question_answer

A block of mass m lying on a rough horizontal plane is acted upon by a horizontal force\[P\]and another force Q is inclined at an angle\[\theta \]to the vertical. The block will remain in equilibrium, if the coefficient of friction between it and the surface is

A) \[\frac{(P+Q\,\sin \theta )}{(mg+Q\cos \theta )}\] 

B) \[\frac{(P\cos \theta +Q)}{(mg-Q\sin \theta )}\]

C) \[\frac{(P+Q\cos \theta )}{(mg+Q\sin \theta )}\]   

D) \[\frac{(P\sin \theta -Q)}{(mg-Q\cos \theta )}\]

Correct Answer: A

Solution :

The free body diagram of the block for critical condition is shown below.

\[F=\mu R\Rightarrow P+Q\,\sin \theta =\mu \,(mg+Q\,\cos \theta )\]

\[\therefore \]      \[\mu =\frac{P+Q\,\sin \theta }{mg+Q\,\cos \theta }\]