Question

The upper half of an inclined plane of inclination q is perfectly smooth while the lower half is rough. A body starting from the rest at top comes back to rest at the bottom if the coefficient of friction for the lower half is given by

• A. m = sin q
• B. m = cot q
• C. m = 2 cos q
• D. m = 2 tan q

Answer 1

The correct option is D

m = 2 tan q

For upper half
$v^2=u^2+\frac{2al}{2}=\frac{2\left(gsin\theta \right)l}{2}=glsin\theta$
For lower half
$\Rightarrow 0=u^2+2g(sin\theta -\mu cos\theta )\frac{1}{2}$

$\Rightarrow -glsin\theta =gl(sin\theta -\mu cos\theta )$
$\Rightarrow \mu cos\theta =2sin\theta \Rightarrow \mu 2tan\theta$