question_answer

The force required to just move a body up an inclined plane is double the force required to just prevent it from sliding down. If \[\theta \] is the angle of friction and \[\phi \] is the angle which the plane makes with horizontal, then:

A) \[\tan \phi  = 2 tan\theta \]    

B)   \[\tan \phi  = 3 tan\theta \]

C) \[\tan \phi  = tan\theta \]            

D) \[\tan \theta  = 3tan\phi \]

Correct Answer: B

Solution :

Force required to just move up the body = \[{{\operatorname{F}}_{up}}=mg\,Sin\theta +\mu mg\,Cos\theta \] Similary \[{{\operatorname{F}}_{\operatorname{down}}}=mg\,Sin\theta -\mu mg\,Cos\theta \] \[{{\operatorname{F}}_{\operatorname{up}}}=2{{\operatorname{F}}_{\operatorname{down}}}\] \[mg\,Sin\phi +\mu mg\,Cos\phi =2\left( mg\,Sin\phi -\mu mg\,Cos\phi  \right)\]\[\tan \phi =3\mu \] As we known \[\mu =\tan \theta \] \[\tan \phi =3\tan \theta \]