Polar moment of inertia of a plane section: If I_x and I_y are the second moments of area about the orthogonal centroidal X and Y axes, what is the polar moment of inertia J about the centroid?

Introduction / Context:
The polar moment of inertia J of a plane area about a point (usually the centroid) is fundamental in torsion (thin sections) and in combined bending problems. It relates to resistance to rotation and to principal moments via the perpendicular axis theorem for flat areas.

Given Data / Assumptions:

• Flat lamina (plane section).
• Centroidal orthogonal axes X and Y lying in the plane of the section.
• Polar axis Z is perpendicular to the section through the same centroid.

Concept / Approach:

The perpendicular axis theorem for plane areas states that the polar second moment of area about Z through the centroid equals the sum of the second moments about any two orthogonal in-plane centroidal axes: J_z = I_x + I_y.

Step-by-Step Solution:

Verification / Alternative check:

For a circle of radius R, I_x = I_y = (π R^4)/4 and J = (π R^4)/2. Indeed, J = I_x + I_y confirms the same result.

Why Other Options Are Wrong:

Differences, ratios, or root-sum-squares have no basis in the perpendicular axis theorem. Option (e) holds only if I_x = I_y, but is not the general identity.

Common Pitfalls:

Confusing moments of inertia (area MOI) with polar mass moment of inertia; mixing centroidal with base axes.

Final Answer:

J = I_x + I_y