In strength of materials, the radius of gyration of a rectangular section is not proportional to which of the following expressions?
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Asquare root of the moment of inertia
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Bsquare root of the inverse of the area
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Csquare root of the moment of inertia divided by area of the section
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Dnone of these
Answer
Correct Answer: square root of the inverse of the area
Explanation
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