A block of mass m lying on a rough horizontal plane is acted upon… - Vidyadip

MCQ

A block of mass $m$ lying on a rough horizontal plane is acted upon by a horizontal force $P$ and another force $Q$ inclined an at an angle $\theta $ to the vertical. The minimum value of coefficient of friction between the block and the surface for which the block will remain in equilibrium is

✓
$\frac{{P\, + \,Q\,\sin \,\theta }}{{mg\, + \,Q\,\cos \,\theta }}$
B
$\frac{{P\,\cos \,\theta \, + \,Q\,}}{{mg\, - \,Q\,\sin \,\theta }}$
C
$\frac{{P\, + \,Q\,\cos \,\theta }}{{mg\, + \,Q\,\sin \,\theta }}$
D
$\frac{{P\,sin\,\theta \, - \,Q\,}}{{mg\, - \,Q\,\cos \,\theta }}$

✓

Answer

Correct option: A.

$\frac{{P\, + \,Q\,\sin \,\theta }}{{mg\, + \,Q\,\cos \,\theta }}$

a
$\mathrm{N}=\mathrm{mg}+\mathrm{Q} \cos \theta$

$\mathrm{P}+\mathrm{Q} \sin \theta \leq \mu \mathrm{N}$

$\Rightarrow \mu \geq \frac{\mathrm{P}+\mathrm{Q} \sin \theta}{\mathrm{N}}, \mu \geq \frac{\mathrm{P}+\mathrm{Q} \sin \theta}{\mathrm{mg}+\mathrm{Q} \cos \theta}$

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