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. $mg\sin \theta$
• 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.
$R=mg\cos \theta$
$F_k=\mu mg\cos \theta$
Also, downward force is
$F^{\prime }=mg\sin \theta$
The resultant of $F^{\prime }$ and $F$ provides the necessary acceleration to the block.
$\therefore$ Resultant force upwards
$\mu mg\cos \theta -mg\sin \theta$