Question

A uniform string of length $20m$ is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the support is (take $g=10m{s}^{-2}$)

• A. $2\pi \sqrt{2}s$
• B. $2s$
• C. $2\sqrt{2}s$
• D. $\sqrt{2}s$

Answer 1

The correct option is C $2\sqrt{2}s$

The tension in a string of linear mass density $\lambda$ as a function of length $x$ measured from the free end is $T_x=\lambda xg$
The velocity of wave when it is at a position $x$ is
$v_x=\sqrt{\frac{T_x}{\lambda }}=\sqrt{gx}$
$\frac{dx}{dt}=\sqrt{gx}$
$T={\int }_0^L\frac{dx}{\sqrt{gx}}=\frac{2\sqrt{L}}{\sqrt{g}}=2\sqrt{2}s$