Question

A uniform rope of mass $M$ and the length $L$ hangs from a ceiling. The speed of transverse waves at a distance $x$ from the lower end will be

• A. $\sqrt{xg}$
• B. $\sqrt{(L-x)g}$
• C. $xg/2$
• D. $\sqrt{2(1-\frac{x}{L})Lg}$

Answer 1

The correct option is A $\sqrt{xg}$

$\mu$ is mass per unit length of rope.

At a distance $x$ from the lower end we will find tension which will be

$T=\mu gx$

So, using $v=\sqrt{\frac{T}{\mu }}$ (speed of transverse wave on a string)

$v=\sqrt{\frac{\mu gx}{\mu }}=\sqrt{gx}$