Question

An Indian rubber cord $L$ metre long and area of cross-section $A$ meter${}^2$ is suspended vertically. Density of rubber is $\rho kg/$ meter${}^3$ and Young's modulus of rubber is $Y$ Newton/metre${}^2$. If the cord extends by $l$ metre under its own weight, then extension $l$ is:

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

Answer 1

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

A small differential element $dx$ at distance $x$ from the bottom of chord

Force acting on this element $=\frac{M}{L}\times x\times g$

If extension in this element is $dl$

Then

$\begin{array}{c}\frac{dl}{dx}=\frac{f}{Ay}=\frac{Mg}{LAy}x\\ {\int }_0^{l}dl=\frac{Mg}{LAy}{\int }_0^{l}xdx\\ l=\frac{Mg{L}^2}{2LAy}=\frac{Mgl}{2Ay}\\ Ifdensityisfthen,\\ \rho =\frac{m}{AL}\\ So,l=\frac{\rho g{L}^2}{2y}\\ HenceoptionBisthecorrectanswer.\end{array}$