Question

A heavy uniform rope hangs vertically from the ceiling, with its lower end free. A disturbance on the rope travelling upwards from the lower end has a velocity \(v\) at a distance \(x\) from the lower end such that :

• A. \( v \propto x \)
• B. \( v \propto \sqrt{x} \)
• C. \( v \propto \frac{1}{x} \)
• D. \( v \propto \frac{1}{\sqrt{x}} \)

Hint:

In hanging rope, tension increases with depth, so wave speed increases upward.

Answer 1

The Correct Option is B

Concept: Wave velocity in a string: \[ v = \sqrt{\frac{T}{\mu}} \]

Step 1: Tension in rope.
At distance \(x\), tension equals weight of portion below: \[ T = \mu g x \]

Step 2: Velocity.
\[ v = \sqrt{\frac{\mu g x}{\mu}} = \sqrt{gx} \] \[ \Rightarrow v \propto \sqrt{x} \]