question_answer

A horizontal force F is applied at the top of an equilateral triangular block having mass m. The minimum coefficient of friction required to topple the block before translation will be

A) \[\frac{2}{\sqrt{3}}\]                 

B)   \[\frac{1}{3}\]

C) \[\frac{1}{\sqrt{3}}\]                             

D) \[\frac{1}{2}\]

Correct Answer: C

Solution :

About C, for toppling \[{{\tau }_{F}}>{{\tau }_{mg}}\] \[Fa\,\cos \,\,30{}^\circ \,\,>\,\,mg\frac{a}{2}\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\,F>\frac{mg}{\sqrt{3}}\] \[If\,\,no\,\,translation,\,\,then\,\,\,{{f}_{s}}\,=\,\,F>\frac{mg}{\sqrt{3}}\] \[but\,\,{{f}_{s}}\le \,\,\mu mg\] \[\frac{mg}{\sqrt{3}}\,\,\le \,\,\mu \,mg\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\mu \ge \frac{1}{\sqrt{3}}\]