M (3 \hat{j})}{3 M}$
$\Rightarrow {\vec{r}}_{com} = \hat{i} + \hat{j}$
$|{\vec{r}}_{com}| = \sqrt{1^{2} + 1^{2}} = \sqrt{2} = \sqrt{x}$
$\Rightarrow x = 2$

Three identical spheres each of mass M are placed at the corners of a right angled triangle with mutually…

Solution