Engineering Mechanics – Area Moment of Inertia\nWhat is the second moment of area (area moment of inertia) of a square plate of side a about one of its diagonals passing through the centroid?

Introduction / Context:
This question tests knowledge of second moment of area (also called area moment of inertia) for standard plane figures used in strength of materials and structural analysis. For a square plate, common reference axes are the centroidal x–y axes (parallel to the sides) and the centroidal axes along the diagonals. Because a square has fourfold symmetry, several centroidal axes become principal axes with equal moments.

Given Data / Assumptions:

• Plane figure: square of side a.
• Axis: along a diagonal, through the centroid (in-plane area moment).
• Homogeneous thin plate; standard engineering definitions apply.

Concept / Approach:
For a square of side a, the centroidal area moments about axes parallel to the sides are Ix = Iy = a^4 / 12. For a square, the principal centroidal axes also include the two diagonals; by symmetry, the in-plane moment about any centroidal principal axis equals a^4 / 12 as well. Another quick route is to use Mohr’s circle for area moments with Ix = Iy and product of inertia Ixy = 0, which implies every centroidal axis is principal with the same value.

Step-by-Step Solution:
Known: Ix = Iy = a^4 / 12 for a square. For a square, diagonal axes are principal centroidal axes because of symmetry. Therefore, Id (about any centroidal diagonal) = a^4 / 12.

Verification / Alternative check:
Using Mohr’s circle: center = (Ix + Iy)/2 = a^4/12, radius = sqrt(((Ix − Iy)/2)^2 + Ixy^2) = 0, so all centroidal orientations have the same value = a^4/12.

Why Other Options Are Wrong:
a^2/8, a^3/12: wrong dimensions; area moments scale with length^4. a^4/16 or a^4/24: do not match the principal value for a square's centroidal axes.

Common Pitfalls:
Confusing polar moment J = Ix + Iy = a^4/6 with an in-plane moment about a single axis. Assuming diagonal gives a different value; for a square it does not.

Final Answer:
a^4 / 12