Q163 / 180 · COMEDK 2015 · Physics
The dimensions of four wires of the same material are given below. In which wire the increase in length will be maximum when the same tension is applied?
- ALength $100 \mathrm{~cm}$, diameter $1 \mathrm{~mm}$
- BLength $200 \mathrm{~cm}$, diameter $2 \mathrm{~mm}$
- CLength $300 \mathrm{~cm}$, diameter $3 \mathrm{~mm}$
- Correct answerDLength $50 \mathrm{~cm}$, diameter $0.5 \mathrm{~mm}$
Solution
We know that, Young's modulus of elasticity,
$$
\begin{array}{ll}
& Y=\frac{F L}{\pi r^{2} l} \quad \text { [where, } l \text { is change in length] } \\
\Rightarrow \quad & l=\frac{F L}{\pi r^{2} Y} \\
\Rightarrow \quad & l \propto \frac{L}{r^{2}} \propto \frac{L}{d^{2}} \quad \text { [where, } d=\text { diameter] } \\
\Rightarrow \quad & l \propto \frac{L}{d^{2}}
\end{array}
$$
Among the given options, $\frac{L}{d^{2}}$ is maximum for the values of $L=50 \mathrm{~cm}$ and $d=0.5 \mathrm{~mm}$.
Hence, the correct answer is option: 'D'.