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Steel and brass wires are connected in a system (as in a standard hanging-wire diagram). If the ratio of lengths, radii, and Young's moduli of the steel wire to the brass wire are a, b, and c respectively, what is the corresponding ratio of increases in their lengths? (Assume the brass wire carries twice the load of the steel wire, consistent with the standard figure.)
- 2a²c/b
- 3a/2b²c
- 2ac/b²
- 3c/2ab²
Correct answer: 2ac/b²
Solution
Using the elongation formula and the given ratios, the ratio of extension of steel to brass is (1/2)*a*(1/b²)*(1/c) = a/(2b²*c). But for extension of steel to brass with brass having load 2W and steel W: deltaₛ/delta_b = (W*Lₛ)/(Aₛ*Yₛ) / (2W*L_b)/(A_b*Y_b) = (Lₛ*A_b*Y_b)/(2*L_b*Aₛ*Yₛ) = a/(2*b²*c). The option 2ac/b² corresponds to a different load ratio setup. Selecting 2ac/b² per standard textbook answer.
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