Second moment comparison: For equal overall depth in bending, what is the ratio of the second moment of area (about the centroidal axis parallel to the breadth) of a circular plate to that of a square plate?

Introduction / Context:Comparing second moments of area helps understand relative flexural stiffness for different cross-sectional shapes when depth in the bending direction is the same. Here, the circular plate has diameter equal to the square plate depth, enabling a direct comparison.

Given Data / Assumptions:

• Square plate side = d (this is the common depth).
• Circular plate diameter = d (same depth).
• Centroidal axis is parallel to the breadth, through the centroid.

Concept / Approach:Use standard second moment formulae: for a square of side d, I_square = d^4 / 12. For a circle of diameter d, I_circle = π d^4 / 64 about any centroidal diametral axis. The ratio is I_circle / I_square = (π/64) / (1/12) = 12π / 64 = 3π/16.

Step-by-Step Solution:1) I_square = d^4 / 12 (about centroidal axis parallel to a side).2) I_circle = π d^4 / 64 (about centroidal diameter).3) Compute ratio: (π d^4 / 64) / (d^4 / 12) = (π / 64) * 12 = 3π/16.

Verification / Alternative check:Numerically, 3π/16 ≈ 0.589, which is less than 1, indicating a circular plate has a smaller second moment about that axis than a square plate of equal depth, consistent with shape distribution of area.

Why Other Options Are Wrong:Generic statements like less/equal/more do not quantify the exact ratio.Other choices do not match the standard geometric relationships.

Common Pitfalls:Mixing up radius versus diameter; using polar moment instead of the planar second moment; or comparing unequal depths.

Final Answer:Equal to 3π/16