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A meter scale of mass $m$ , Young modulus $Y$ and cross section area $A$ is hanged vertically from ceiling at zero mark. Then separation between $30\ cm$ and $70\ cm$ mark will be 🙁 $\frac{{mg}}{{AY}}$ is dimensionless)
$30\frac{{mg}}{{AY}}\ cm + 40\ cm$
$20\frac{{mg}}{{AY}}\ cm + 40\ cm$
$50\frac{{mg}}{{AY}}\ cm + 40\ cm$
$40\frac{{mg}}{{AY}}\ cm + 40\ cm$
Solution
Tension is rod at a distnace $x$ from lower end is $\frac{\mathrm{mxg}}{\ell}$
$Y$ is young modulus of elasticity then change in length in $dx$ element is $dy$ $\mathrm{Y} \times$ strain $=$ stress
$\mathrm{Y} \times \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{T}}{\mathrm{A}}$
$\mathrm{Y} \times \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{mgx}}{\ell \times \mathrm{A}}$
$\int_{0}^{y} Y d y=\int_{30}^{70} \frac{m g}{\ell A} \times x d x$
$\mathrm{Y} \mathrm{y}=\frac{\mathrm{mg}}{\ell \mathrm{A}}\left[\frac{(70)^{2}-(30)^{2}}{2}\right]$
$\mathrm{y}=\frac{\mathrm{mg}}{\mathrm{AY} \times 100} \times 2000$
$\mathrm{y}=\frac{\mathrm{mg}}{\mathrm{AY}} \times 20$
Total length is $=40 \mathrm{cm}=20 \frac{\mathrm{mg}}{\mathrm{AY}} \mathrm{cm}$
