Question
Download Solution PDFThe maximum value of the radius of gyration is obtained if a material _____.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Radius of gyration
- The radius of gyration (k) is a property that characterizes the distribution of an object's area or mass about an axis. It is defined as the square root of the ratio of the moment of inertia to the area (for area) or mass (for mass).
- The equation for the radius of gyration is given by:
\(k=\sqrt{\frac{I}{A}}\)
where,
k is the radius of gyration, I is the moment of inertia about the axis, A is the area (for the case of mass distribution).
Explanation:
Is placed at the centroid:
- Placing the material at the centroid minimizes the distance of each element of the material from the axis of rotation.
- This minimizes the moment of inertia and results in the minimum value for the radius of gyration.
Has the maximum area:
- The radius of gyration is inversely proportional to the square root of the area. Therefore, a larger area results in a smaller radius of gyration.
Is placed farthest from the centroid:
- Placing the material farther from the centroid increases the distance of each element from the axis of rotation, resulting in a larger moment of inertia.
- This leads to the maximum value of the radius of gyration.
Is placed nearest the centroid:
- Placing the material nearest to the centroid minimizes the moment of inertia and leads to the minimum value of the radius of gyration.
In summary, "Is placed farthest from the centroid," results in the maximum value of the radius of gyration.
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