The angle of twist (wind-up) of a close-coiled helical spring under an axial torque T is given by (D = mean coil diameter, d = wire diameter, n = number of coils):
Try it first: pick an option, then read the explanation.
Correct answer: B. 64TDn/(Ed⁴)
Asked in SSC JE 2012
Detailed Explanation
Under an axial couple the whole wire is in pure bending with M = T. The rotation of one end relative to the other is ML/EI.
\[ \theta = \frac{ML}{EI} = \frac{T\,(\pi D n)}{E\,(\pi d^4/64)} = \frac{64\,T D n}{E d^4} \]
Why “64TDn/(Ed⁴)” Is Correct
The angle of twist is 64TDn/(Ed⁴).
| Close-coiled helical spring | Axial load W | Axial torque (couple) T |
|---|---|---|
| Main action on the wire | Twisting (torsion) T = WR | Bending M = T |
| Maximum stress | τ = 16WR/(πd³) | σ = 32T/(πd³) |
| Deformation | δ = 64WR³n/(Gd⁴) | θ = 64TDn/(Ed⁴) = TL/(EI), L = πDn |
| Stiffness | k = W/δ = Gd⁴/(64R³n) | T/θ = Ed⁴/(64Dn) |
R = mean coil radius, D = 2R mean coil diameter, d = wire diameter, n = number of active coils.
Why the Other Options Are Wrong
- A. 32TDn/(Ed⁴): 32 comes from the polar J (πd⁴/32); bending uses I = πd⁴/64.
- C. 64Tdn/(ED⁴): Swaps D and d — wire diameter must be in the denominator to the fourth power.
- D. 32Tdn/(ED⁴): Wrong constant and swapped diameters.
Key Point
Axial torque: θ = TL/EI with L = πDn, I = πd⁴/64 → 64TDn/Ed⁴.
Key Concepts for Students
- Uses E because the wire bends.
- Axial load deflection uses G: δ = 64WR³n/Gd⁴.
- Thicker wire is much stiffer (d⁴).
Practice more: this is question 8 in Spring MCQs – Part 1. Try the whole set and check every answer instantly.