question_answer

Three identical spheres of mass M each are placed at the corners 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) \[\frac{{\hat{i}}}{\sqrt{3}}+\hat{j}\]

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

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

Correct Answer: D

Solution :

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