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(D) Let the actual length of the string be $L$ and the force constant be $k = \frac{AY}{L_0}$.According to Hooke's Law, the extension is proportional

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$\frac{F_2 L_1-F_1 L_2}{F_2-F_1}$

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(D) Let the actual length of the string be $L$ and the force constant be $k = \frac{AY}{L_0}$.According to Hooke's Law, the extension is proportional to the applied force: $F = k \Delta L$.For the first case: $F_1 = k(L_1 - L)$.For the second case: $F_2 = k(L_2 - L)$.Dividing the two equations: $\frac{F_1}{F_2} = \frac{L_1 - L}{L_2 - L}$.Cross-multiplying gives: $F_1(L_2 - L) = F_2(L_1 - L)$.$F_1 L_2 - F_1 L = F_2 L_1 - F_2 L$.Rearranging to solve for $L$: $F_2 L - F_1 L = F_2 L_1 - F_1 L_2$.$L(F_2 - F_1) = F_2 L_1 - F_1 L_2$.Therefore, $L = \frac{F_2 L_1 - F_1 L_2}{F_2 - F_1}$.

If a string, suspended from the ceiling, is given a downward force $F_1$, its length becomes $L_1$. Its length becomes $L_2$ if the downward force is $F_2$. What is its actual length?

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