Question

A thick rubber rope of density p and length L is suspended from a rigid support. If Y is the Young's modulus of elasticity of the material of the rope, then the increase in length of the rope due to its own weight is

• A. $\frac{1}{2}\rho g{L}^{2}Y$
• B. $\frac{\rho g{L}^2}{2Y}$
• C. $\frac{\rho gL}{Y}$
• D. $\frac{\rho gL}{6Y}$

Answer 1

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

$y=\frac{stress}{strain}$
Let area be A.
$Stress=\frac{pAg(L-x)}{A}$
$y=\frac{pg(L-x)}{\sqrt{\frac{\Delta l}{dx}}}$

$\Delta e=\frac{pg{\int }_0^L(L-x)dx}{Y}=\frac{pg}{Y}(L^{2}x-\frac{x^2}{2}){\int }_0^L$
$\Delta l=\frac{pg{L}^2}{Y^2}=\frac{pg{L}^2}{2Y}$