Based on the provided reference information, the radius of gyration of a uniform solid sphere of radius r about its diameter is √25 r.
Understanding the Radius of Gyration
The radius of gyration is a concept used in rotational mechanics to describe how the mass of an object is distributed around an axis of rotation. It is represented by the symbol k and is defined by the relationship between the moment of inertia (I), the total mass (M) of the object, and the radius of gyration:
- *I = M k²**
This means that if the entire mass of the object were concentrated at a distance k from the axis of rotation, it would have the same moment of inertia as the actual object with its mass distributed in space. The radius of gyration is essentially a measure of the effective distance of the mass from the axis of rotation.
Several factors influence the radius of gyration:
- Mass distribution: How the mass is spread out relative to the axis.
- Axis of rotation: The specific line about which the object rotates.
- Shape of the object: Different shapes have different mass distributions.
Radius of Gyration for a Uniform Solid Sphere
The question asks about the radius of gyration of a sphere of radius r. To provide a specific value, we need to know the axis of rotation. The provided reference specifies the scenario:
"The radius of gyration of a uniform solid sphere of radius R is √25 R for rotation about its diameter."
This tells us:
- Object Type: Uniform solid sphere (mass is evenly distributed throughout its volume).
- Radius: Given as 'R' in the reference, which corresponds to 'r' in your question.
- Axis of Rotation: About its diameter (an axis passing through the center of the sphere).
According to the reference, the radius of gyration (k) for this specific case (radius R, about diameter) is √25 R.
Applying this information to a sphere of radius r, we substitute R with r.
Therefore, the radius of gyration of a uniform solid sphere of radius r about its diameter, based on the provided reference, is:
- k = √25 r
Simplifying the value from the reference, √25 is equal to 5.
So, based on the reference:
- k = 5r
Summary Table
| Property | Value | Notes |
|---|---|---|
| Object Type | Sphere | Uniform solid sphere specified by reference |
| Radius | r | Corresponds to R in the reference |
| Axis of Rotation | Diameter | Specified in the reference |
| Radius of Gyration (k) | √25 r | Value directly from the reference |
| Simplified Value | 5r | Simplification of √25 |
This table summarizes the specific conditions and the resulting radius of gyration value as derived from the provided reference.
