Torsional rigidity identification: For a prismatic circular shaft, which expression represents torsional rigidity (torque per radian of twist over the full length)?
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AT / J
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BT / θ
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CT / r
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DT / G
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EG * J / L
Answer
Correct Answer: T / θ
Explanation
Introduction / Context:Torsional rigidity characterizes how resistant a shaft is to twisting under applied torque. It is central to shaft design for power transmission, where angular deflection limits are specified in addition to strength.
Given Data / Assumptions:
- Straight, prismatic shaft, length L, polar second moment J, shear modulus G.
- Linear elastic torsion: θ = T * L / (G * J).
Concept / Approach:Rearranging the torsion formula shows torque per radian of twist (stiffness) equals GJ/L. Since T/θ = GJ/L, the expression T/θ directly represents torsional rigidity over the entire length (units: N·m/rad).
Step-by-Step Solution:
θ = (T * L) / (G * J)T / θ = (G * J) / LHence torsional rigidity = T / θVerification / Alternative check:Dimension check: GJ/L has dimensions of torque per radian, consistent with stiffness. Numerical examples confirm proportionality between T and θ for given G, J, and L.
Why Other Options Are Wrong:
- T/J and T/r are not stiffness measures.
- T/G has incorrect dimensions.
- G * J / L is a correct explicit form, but the option asked for an expression in terms of T and θ; thus T/θ is the matching choice.
Common Pitfalls:Confusing section property J with polar radius r; mixing strength (τ = T r / J) with stiffness (θ relation).
Final Answer:T / θ