Question

A body is moving up an inclined plane of angle $\theta$ with an initial kinetic energy E. The coefficient of friction between the plane and the body is $\mu$. The work done against friction before the body comes to rest is

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

Answer 1

The correct option is D

$\frac{\mu Ecos\theta }{\mu cos\theta +sin\theta }$

Let the body slide a distance $s$ along the incline before coming to rest. So, it ascends by a vertical height $h=ssin\theta$.


From the work-energy theorem, $\Delta W_{fr}+\Delta W_{mg}=\Delta K\Rightarrow -\mu mg\left(cos\theta \right)s-mgssin\theta =0-E\Rightarrow \mu mgscos\theta +mgssin\theta =E\Rightarrow s=\frac{E}{mg(\mu cos\theta +sin\theta )}$
So, the work done by friction, $\Delta W_{fr}=-\mu mgscos\theta =\frac{-\mu Ecos\theta }{\mu cos\theta +sin\theta }$
Hence, the work done against friction is $\frac{\mu Ecos\theta }{\mu cos\theta +sin\theta }$ and the correct choice is (d).