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(B) The formula for Young's modulus $Y$ is given by $Y = \frac{F l}{A \Delta l}$,where $A = \pi r^2$.Rearranging for the change in length $\Delta l$,w

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$1:1$

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(B) The formula for Young's modulus $Y$ is given by $Y = \frac{F l}{A \Delta l}$,where $A = \pi r^2$.Rearranging for the change in length $\Delta l$,we get $\Delta l = \frac{F l}{\pi r^2 Y}$.Since the material is the same,$Y$ is constant. Given that the force $F$ is also the same,we have $\Delta l \propto \frac{l}{r^2}$.Let the lengths be $l_1 = l$ and $l_2 = 2l$,and the radii be $r_1 = r$ and $r_2 = \sqrt{2}r$.The ratio of the increase in lengths is $\frac{\Delta l_1}{\Delta l_2} = \frac{l_1}{r_1^2} 	imes \frac{r_2^2}{l_2} = \frac{l}{r^2} 	imes \frac{(\sqrt{2}r)^2}{2l} = \frac{l}{r^2} 	imes \frac{2r^2}{2l} = 1$.Thus,the ratio is $1:1$.

The ratio of the lengths of two wires of the same material is $1:2$ and the ratio of their radii is $1:\sqrt{2}$. If they are stretched by the same force,what is the ratio of the increase in their lengths?

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