Question

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

• A. $\sqrt{\frac{8 \ell _{0}}{ g}}$
• B. $\sqrt{\frac{4 \ell _{0}}{ g}}$
• C. $\sqrt{\frac{2 \ell _{0}}{ g}}$
• D. None

Answer 1

Answer: (A)

$m=\displaystyle \int _{0}^{x}\mu _{0}xdx=\frac{\mu _{0} x^{2}}{2}$

$V=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{\mu _{0} x^{2} g}{2 . \mu _{0} x}}$
$V=\sqrt{\frac{xg}{2}}$
$a=\frac{vdv}{dx}=\sqrt{\frac{xg}{2}}\sqrt{\frac{g}{2}}\cdot \frac{1}{2 \sqrt{x}}=\frac{g}{4}m/s^{2}$
$\ell_{0}=\frac{1}{2} \times \frac{\mathrm{g}}{4} \mathrm{t}^{2} \Rightarrow \mathrm{t}=\sqrt{\frac{8 \ell_{0}}{\mathrm{~g}}}$