The following four wires are made of the same material. Which of these will have the smallest extens

Question

The following four wires are made of the same material. Which of these will have the smallest extension when the same tension is applied?

Professor PC Thomas Classes · Answer

(a) : Young's modulus $Y = \frac{FL}{A \Delta L} = \frac{4FL}{\pi D^2 \Delta L}$. Therefore, extension $\Delta L = \frac{4FL}{\pi D^2 Y}$. For wires of the same material under the same tension, $\Delta L \propto \frac{L}{D^2}$. Let's evaluate $\frac{L}{D^2}$ for each option: (a) $\frac{300}{0.3^2} = 3.33 	imes 10^3 \mathrm{~cm}^{-1}$; (b) $\frac{200}{0.2^2} = 5 	imes 10^3 \mathrm{~cm}^{-1}$; (c) $\frac{50}{0.05^2} = 20 	imes 10^3 \mathrm{~cm}^{-1}$; (d) $\frac{100}{0.1^2} = 10 	imes 10^3 \mathrm{~cm}^{-1}$. The smallest value of $\frac{L}{D^2}$ corresponds to the smallest extension, which is option (a).