question_answer

A block of mass m rests on a rough horizontal surface as shown in the figure. Coefficient of friction between the block and the surface is p. A force \[F=mg\] acting at angle \[\theta \] with the vertical side of the block pulls it. In which of the following cases can the block be pulled along the surface?

A) \[\tan \,\theta \,\,\ge \,\,\mu \,\]                

B) \[\cot \,\theta \,\,\ge \,\,\mu \,\]  

C) \[\tan \,\,\frac{\theta }{2}\,\,\ge \,\,\mu \,\]            

D) \[cot\,\,\frac{\theta }{2}\,\,\ge \,\,\mu \,\]

Correct Answer: D

Solution :

  \[N\,\,=\,\,Mg\,\,-\,\,F\,cos\,\theta \] \[N\,\,=\,\,Mg\left( 1-cos\,\theta  \right)\] \[f=\mu N\,\,=\mu Mg\,\left( 1-cos\,\theta  \right)\] To move the block \[f\,\,\le \,\,F\,sin\,\theta \] \[\mu Mg(1-cos\,\theta )\,\le \,Mgsin\theta \] \[\mu \,\,\le \,\,\frac{\sin \,\theta }{1-\,\cos \,\theta }\] \[\mu \,\,\le \,\,\frac{2\,\sin \,\frac{\theta }{2}\,\cos \,\frac{\theta }{2}}{2\,{{\sin }^{2}}\frac{\theta }{2}}\] \[\mu \,\,\le \,\,\cot \,\frac{\theta }{2}\]