Question

A rope length $l$ and mass $m$ hanges freely from the ceiling. The velocity of transverse wave as a function of position $x$ along the rope is proportional to

• A. 1 / $\sqrt{x}$
• B. $\sqrt{x}$
• C. $x$
• D. $x^0$

Answer 1

The correct option is B $\sqrt{x}$

$\text{Mass per unit length of rope}=\frac{m}{l}$
Mass of rope of length $x=\frac{m}{l}x$
Tension, $T=\frac{m}{l}xg$
Velocity, $v=\sqrt{\frac{T}{m}}=\sqrt{\frac{mgl}{lm}x}$ which is directly proportional to $\sqrt{x}$