Definition check: radius of gyration True or False — “The radius of gyration is the distance where the whole mass (or area) of a body is assumed to be concentrated.”

Introduction / Context:
Radius of gyration is fundamental in strength of materials and dynamics because it links mass distribution to rotational inertia. Misstatements often omit the crucial reference to an axis, which this question highlights.

Given Data / Assumptions:

• Standard engineering definition is intended.
• Moment of inertia I and total mass m (or area A) exist for a chosen axis.

Concept / Approach:
The radius of gyration k about a given axis is defined by I = m * k^2 (or I = A * k^2 for area moments). It is the distance from that axis at which the entire mass (or area) could be imagined concentrated so that it would have the same moment of inertia as the actual distribution. Reference to an axis is essential.

Step-by-Step Solution:

Verification / Alternative check:
For a thin ring of radius R about its central axis, I = m * R^2, so k = R. If you change the axis (e.g., a diameter), I changes and so does k, confirming the axis dependence.

Why Other Options Are Wrong:

• Correct / Partly correct: Both fail because they ignore the axis requirement.
• Correct only for point masses: Trivializes the concept; definition applies to distributed mass.
• Meaningless without density: Density is not required for the definition; I and m (or A) suffice.

Common Pitfalls:
Quoting radius of gyration without specifying the axis, leading to misuse in column buckling (k = r) and shaft design formulas.

Final Answer:
Incorrect