Question

A rubber cord of density $d$, Young's modulus $Y$ and length $L$ is suspended vertically. If the cord extends by a length $0.5 \, L$ under its own weight, then $L$ is

• A. $\frac{Y}{2dg}$
• B. $\frac{Y}{dg}$
• C. $\frac{2Y}{dg}$
• D. $\frac{dg}{2Y}$
• E. $\frac{dg}{Y}$

Hint:

The extension of a wire under its own weight is exactly half the extension it would experience if the total weight were applied as a load at the end.

Answer 1

The Correct Option is B

Concept:
For a cord hanging under its own weight, the extension $\Delta L$ is given by: \[ \Delta L = \frac{dgL^2}{2Y} \] This formula accounts for the fact that the tension varies from zero at the bottom to maximum at the top.

Step 1: Applying the given condition.
The problem states that the extension $\Delta L = 0.5 L = \frac{L}{2}$. \[ \frac{L}{2} = \frac{dgL^2}{2Y} \]

Step 2: Solving for $L$.
Cancel $L$ and 2 from both sides of the equation: \[ 1 = \frac{dgL}{Y} \implies L = \frac{Y}{dg} \]