In strength of materials, the radius of gyration of a rectangular section is not proportional to which of the following expressions?

Introduction / Context:
The radius of gyration (k) is an important geometric property used in buckling and structural stability calculations. It links the moment of inertia (I) to the cross-sectional area (A).

Given Data / Assumptions:

• Rectangular cross-section.
• Radius of gyration definition: k = sqrt(I / A).
• I and A are in consistent units.

Concept / Approach:
From definition: k = sqrt(I / A). Therefore, k is proportional to sqrt(I) and inversely proportional to sqrt(A). It is not proportional to sqrt(1/A) alone, because I must also be considered.

Step-by-Step Solution:
Start with k = sqrt(I / A).Break down: k ∝ sqrt(I) and k ∝ 1 / sqrt(A).Thus, any expression ignoring I (like sqrt(1/A)) is not correct.

Verification / Alternative check:
For a rectangle b × d, I = bd^3 / 12. Substituting confirms k depends on both I and A, not A alone.

Why Other Options Are Wrong:

• Square root of moment of inertia: partly correct, as I is included.
• Square root of I/A: correct by definition.
• None of these: incorrect, since one option is truly wrong.

Common Pitfalls:

• Confusing proportionalities; radius of gyration is not just a function of area or inertia alone.

Final Answer:
square root of the inverse of the area