Question

A block of mass m is lying a horizontal surface of coefficient of friction $\mu$. A force F is applied to the block at an angle $\theta$ with the horizontal. The block will move with a minimum force F if

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

Answer 1

The correct option is A

$\mu =tan\theta$

The applied force F can be resolved into two components: $Fsin\theta$ in vertical and $Fcos\theta$ in horizontal.
Now, $R+Fsin\theta =mg\Rightarrow R=mg-Fsin\theta$
Frictional force, $f=\mu R=\mu (mg-Fsin\theta )\Rightarrow Fcos\theta =\mu (mg-Fsin\theta )\Rightarrow F=\frac{\mu mg}{(\mu sin\theta +cos\theta )}$
F will be minimum if the denominator is maximum, i.e. if $\frac{d}{d\theta }(\mu sin\theta +cos\theta )=0\Rightarrow \mu cos\theta -sin\theta =0\Rightarrow \mu =tan\theta$
Hence, the correct choice is (a).