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. $\frac{1}{4}\rho g{L}^2/Y$
• B. $\frac{1}{2}\rho g{L}^2/Y$
• C. $\rho g{L}^2/Y$
• D. $\rho$gL/Y

Answer 1

Answer: (B)

Take a small section $\delta y$ at distance y from top as shown in figure. Let area of rope be A
Stress due to this section $=\rho Ag\delta y/A=\rho d\delta y$
Strain on section $y=\delta l/y$
Using Stress $=Y\backslash$ times Strain
$\Rightarrow \rho g\delta y=Y\times \frac{\delta 1}{y}$
$\Rightarrow \Delta l=\frac{\rho g}{y}{\int }_0^{L}y\delta y$
$\Rightarrow \Delta l=\frac{\rho g{L}^2}{2Y}$
Answer. $\frac{\rho g{L}^2}{2Y}$