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)

$m={\int }_0^x{\mu }_{0}xdx=\frac{{\mu }_0{x}^2}{2}$

$V=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{{\mu }_0{x}^{2}g}{2.{\mu }_{0}x}}$
$V=\sqrt{\frac{xg}{2}}$
$a=\frac{vdv}{dx}=\sqrt{\frac{xg}{2}}\sqrt{\frac{g}{2}}\cdot \frac{1}{2\sqrt{x}}=\frac{g}{4}m/s^2$
${\ell }_0=\frac{1}{2}\times \frac{\mathrm{g}}{4}{\mathrm{t}}^2\Rightarrow \mathrm{t}=\sqrt{\frac{8{\ell }_0}{\mathrm{g}}}$