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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.

Difficulty: Easy

Correct Answer: k_eq = (k₁ k₂)/(k₁ + k₂)

Explanation:

Given data

  • Two shaft segments in series with torsional spring constants k₁ and k₂.


Concept/Approach
For springs in series (same torque T), total twist θ = θ₁ + θ₂ = T/k₁ + T/k₂.


Step-by-step
Define k_eq by T/θ = k_eq ⇒ 1/k_eq = 1/k₁ + 1/k₂.Therefore, k_eq = (k₁ k₂)/(k₁ + k₂).


Verification
If k₁ = k₂ = k, then k_eq = k/2 (softer than either alone), which is consistent for series combination.


Final Answer
k_eq = (k₁ k₂)/(k₁ + k₂)

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