The upper half of an inclined plane of inclination is perfectly s… - Vidyadip

MCQ

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

A
$\mu = \frac{1}{{tan\theta }}$
B
$\;\mu = \frac{2}{{tan\theta }}$
✓
$\;\mu = $$ 2$$\;tan\theta $
D
$\;\mu = $ $\;tan\theta $

✓

Answer

Correct option: C.

$\;\mu = $$ 2$$\;tan\theta $

c
$\begin{array}{l}
Let\,m\,be\,mass\,of\,the\,block\,and\,L\,be\,lenght\,of\\
the\,inclined\,plane.\\
According\,to\,work - energy\,theorem\\
\,\,\,\,\,\,\,\,W = \Delta K = 0\,\,(Initial\,and\,final\,speeds\,are\,zero)\\
\therefore \,Wrok\,done\,by\,friction\, + Work\,done\,by\,gravity = 0\\
\,\,\,\,\,\,\,\,\,\, - \mu mg\cos \theta \frac{L}{2} + mg\sin \theta L = 0\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{\mu }{2}\cos \theta = \sin \theta \\
\,\,\,\,\,\,\,\,\,\,\,\,\mu \, = \frac{{2\sin \theta }}{{\cos \theta }} = 2\tan \,\theta
\end{array}$

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