Question

The upper half of an inclined plane with an angle of inclination $\varphi$ is perfectly smooth, while the lower half is rough. A body starting from rest at the top will again comes to rest at the bottom .If the coefficient of friction for the lower half is :

• A. $2tan\varphi$
• B. 2 cos $\varphi$
• C. none of above
• D. tan $\varphi$

Answer 1

Answer: (A)

$2tan\varphi$

Let the length of inclined plane be $2l$ and height be $h$.

$h=2l\sin \varphi .......\left(i\right)$

Normal reaction, $N=mg\cos \left(\varphi \right)$

Frictional force, $F_r=\mu N=\mu mg\cos \left(\varphi \right)$

Then, applying work energy theorem, the total work done by friction and gravity must be equal to the change in kinetic energy, that is equal to zero.

$mgh=\mu mg\cos \varphi \left(l\right).............\left(ii\right)$

From (i) and (ii),

$2l\sin \varphi =\mu l\cos \varphi$

$\mu =2\tan \varphi$