Units check (geometric vs mass moment): The moment of inertia of an area (second moment of area) has which SI units?

Introduction / Context:
Two commonly confused quantities are the geometric (second) moment of area and the mass moment of inertia. Both are called “moment of inertia” in practice, but they have different dimensions and applications. The geometric moment of area measures resistance to bending (beam theory), while the mass moment of inertia measures resistance to angular acceleration (dynamics).

Given Data / Assumptions:

• We are asked specifically about the moment of inertia of an area, not of mass.
• Standard SI base units are to be used.

Concept / Approach:
The second moment of area I for a plane area A relative to an axis is defined by an integral of the form I = ∫ y^2 dA (or ∫ r^2 dA), where y (or r) is a length. Since dA has units of m^2 and y^2 has units of m^2, the product yields m^4. Conversely, mass moment of inertia involves ∫ r^2 dm and hence has units kg·m^2.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis of standard formulas, e.g., rectangular section about centroidal axis: I = b h^3 / 12 has units (m) * (m^3) = m^4, confirming the result.

Why Other Options Are Wrong:

• m^3 and kg/m^2: Not consistent with ∫ y^2 dA.
• kg·m^2: Units of mass moment of inertia, not area moment.
• N·m^2: Units of torque times length, irrelevant here.

Common Pitfalls:
Confusing geometric with mass moment; forgetting that area moment of inertia is purely geometric and independent of material density.

Final Answer:
m^4