Inertia about axes inclined to the principal axes: The locus of the area moment of inertia is what?

Introduction:
This conceptual question checks whether you recognize the geometric representation of second moments of area about axes rotated from the principal directions. The variation of moment of inertia with axis orientation has a characteristic locus used in advanced section analysis.

Given Data / Assumptions:

• A plane area with well-defined principal moments I1 and I2 and product of inertia zero in principal axes.
• We consider axes through the centroid rotated by an angle θ relative to principal axes.

Concept / Approach:
The transformation of second moments for a rotated axis is I(θ) = (I1 + I2)/2 + (I1 − I2)/2 * cos(2θ). When you plot I(θ) in orthogonal components corresponding to the two axis directions, the tip of the vector describing the second moment traces an ellipse, known as the ellipse of inertia.

Step-by-Step Solution:
Start from inertia transformation equations for rotated axes.Note the cos(2θ) and sin(2θ) terms that generate elliptical parametric relations.Conclude that the locus of I for all θ is an ellipse whose axes align with the principal directions.

Verification / Alternative check:
In the special case I1 = I2 (axisymmetric area), the ellipse degenerates to a circle, which is consistent with the general elliptical form.

Why Other Options Are Wrong:
Straight line / parabola / hyperbola: Do not match the trigonometric (cos 2θ, sin 2θ) parametric form of inertia transformation.Circle: Only for I1 = I2; not general.

Common Pitfalls:
Confusing Mohr’s circle for stresses with inertia transformations; although both use angle-doubling relations, inertia locus is the ellipse of inertia, not a circle except in the symmetric case.

Final Answer:
ellipse.