A block of mass m is lying on a horizontal surface. The coefficient of friction between block and the surface is $\mathrm{\mu }$ . The minimum possible force needed to move the block on the surface is

$N=mg-F\sin \theta$

$F\cos \theta =f=\mu N$

$F\cos \theta =\mu mg-\mu F\sin \theta$

$F(\cos \theta +\mu \sin \theta )=\mu mg$

$F=\mu mg/(\cos \theta +\mu \sin \theta )$

$\mu =\tan \varphi \varphi -angleoffriction$

$F=\frac{mg\sin \varphi /\cos \varphi }{\cos \theta +\frac{\sin \varphi \sin \theta }{\cos \varphi }}$

$F=\frac{mg\sin \varphi }{\cos \theta \cos \varphi +\sin \theta \sin \varphi }$

$F=\frac{mg\sin \varphi }{\cos (\theta -\varphi )}$

F to be minimum if $\cos (\theta -\varphi )=1\Rightarrow \theta -\varphi =0$

$\theta =\varphi$

$\tan \varphi =\tan \theta =\mu$

$F_{\min }=mg\sin \theta$

$F_{\min }=\frac{\mu mg}{\sqrt{1+{\mu }^2}}$