Question

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

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

Answer 1

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

$\begin{array}{c}\text{Given,}\\ \text{P= density of rope}\\ \text{Y= Young's}modules\end{array}$

$\begin{array}{c}\text{Let}A\text{be the cross sectional area of rope}\\ \text{then,}\\ Y=\frac{\text{stress}}{\text{strain}}\\ Y=\frac{FL}{A\Delta L}\end{array}$

$\begin{array}{cc}& F=mg\text{(due to own weight)}\\ F& =fvg\\ F& =fALg\end{array}$

$\begin{array}{c}\text{Now,}\\ Y=\frac{\left(\rho ALg\right)L}{A\Delta L}\\ \Delta L=\frac{P{L}^{2}g}{Y}\end{array}$