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(A) The Young's modulus $Y$ is given by the formula $Y = \frac{FL}{Al} = \frac{fL}{\pi r^2 l}$.From this,the increase in length $l$ is $l = \frac{fL}{

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$l$

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(A) The Young's modulus $Y$ is given by the formula $Y = \frac{FL}{Al} = \frac{fL}{\pi r^2 l}$.From this,the increase in length $l$ is $l = \frac{fL}{\pi r^2 Y}$.For the second wire,the length is $L' = 2L$,the radius is $r' = 2r$,and the force is $f' = 2f$.The new increase in length $l'$ is given by $l' = \frac{f' L'}{A' Y} = \frac{(2f)(2L)}{\pi(2r)^2 Y}$.Simplifying this,we get $l' = \frac{4fL}{4\pi r^2 Y} = \frac{fL}{\pi r^2 Y}$.Since $l = \frac{fL}{\pi r^2 Y}$,it follows that $l' = l$.

$A$ wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$,its length increases by $l$. Another wire of the same material of length $2L$ and radius $2r$ is pulled by a force $2f$. Find the increase in length of this wire.

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