Question

The minimum force required to move a body of weight $w$ placed on a rough horizontal plane surface is

• A. $w\sin \lambda$
• B. $w\cos \lambda$
• C. $w\tan \lambda$
• D. $w\cot \lambda$

Answer 1

The correct option is A $w\sin \lambda$

We have, $P\cos \theta =\mu R$
$R+P\sin \theta =w$
$\Rightarrow P\cos \theta =\mu (w-P\sin \theta )$
$\Rightarrow P(\cos \theta +\mu \sin \theta )=\mu w$
$\Rightarrow P=\frac{\mu w}{\cos \theta +\mu \sin \theta }$
$=\frac{w\tan \lambda }{\cos \theta +\tan \lambda \sin \theta }$ $(\because \mu =\tan \lambda )$
$=\frac{w\sin \lambda }{\cos \theta \cos \lambda +\sin \theta \sin \lambda }$
$=\frac{w\sin \lambda }{\cos (\theta -\lambda )}$
$P$ is minimum, when $\cos (\theta -\lambda )$ is maximum i.e., $\cos (\theta -\lambda )=1$.
Hence, minimum $P=w\sin \lambda$.