Question

A thick rope of density $\rho$ and length $L$ is hung from a rigid support. The Young's modulus of the material of rope is $Y$. The increase in length of the rope due to its own weight is

• A. $\rho g{L}^2/4Y$
• B. $\rho g{L}^2/2Y$
• C. $\rho g{L}^2/Y$
• D. $\rho gL/Y$

Answer 1

The correct option is B $\rho g{L}^2/2Y$

Take a small section $\delta y$ at distance $Y$ from top as shown in the figure.Let area of rope be $A$

Stress due to this section $=\rho Ag\frac{\delta y}{A}=\rho d\delta y$

Strain on section$Y=\frac{\delta l}{y}$

Using stress$=Y\times$ strain

$\Rightarrow \rho g\delta y=Y\times \frac{\delta l}{y}$

$\Rightarrow \Delta l=\frac{\rho g}{y}{\int }_0^{L}y\delta y$

$\Rightarrow \Delta l=\frac{\rho g{L}^2}{2Y}$