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Q. 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 :

Mechanical Properties of Solids

Solution:

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
$\Rightarrow \rho g \delta y=Y \times \frac{\delta 1}{y}$
$\Rightarrow \Delta l=\frac{\rho g}{y} \int_{0}^{ L } y \delta y$
$\Rightarrow \Delta l=\frac{\rho g L^{2}}{2 Y}$
Answer. $\frac{\rho g L^{2}}{2 Y }$