About the principal axes of a section:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 02/03/2017 (Evening)
Detailed Explanation
For any pair of perpendicular axes through a point, the area has two moments of inertia (I_x, I_y) and a product of inertia I_xy = ∫xy dA. Rotating the axes changes all three.
\[ \tan 2\theta_p = \frac{-2I_{xy}}{I_x - I_y},\qquad I_{1,2} = \frac{I_x+I_y}{2} \pm \sqrt{\left(\frac{I_x-I_y}{2}\right)^2 + I_{xy}^2} \]
Why “the product of inertia is zero” Is Correct
At the principal orientation the product of inertia is zero, and the two moments of inertia are the maximum and minimum possible values. Any axis of symmetry is a principal axis.
Why the Other Options Are Wrong
- A. the sum of the moments of inertia is zero: Moments of inertia are always positive, so their sum cannot be zero; it is constant for all rotations.
- B. the difference of the moments of inertia is zero: Equal moments of inertia happen only for special shapes like a circle or square.
- D. none of these: Zero product of inertia is the defining property, so (c) is correct.
Key Point
Principal axes: I_xy = 0; I is maximum about one and minimum about the other.
Key Concepts for Students
- I_x + I_y is invariant under rotation (= polar I).
- Any axis of symmetry is a principal axis.
- Unsymmetric sections (angles) bend about their principal axes.
Practice more: this is question 21 in Bending and Shear Stress MCQs – Part 2. Try the whole set and check every answer instantly.