Question

A block of mass m lying on a horizontal surface (coefficient of static friction = ${\mu }_s$) is to be brought into motion by a pulling force F. At what angle $\theta$ with the horizontal should the force F be applied so that its magnitude is minimum? Also find this minimum magnitude.

Answer 1

Let us first calculate the force F required to bring m into motion in terms of angle $\theta$. Equation using laws of motion are as follows:
$N=mg-Fsin\theta$
and $Fcos\theta ={\mu }_{s}N$
$Fcos\theta =\mu (mg-Fsin\theta )$

$\Rightarrow F=\frac{\mu mg}{cos\theta +{\mu }_{s}sin\theta }$

We have to find the angle $\theta$ for which this force F is minimum.

Substituting ${\mu }_s=tan\varphi$ ( for simplification), we get

$F=\frac{mgtan\varphi }{cos\theta +tan\varphi sin\theta }=\frac{mgsin\varphi }{cos(\theta -\varphi )}$

F is minimum if $cos(\theta -\varphi )$ is maximum. Hence, F is minimum for $\theta =\varphi =ta{n}^{-1}{\mu }_s$ and $F_{min}=mgsin\varphi$.

To bring m into motion with least effort, force should be applied at an angle $ta{n}^{-1}{\mu }_s$ and should have a magnitude equal to
$F_{min}=mgsin\varphi =\frac{{\mu }_{s}mg}{\sqrt{1+{\mu }_s^2}}$