First slide

Friction on inclined plane of angle more than angle of repose

Question

The upper half of an incline plane with inclination $ϕ$ is perfectly smooth while the lower half is rough. A body starting from rest at top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

Moderate

A

$2 \tan ϕ$
B

$\tan ϕ$
C

$2 \sin ϕ$
D

$2 \cos ϕ$

Solution

For upper half, we have
$v^{2} = (0)^{2} + (2 gsin ϕ) l$
For lower half
$\begin{array}{rrlrlrlrlrlrl} 0 = v^{2} + 2 g (\sin ϕ - μcos ϕ) l \\ or & 2 gsin ϕ . l + 2 gsin ϕl = & μ 2 gcos ϕl \\ μ = & 2 \tan ϕ \end{array}$

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