question_answer

Three idential spheresof mass M each are placed at the comers of an equilateral triangle of side \[2m\]. Taking one of the comer as the origin, the position vector of the centre of mass is

A)  \[\sqrt{3}\,\,(\hat{i}-\hat{j})\]

B)  \[\frac{i}{\sqrt{3}}+\hat{j}\]

C)  \[\frac{\hat{i}+\hat{j}}{3}\]

D)  \[\hat{i}+\frac{{\hat{j}}}{\sqrt{3}}\]

Correct Answer: D

Solution :

The x coordinate of centre of mass is \[\bar{x}=\frac{\Sigma {{m}_{i}}{{x}_{i}}}{\Sigma {{m}_{i}}}\] \[=\frac{m\times 0+m\times 1+m\times 2}{m+m+m}=1\] \[\bar{y}=\frac{\Sigma {{m}_{i}}{{y}_{i}}}{\Sigma {{m}_{i}}}\] \[=\frac{m\times 0+m(2\,\sin \,{{60}^{o}})+m\times 0}{m+m+m}\] \[\bar{y}=\frac{\sqrt{3}m}{3m}=\frac{1}{\sqrt{3}}\] Position vector of centre of mass is \[\left( \hat{i}+\frac{{\hat{j}}}{\sqrt{3}} \right).\]