Principal Axes of an Area – Property of Product of Inertia For the principal centroidal axes of a plane section, which statement about the second moments is correct?

Introduction / Context:
Principal axes are the orthogonal centroidal axes for which the product of inertia vanishes. They align with directions where the area's inertia matrix is diagonal, simplifying bending and torsion analysis.

Given Data / Assumptions:

• Plane area with centroidal axes x and y.
• Principal axes are obtained by rotating axes so that coupling terms disappear.

Concept / Approach:
The product of inertia I_xy measures coupling of bending about x and y. On principal axes, the inertia matrix is diagonal, thus I_xy = 0, while I_x and I_y are generally nonzero and unequal unless the shape is isotropic.

Step-by-Step Solution:
Inertia matrix about centroid: [[I_x, -I_xy], [-I_xy, I_y]]Rotate axes to eliminate off-diagonal termsOn principal axes: I_xy = 0; I_x and I_y are extremal

Verification / Alternative check:
Mohr's circle of area moments shows that principal directions occur where the circle intersects the horizontal axis, i.e., I_xy = 0.

Why Other Options Are Wrong:

• Sum or difference zero: not generally true; depends on geometry.
• None of these / both product and sum zero: contradicts standard inertia properties.

Common Pitfalls:
Confusing principal axes (I_xy = 0) with axes of symmetry (which may coincide but are not guaranteed).

Final Answer:
Product of moment of inertia is zero