The strain energy stored per unit volume in a hollow shaft under pure torque, when the maximum shear stress is fs, is 17fs²/(64N), where N is the modulus of rigidity. The ratio of inner diameter to outer diameter is:
Try it first: pick an option, then read the explanation.
Correct answer: C. 1/4
Asked in SSC JE 27/01/2018 (Morning)
Detailed Explanation
Because τ varies linearly with radius, the average energy density in a hollow shaft depends on the diameter ratio.
\[ \frac{U}{V} = \frac{f_s^2}{4N}\cdot\frac{D^2 + d^2}{D^2} = \frac{17f_s^2}{64N} \;\Rightarrow\; 1 + \frac{d^2}{D^2} = \frac{17}{16} \;\Rightarrow\; \frac{d^2}{D^2} = \frac{1}{16} \;\Rightarrow\; \frac{d}{D} = \frac14 \]
Why “1/4” Is Correct
The diameter ratio is 1/4.
| Quantity | Solid shaft (diameter D) | Hollow shaft (D outer, d inner) |
|---|---|---|
| Polar moment of inertia J | πD⁴/32 | π(D⁴ − d⁴)/32 |
| Polar modulus Zp = J/R | πD³/16 | π(D⁴ − d⁴)/(16D) |
| Maximum shear stress τ = T/Zp | 16T/(πD³) | 16TD/(π(D⁴ − d⁴)) |
| Angle of twist θ | TL/(GJ) | TL/(GJ) |
| Strain energy per unit volume | τ²/(4G) | (τ²/4G)·(D² + d²)/D² |
Torsion equation: T/J = τ/R = Gθ/L. GJ is the torsional rigidity; GJ/L is the torsional stiffness.
Why the Other Options Are Wrong
- A. 1/2: d/D = 1/2 gives (1 + 1/4)/4 = 5/16 = 20/64, not 17/64.
- B. 1/3: d/D = 1/3 gives (1 + 1/9)/4 = 10/36 ≈ 17.8/64.
- D. 1/5: d/D = 1/5 gives (1 + 1/25)/4 = 26/100 ≈ 16.6/64.
Key Point
Hollow shaft: U/V = (τ²/4G)(D² + d²)/D²; solid shaft: τ²/4G.
Key Concepts for Students
- Solid shaft stores only half the energy of uniform shear (τ²/2G).
- Hollow shafts use the material more effectively.
- N and G both denote the modulus of rigidity.
Practice more: this is question 8 in Torsion MCQs – Part 1. Try the whole set and check every answer instantly.