A rubber cord of density $d,$Young's modulus $Y$ and length $L$ is suspended vertically. If the cord extends by a length $0.5L$ under its own weight, then $L$ is

• A

  $\frac{Y}{2dg}$
• Correct answer
  B

  $\frac{Y}{dg}$
• C

  $\frac{2Y}{dg}$
• D

  $\frac{dg}{2Y}$

Solution

$\because$ Young's modulus $Y=\frac{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}}{\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{n}}$

$Y=\frac{\left(\frac{F}{A}\right)}{\left(\frac{∆L}{L}\right)}$

  $\Rightarrow ∆l=\frac{FL}{AY}$

extennsion in $dx$ at $‘x’$

$de=\frac{x d A g\left(dx\right)}{AY}$

total extension $e= \underset{0}{\overset{L}{\int }}\frac{x d Ag \left(dx\right)}{AY}$

$=\frac{dg}{Y}\frac{L^2}{2}$

$0.5 L=\frac{dg{L}^2}{{\mathrm{Y\; 2}}_{}} \Rightarrow L=\frac{Y}{dg}$

Hence, the correct answer is option: 'B'.