Q. The upper half of an inclined plane with inclination $\varnothing$ is perfectly smooth, while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom. If the coefficient of the friction for the lower half is given by
A) 2$\sin\phi$
B) $2\cos\phi$
C) $2\tan\phi$
D) $\tan\phi$
Explanation
(c) According to work-energy theorem,
$W = \mathrm{\Delta}K = 0$
$(\because Initial\ and\ final\ speeds\ are\ zero)$
$\therefore$ work done by friction +work done by gravity=0
$- \left( \mu\ mg\ \cos\varnothing \right)\ \frac{l}{2} + mgl\ \sin{\varnothing = 0}$
$or\ \ \frac{\mu}{2}\cos{\varnothing = \sin\varnothing}\ \ $
$\therefore\ \ \mu = 2\tan\varnothing$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.