First slide

Friction on inclined plane of angle more than angle of repose

Question

To determine the coefficient of friction between a rough surface and a block, the surface is kept inclined at 45° and the block is released from rest. The block takes a time t in moving a distance d. The rough surface is then replaced by a smooth surface and the same experiment is repeated. The block now takes a time t/2 in moving down the same distance d. The coefficient of friction is

Moderate

A

3/4
B

5/4
C

1/2
D

$1 / \sqrt{2}$

Solution

If the same wedge is made rough, then time taken by it to come down becomes n times more.
For rough surface, $t = \sqrt{\frac{2 s}{g \sin θ - μ g \cos θ}} . . . . . . . . (i)$
For smooth surface, $\frac{t}{n} = \sqrt{\frac{2 s}{g \sin θ}} . . . . . . (ii)$
Squaring Eqs. (i) and (ii) and dividing them, we get
$μ = (1 - \frac{1}{n^{2}}) \tan θ$
$∴ μ = [1 - \frac{1}{n^{2}}] \tan θ = [1 - \frac{1}{2^{2}}] \tan 45 °$
$\Rightarrow μ = \frac{4 - 1}{4} = \frac{3}{4}$

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