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Question

Answers

$ \begin{align}

& \text{A}\text{. ta}{{\text{n}}^{-1}}\mu \\

& \text{B}\text{. }2\theta \\

& \text{C}\text{. }\theta \\

& \text{D}\text{. }\dfrac{\theta }{2} \\

\end{align} $

Answer
Verified

Formula used:

Relation between frictional force and normal reaction,

$ f=\mu N $

Motion of a block on an inclined plane is the interplay of different force types and the characterising features of the inclined plane. An inclined plane is a surface whose one end is raised. The raised surface forms angle $ \theta $ with the horizontal. The block placed on the inclined surface is acted upon by the force of gravity and contact forces, normal force and frictional force.

Forces acting on the block of mass $ m $ are:

Gravitational force $ mg $ in downward direction,

Components of the gravitational force will be:

$ mg\cos \theta $ in the direction parallel to the inclined plane and $ mg\sin \theta $ in the direction perpendicular to the inclined plane

Frictional force $ f $ in the direction opposite to the tending motion of the block

Force-body diagram (FBD) of block:

As we are given that the system is at rest,

It means that all the forces on the block are balancing each other.

In direction perpendicular to the inclined plane, we have Normal force $ N $ and component of gravitational force $ mg\cos \theta $

These two forces will balance each other,

$ N=mg\cos \theta $

In direction perpendicular to the inclined plane, we have frictional force $ f $ and component of gravitational force $ mg\sin \theta $

These two forces will balance each other,

$ f=mg\sin \theta $

We have,

$ f=\mu N $

Where,

$ f $ is the frictional force

$ \mu $ is the coefficient of friction

$ N $ is the normal force

And, $ N=mg\cos \theta $

$ \begin{align}

& \mu N=mg\sin \theta \\

& \mu mg\cos \theta =mg\sin \theta \\

& \mu =\dfrac{\sin \theta }{\cos \theta } \\

& \mu =\tan \theta \\

& \theta ={{\tan }^{-1}}\mu \\

\end{align} $

The angle of friction between the block and the wedge is $ {{\tan }^{-1}}\mu $

Hence, the correct option is A.

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