Question

A block rests on a fixed wedge inclined at an angle $\theta$. The coefficient of friction between the block and plane is $\mu$. The maximum value of $\theta$ for the block to remain motionless on the wedge is

• A. $\mu =\tan \theta$
• B. $\mu =\sin \theta$
• C. $\mu =\cos \theta$
• D. $\mu =\cot \theta$

Answer 1

Answer: (A)

The given condition can be shown with a proper $FBD$ of the block, as below

Let us suppose the mass of the block is $m$ and acceleration due to gravity is $g$.
Normal reaction, $N=mg\cos \theta$
Limiting friction, $(fr{)}_{\lim }={\mu }_{s}N=\mu mg\cos \theta$
Net driving force, $(F{)}_{\text{net driving }}=mg\sin \theta$
If the block remain motionless on the wedge, then
$(F{)}_{\text{net driving }}\le (fr{)}_{\text{lim }}$
$\Rightarrow mg\sin \theta \le \mu mg\cos \theta$
$\sin \theta \le \mu \cos \theta$
$\Rightarrow \tan \theta \le \mu$
$\Rightarrow \tan {\theta }_{\max }=\mu$