question_answer

Three identical spheres of radius r and mass m touch each other (as shown). If the centre of mass of one of them is (0,0) then the centre of mass of the system is

A)  \[r,r/2\]

B)  \[r,r/3\]

C)  \[\frac{\sqrt{{}}3}{2}r,r\]

D)  \[r,r\sqrt{{}}3\]

Correct Answer: D

Solution :

: Let A be origin (0, 0) Coordinates of\[B=(2r,\text{ }0)\] Coordinates of \[C=(r,\sqrt{3}r)\] \[\overline{x}=\frac{{{m}_{1}}{{x}_{1}}+{{m}_{2}}{{x}_{2}}+{{m}_{3}}{{x}_{3}}}{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}}\] \[=\frac{0+(m\times 2r)+(m\times r)}{m+m+m}=\frac{3mr}{3m}=r\] \[\overline{y}=\frac{{{m}_{1}}{{y}_{1}}+{{m}_{2}}{{y}_{2}}+{{m}_{3}}{{y}_{3}}}{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}}=\frac{0+0+m\times \sqrt{3}r}{m+m+m}\] \[=\frac{\sqrt{3}mr}{3m}=\frac{r}{\sqrt{3}}\] \[\therefore \]Centre of mass\[=(r,r/\sqrt{3})\].