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A wire of variable mass per unit length $μ = μ_{0} x$ , is hanging from the ceiling as shown in figure. The length of wire is $l_{0}$ . A small transverse disturbance is produced at its lower end. Find the time after which the disturbance will reach to the other ends.

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$\sqrt{\frac{10 l_{0}}{g}}$

$\sqrt{\frac{6 l_{0}}{g}}$

$\sqrt{\frac{8 l_{0}}{g}}$

$\sqrt{\frac{9 l_{0}}{g}}$

answer is B.

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Detailed Solution

Tension at a distance 'x' from the free end is given by

$T = \int μgdx = \int_{0}^{x} μ_{0} gxdx = \frac{μ_{0} {gx}^{2}}{2}$
Speed of wave $V = \sqrt{T / μ}$
$\begin{array}{l} V = \sqrt{\frac{μ_{0} {gx}^{2}}{2 μ_{0} x}} = \sqrt{\frac{gx}{2}} \end{array}$

$Also \frac{dx}{dt} = \sqrt{gx / 2}$

$\int_{0}^{L} \frac{dx}{\sqrt{x}} = \int_{0}^{T} \sqrt{g / 2};$

$2 \sqrt{L} = \sqrt{\frac{g}{2}} T,$

$T = \sqrt{\frac{8 L}{g}}$

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