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Q.

A block of mass m is moving along positive x direction on a smooth horizontal surface with velocity u. It enters a rough horizontal region at x = 0. The coefficient of friction in this rough region varies according to $μ$ = ax, where ‘a’ is a positive constant and x is displacement of the block in the rough region. Find the time for which the block will slide in this rough region.

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a

$\frac{π}{\sqrt{a g}}$

b

$\frac{π}{4} \sqrt{\frac{a}{g}}$

c

$\frac{π}{2 \sqrt{a g}}$

d

$\frac{2 π}{\sqrt{a g}}$

answer is D.

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Detailed Solution

$\begin{array}{l} m \frac{d^{2} x}{{dt}^{2}} = - μmg \\ \frac{d^{2} x}{{dt}^{2}} = - agx \end{array}$
This is equation of SHM whose time period is
$T = 2 π \sqrt{\frac{1}{ag}}$
$∴$ The block will slide for a time
$t = \frac{T}{4} = \frac{π}{2} \sqrt{\frac{1}{ag}}$

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