The identical spheres, each of mass 2M, are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to 4 m each. Taking the point of intersection of these two sides as the origin, the magnitude of the position vector of th

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Solution

Q.
The identical spheres, each of mass 2M, are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to 4 m each. Taking the point of intersection of these two sides as the origin, the magnitude of the position vector of the center of mass of the system is $\frac{4 \sqrt{2}}{x}$ , where the value of $x$ is ______. [2024]

Ans.
(3)

$Position vector {\vec{r}}_{C O M} = \frac{m_{1} {\vec{r}}_{1} + m_{2} {\vec{r}}_{2} + m_{3} {\vec{r}}_{3}}{m_{1} + m_{2} + m_{3}}$

${\vec{r}}_{C O M} = \frac{2 M \times 0 + 2 M \times 4 \hat{i} + 2 M \times 4 \hat{j}}{6 M}$

$\vec{r} = \frac{4}{3} \hat{i} + \frac{4}{3} \hat{j}$

$| \vec{r} |$ $= \sqrt{{(\frac{4}{3})}^{2} + {(\frac{4}{3})}^{2}} = \frac{4}{3} \sqrt{2} = \frac{4 \sqrt{2}}{x}$

$x = 3$

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