Question

A string is hanging from a rigid support. A transverse wave pulse is setup at the free end. The velocity $v$ of the pulse related to the distance $x$ covered by it is

• A. $v\propto \frac{1}{x}$
• B. $v\propto \frac{1}{x^2}$
• C. $v\propto \sqrt{x}$
• D. $v\propto x$

Answer 1

The correct option is C $v\propto \sqrt{x}$


Let the mass of the string is $m$ and length is $l$.
The linear density of the string becomes
$\Rightarrow \frac{m}{l}=\mu$
Tension $\left(T\right)$ at distance $x$ from free end
$T=\frac{m}{l}x=\mu xg$
$\Rightarrow \frac{T}{\mu }=xg$
$\Rightarrow \sqrt{\frac{T}{\mu }}=\sqrt{xg}$
$\Rightarrow v=\sqrt{xg}$
$\therefore v\propto \sqrt{x}$
Hence, option (a) is correct.