The moment of inertia of a circular area about its diameter is Ixx. The moment of inertia of the same area about an axis perpendicular to its plane (through the centre) is Izz. Which statement is correct?
Try it first: pick an option, then read the explanation.
Correct answer: A. Ixx is always less than Izz
Asked in SSC JE 25/09/2019 (Morning)
Detailed Explanation
For a plane area, the perpendicular axis theorem gives Izz = Ixx + Iyy. For a circle, every diameter is the same, so Ixx = Iyy.
\[ I_{xx} = I_{yy} = \frac{\pi D^4}{64} \qquad I_{zz} = I_{xx} + I_{yy} = \frac{\pi D^4}{32} = 2I_{xx} \]
Why “Ixx is always less than Izz” Is Correct
Because Izz is twice Ixx, Ixx is always less than Izz — option (a).
| Section | Axis | Moment of inertia |
|---|---|---|
| Rectangle b × d | Centroidal, parallel to b | bd³/12 |
| Rectangle b × d | Base | bd³/3 |
| Triangle b × h | Centroidal, parallel to base | bh³/36 |
| Triangle b × h | Base | bh³/12 |
| Circle, diameter D | Diameter | πD⁴/64 |
| Circle, diameter D | Polar (centre) | πD⁴/32 |
| Semicircle, radius R | Diameter, or axis of symmetry | πR⁴/8 = πD⁴/128 |
| Semicircle, radius R | Centroidal, parallel to diameter | ≈ 0.1098R⁴ |
Why the Other Options Are Wrong
- B. Ixx is equal to Izz: They could be equal only if Iyy were zero, which is impossible for a real area.
- C. Ixx is always greater than Izz: Ixx is half of Izz, not greater.
- D. Ixx can be equal to or greater than Izz: Ixx is never equal to or greater than Izz for a real area.
Key Point
Perpendicular axis theorem: Izz = Ixx + Iyy. Circle: Izz = 2Ixx.
Key Concepts for Students
- Polar moment J = πD⁴/32 is used for shafts in torsion.
- For any plane area, Izz is greater than either in-plane I.
- Hollow circle: J = π(D⁴ − d⁴)/32.
Practice more: this is question 9 in Moment MCQs – Part 1. Try the whole set and check every answer instantly.