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(C) The formula for the extension $l$ in a wire is given by $l = \frac{FL}{\pi r^2 Y}$,where $F$ is the force,$L$ is the length,$r$ is the radius,and

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$1:1$

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(C) The formula for the extension $l$ in a wire is given by $l = \frac{FL}{\pi r^2 Y}$,where $F$ is the force,$L$ is the length,$r$ is the radius,and $Y$ is Young's modulus.Since the material is the same,$Y$ is constant. Given that the forces $F$ are equal,we have $l \propto \frac{L}{r^2}$.Therefore,the ratio of the increase in lengths is $\frac{l_1}{l_2} = \frac{L_1}{L_2} 	imes \left( \frac{r_2}{r_1} ight)^2$.Given $\frac{L_1}{L_2} = \frac{1}{2}$ and $\frac{r_1}{r_2} = \frac{1}{\sqrt{2}}$,we have $\frac{r_2}{r_1} = \sqrt{2}$.Substituting these values: $\frac{l_1}{l_2} = \frac{1}{2} 	imes (\sqrt{2})^2 = \frac{1}{2} 	imes 2 = 1$.Thus,the ratio is $1:1$.

Two wires of the same material have lengths in the ratio $1:2$ and their radii are in the ratio $1:\sqrt{2}$. If they are stretched by applying equal forces,the increase in their lengths will be in the ratio:

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