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(D) Let the three spheres be located at coordinates $(0, 0)$,$(2, 0)$,and $(0, 2)$ in the $xy$-plane.Since all spheres have the same mass $M$,the coor

Vedclass · Accepted Answer

$\frac{2}{3}(\hat{i}+\hat{j})$

Vedclass · Answer

(D) Let the three spheres be located at coordinates $(0, 0)$,$(2, 0)$,and $(0, 2)$ in the $xy$-plane.Since all spheres have the same mass $M$,the coordinates of the center of mass $(x_{cm}, y_{cm})$ are given by:$x_{cm} = \frac{M(0) + M(2) + M(0)}{M + M + M} = \frac{2M}{3M} = \frac{2}{3} \; m$$y_{cm} = \frac{M(0) + M(0) + M(2)}{M + M + M} = \frac{2M}{3M} = \frac{2}{3} \; m$Therefore,the position vector of the center of mass is $\vec{r}_{cm} = x_{cm}\hat{i} + y_{cm}\hat{j} = \frac{2}{3}(\hat{i} + \hat{j}) \; m$.

Three identical spheres,each of mass $M$,are placed at the corners of a right-angled triangle with mutually perpendicular sides equal to $2 \; m$ (see figure). Taking the point of intersection of the two mutually perpendicular sides as the origin,find the position vector of the center of mass.

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