A metal string A is suspended from a rigid support and supports a block of mass M. A second block of mass 2M is suspended below the first block by another string B. The two strings have equal cross-sectional area. The ratio of their lengths is L(A):L(B) = 2:1, and their Young's moduli satisfy Y(A) /Y(B) = 0.5. What is the ratio of their elongations ΔL(A) / ΔL(B)?
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Question

A metal string A is suspended from a rigid support and supports a block of mass M. A second block of mass 2M is suspended below the first block by another string B. The two strings have equal cross-sectional area. The ratio of their lengths is L(A):L(B) = 2:1, and their Young's moduli satisfy Y(A) /Y(B) = 0.5. What is the ratio of their elongations ΔL(A) / ΔL(B)?

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G1 Physics · Accepted Answer

Let the cross-sectional area of each string be A₀. The upper string A supports both masses M and 2M, so its tension is T(A) = 3Mg. The lower string B supports only the 2M block: T(B) = 2Mg. Let L(A) = L, L(B) = L ÷ 2. Given, Y(A) ÷ Y(B) = 0.5, so Y(B) = 2Y(A). For a stretched wire, ΔL = TL ÷ (AY). Hence, ΔL(A) = 3MgL ÷ [A₀Y(A)]. For string B: ΔL(B) = 2Mg(L ÷ 2) ÷ [A₀·2Y(A)] = MgL ÷ [2A₀Y(A)]. Therefore, ΔL(A) ÷ ΔL(B) = 3 ÷ (1 ÷ 2) = 6. Correct option: D.