Question

The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is

• A. $\mu =2tan\theta$
• B. $\mu =tan\theta$
• C. $\mu =\frac{2}{tan\theta }$
• D. $\mu =\frac{1}{tan\theta }$

Answer 1

The correct option is A $\mu =2tan\theta$

Let the length of upper half of the plane be l. Now, by work energy theorem, total work done must be equal to the change in kinetic energy, i.e. 0.

Since normal force is $N=mgcos\left(\theta \right)$ and $f=\mu N$

Hence, $mg\left(2lsin\right(\theta \left)\right)-\left(\mu mgcos\right(\theta \left)\right)l=0$

$\mu =2tan\left(\theta \right)$