If Ix and Iy are the moments of inertia of a section about the X and Y axes, the polar moment of inertia of the section is:
Try it first: pick an option, then read the explanation.
Correct answer: C. Ix + Iy
Asked in SSC JE 27/01/2018 (Morning)
Detailed Explanation
For a plane area, the moment of inertia about an axis perpendicular to the plane through O equals the sum of the moments of inertia about two perpendicular in-plane axes through O.
\[ J = \int r^2\,dA = \int (x^2 + y^2)\,dA = I_y + I_x \]
Why “Ix + Iy” Is Correct
So the polar moment of inertia is Ix + Iy.
Why the Other Options Are Wrong
- A. (Ix + Iy)/2: Half the sum is the average, not J.
- B. (Ix − Iy)/2: The difference has no physical meaning here (it appears in Mohr's circle for inertia).
- D. Ix/Iy: A ratio is dimensionless; J has units of length⁴.
Key Point
J = Ix + Iy (perpendicular axis theorem).
Key Concepts for Students
- For a circle Ix = Iy = πD⁴/64, so J = πD⁴/32.
- Holds for any plane area.
- Do not confuse with the parallel axis theorem.
Practice more: this is question 7 in Torsion MCQs – Part 1. Try the whole set and check every answer instantly.