Question
A string A of length 0.314 m of Young's modulus $2 \times 10^{10}\text{ }N/m^{2}$ is connected to another string B of length and Young's modulus both twice of those of A. This series combination of strings is then suspended from a rigid support and its free end is fixed to a load of mass 0.8 kg . The net change in length of the combination is $\_\_\_\_$ mm .
(radius of both the strings is 0.2 mm and acceleration due to gravity $= 10\text{ }m/s^{2}$ )
(Mass of both strings is to be neglected as compared to the mass of load)
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Solution
Sol. [Key Concept: Net elongation $\Delta\mathcal{l = \sum}\frac{mg\mathcal{l}}{AY}$; add both segments using their own $\mathcal{l}$ and $Y$.]
$$\begin{matrix} \Delta\mathcal{l} & \ \Rightarrow \Delta\mathcal{l}_{1} + \Delta\mathcal{l}_{2} \\ \Rightarrow & \frac{mg\mathcal{l}}{AY} + \frac{mg(2\mathcal{l})}{A(2Y)} = \frac{2mg\mathcal{l}}{AY} \\ & \ = \frac{2 \times 0.8 \times 10 \times (0.314)}{3.14 \times \left( 2 \times 10^{- 4} \right)^{2} \times 2 \times 10^{10}} = 2\text{ }mm \end{matrix}$$
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