A wire of length $L$ and radius $r$ is clamped at one end. If its other end is pulled by a force $F$, its length increases by $l$. If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant, the increase in length will become.
JEE MAIN 2024, Diffcult
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$ \mathrm{Y}=\frac{\text { stress }}{\text { strain }} $
$ \mathrm{Y}=\frac{\frac{\mathrm{F}}{\mathrm{r}^2}}{\frac{\ell}{\mathrm{L}}} $
$ \mathrm{F}=\mathrm{Y} \pi \mathrm{r}^2 \times \frac{\ell}{\mathrm{L}} $ $...........(i)$
$ \mathrm{Y}=\frac{\frac{\pi \mathrm{r}^2 / 4}{\Delta \ell}}{\mathrm{L}} $
$ \mathrm{F}=\mathrm{Y} \frac{\Delta \ell}{\mathrm{L}} \times 2 \times \frac{\pi \mathrm{r}^2}{4} $
$ \text { From }(\mathrm{i}) $
$ \mathrm{Y} \pi \mathrm{r}^2 \frac{\ell}{\mathrm{L}}=\mathrm{Y} \frac{\Delta \ell}{\mathrm{L}} \frac{\pi \mathrm{r}^2}{2} $
$ \Delta \ell=2 \ell$
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