Question
A metal string A is suspended from a rigid support and its free end is attached to a block of mass M. Second block having mass 2 M is suspended at the bottom of the first block using a string B . The area of cross sections of strings A and B are same. The ratio of lengths of strings of $A$ to $B$ is 2 and the ratio of their Young's moduli $\left( Y_{A}/Y_{B} \right)$ is 0.5 . The ratio of elongations in A to B is $\_\_\_\_$ .
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Solution
Sol. [Key Concept: Elongation $\Delta\mathcal{l =}\frac{T\mathcal{l}}{AY}$; compute $\Delta\mathcal{l}_{1}$ (load $3mg$) and $\Delta\mathcal{l}_{2}$ using each string's $\mathcal{l}$ and $Y$.]
$$\begin{matrix} & \Delta\mathcal{l} = \frac{T\mathcal{l}}{YA} \\ & \Delta\mathcal{l}_{1} = \frac{3mg\mathcal{l}}{YA} \\ & \Delta\mathcal{l}_{2} = \frac{2mg(\mathcal{l}/2)}{2YA} = \frac{mg\mathcal{l}}{2YA} \\ & \frac{\Delta\mathcal{l}_{1}}{\Delta\mathcal{l}_{2}} = 6 \end{matrix}$$
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