Question:
Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is 1 : 4, the ratio of their diameters is:
Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is 1 : 4, the ratio of their diameters is:
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Energy per unit volume is \(\frac{\sigma^2}{2Y}\); with the same load, \(\sigma \propto \frac{1}{d^2}\), so \(u \propto \frac{1}{d^4}\).
Updated On: Nov 5, 2026
- 1 : 2
- 2 : 1
- \(1 : \sqrt{2}\)
- \(\sqrt{2} : 1\)
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The Correct Option is D
Solution and Explanation
Energy stored per unit volume: \(u = \frac{1}{2} \times \text{stress} \times \text{strain} = \frac{\sigma^2}{2Y}\).
Same load F and same material (same Y): \(\sigma = \frac{F}{A} \propto \frac{1}{d^2}\), so \(u \propto \frac{1}{d^4}\).
\(\frac{u_1}{u_2} = \left(\frac{d_2}{d_1}\right)^4 = \frac{1}{4} \Rightarrow \frac{d_2}{d_1} = \frac{1}{\sqrt{2}}\).
Hence \(\frac{d_1}{d_2} = \sqrt{2}\), so \(d_1 : d_2 = \sqrt{2} : 1\).
Answer: (4) \(\sqrt{2} : 1\)
Same load F and same material (same Y): \(\sigma = \frac{F}{A} \propto \frac{1}{d^2}\), so \(u \propto \frac{1}{d^4}\).
\(\frac{u_1}{u_2} = \left(\frac{d_2}{d_1}\right)^4 = \frac{1}{4} \Rightarrow \frac{d_2}{d_1} = \frac{1}{\sqrt{2}}\).
Hence \(\frac{d_1}{d_2} = \sqrt{2}\), so \(d_1 : d_2 = \sqrt{2} : 1\).
Answer: (4) \(\sqrt{2} : 1\)
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