pendulum made of a uniform wire of cross sectional area $A$ has time period $T$. When an additional mass $M$ is added to its bob, the time period changes to $T_M$. If the Young's modulus of the material of the wire is $Y$ then $\frac{1}{Y}$ is equal to : ($g$ = gravitational acceleration)
JEE MAIN 2015, Diffcult
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As we know, time period, $T=2 \pi \sqrt{\frac{\ell}{g}}$
$ When\, additional \,mass \mathrm\,{M}$ is added then
$\mathrm{T}_{\mathrm{M}}=2 \pi \sqrt{\frac{\ell+\Delta \ell}{\mathrm{g}}}$
$T_{\frac{M}{T}}=\sqrt{\frac{\ell+\Delta \ell}{\ell}}$ or $\left(\frac{T_{M}}{T}\right)^{2}=\frac{\ell+\Delta \ell}{\ell}$
or, $\left(\frac{T_{M}}{T}\right)^{2}=1+\frac{M g}{A y}\left[\because \Delta \ell=\frac{M g \ell}{A y}\right]$
$\therefore \frac{1}{y}=\left[\left(\frac{T_{M}}{T}\right)^{2}-1\right] \frac{A}{M g}$
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