Question

A small mass slides down an inclined plane of inclination $\theta$ with the horizontal. The coefficient of friction is $\mu ={\mu }_{0}x$, where $x$ is the distance through which the mass slides down, and ${\mu }_0$ a constant. Then, the speed is maximum after the mass covers a distance of

• A. $\frac{\cos \theta }{{\mu }_0}$
• B. $\frac{\sin \theta }{{\mu }_0}$
• C. $\frac{\tan \theta }{{\mu }_0}$
• D. $\frac{2\tan \theta }{{\mu }_0}$

Answer 1

The correct option is C $\frac{\tan \theta }{{\mu }_0}$


$f_k=\mu N={\mu }_{0}xmg\cos \theta$
$ma=mg\sin \theta -{\mu }_{0}xmg\cos \theta$
$v\frac{dv}{dx}=g[\sin \theta -{\mu }_{0}x\cos \theta ]$
when $v$ is maximum, $\frac{dv}{dx}=0$
$\Rightarrow \sin \theta ={\mu }_{0}x\cos \theta$
$\Rightarrow x=\frac{\tan \theta }{{\mu }_0}$