First slide

Force

Question

A smooth block is released at rest on a 45° incline and then slides a distance d. The time taken to slide is n times as much to slide on rough incline than on a smooth incline. The coefficient of friction is

Moderate

A

$μ_{s} = 1 - \frac{1}{n^{2}}$
B

$μ_{s} = \sqrt{(1 - \frac{1}{n^{2}})}$
C

$μ_{k} = 1 - \frac{1}{n^{2}}$
D

$μ_{k} = \sqrt{(1 - \frac{1}{n^{2}})}$

Solution

See fig.
Let t be the time taken by the block to cover a distance d.
Then using the formula, $s = ut + (1 / 2) g^{'} t^{2},$ we get
$d = \frac{1}{2} (g \sin 45^{\circ}) t^{2}$ …(1)
$(∵ u = 0 and g^{'} = gsin 45^{\circ})$
Let t' be the time taken for rough plane.
Then $d = \frac{1}{2} g (\sin 45^{\circ} - μ_{k} \cos 45^{\circ}) t^{' 2}$
$or d = \frac{1}{2} g (\sin 45^{\circ} - μ_{k} \cos 45^{\circ}) (nt)^{2}$ …(2)
Equating equations (1) and (2), we get
$\sin 45^{\circ} = (\sin 45^{\circ} - μ_{k} \cos 45^{\circ}) n^{2}$
$or \frac{1}{\sqrt{2}} \times \frac{1}{n^{2}} = \frac{1}{\sqrt{2}} - \frac{μ_{k}}{\sqrt{2}} or μ_{k} = 1 - \frac{1}{n^{2}}$ .

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