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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.
$3$ times
$3 / 2$ times
$4$ times
$2$ times
Solution
$ \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$
