Question

A block of mass $m$ rests on a rough horizontal surface with a rope tied to it. The co-efficient of friction between the surface and the block is $\mu$. A monkey of the same mass climbs at the free end of the rope. The maximum acceleration with which the monkey can climb without moving the block is

• A. $\frac{\mu g}{\mu \sin \theta +\cos \theta }-g$
• B. $\frac{\mu g}{\mu \sin \theta -\cos \theta }+g$
• C. $\frac{\mu g}{\mu \tan \theta -\cos \theta }+g$
• D. $\frac{\mu g}{\mu \tan \theta +\cos \theta }-g$

Answer 1

The correct option is A $\frac{\mu g}{\mu \sin \theta +\cos \theta }-g$

By the free body diagram of the block and the monkey,
We write the force equation for the block in equilibrium,

$N=mg-T\sin \theta$, and

$T\cos \theta =\mu N=\mu (mg-T\sin \theta )$

The maximum tension for which block doesnt move is,

$T_{max}=\frac{\mu mg}{(\mu \sin \theta +\cos \theta )}$

Hence the maximum acceleration will be when tension is maximum , so by the force equation on monkey we get,

$a_{max}=\frac{T_{max}-mg}{m}=\frac{\mu g}{\mu \sin \theta +\cos \theta }-g$

Hence option A is the correct answer.