Composite torsional shaft: Equivalent torsional spring constant for two serial segments between rigid ends. Given two stepped portions with torsional spring constants k₁ and k₂ in series (same torque, angular twists add). Choose the correct expression for the equivalent spring constant k_eq.
Mechanical Engineering
Strength of Materials
Difficulty: Easy
Choose an option
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Ak_eq = (k₁ k₂)/(k₁ + k₂)
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Bk_eq = k₁ + k₂
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Ck_eq = (k₁ + k₂)/2
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Dk_eq = (k₁ k₂)/(k₂ − k₁)
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Ek_eq = √(k₁ k₂)
Answer
Correct Answer: k_eq = (k₁ k₂)/(k₁ + k₂)
Explanation
Given data
- Two shaft segments in series with torsional spring constants k₁ and k₂.
Concept/ApproachFor springs in series (same torque T), total twist θ = θ₁ + θ₂ = T/k₁ + T/k₂.
Step-by-stepDefine k_eq by T/θ = k_eq ⇒ 1/k_eq = 1/k₁ + 1/k₂.Therefore, k_eq = (k₁ k₂)/(k₁ + k₂).
VerificationIf k₁ = k₂ = k, then k_eq = k/2 (softer than either alone), which is consistent for series combination.
Final Answerk_eq = (k₁ k₂)/(k₁ + k₂)