Question

An inclined plane has an inclination $ \theta $ with horizontal. A body of mass m rests on it. If the coefficient of friction between the body and the plane is $ \mu , $ then the minimum force that needs to be applied parallel to the inclined plane is

• A. $m g\, \sin\, \theta$
• B. $ \mu \,mg\,\cos \,\theta $
• C. $ \mu \,mg\,\cos \,\theta +mg\,\sin \,\theta $
• D. $ \mu\, mg\,\cos \,\theta -mg\,\sin \,\theta $

Answer 1

Answer: (D)

The free body diagram showing the various forces acting on the block are as follows
The frictional force $\left(F_{k}\right)$ acts opposite to direction of motion of block, given by
$F_{k}=\mu R$
where $R$ is reaction of the plane on block.
$R=m g \cos \theta$
$ F_{k}=\mu m g \cos \theta$
Also, downward force is
$F'=m g \sin \theta$
The resultant of $F'$ and $F$ provides the necessary acceleration to the block.
$\therefore $ Resultant force upwards
$\mu m g \cos \theta-m g \sin \theta$