To find the radius of gyration of a sphere, you typically determine its moment of inertia relative to a specific axis and then use the relationship between moment of inertia, mass, and the radius of gyration.
The radius of gyration ($k$) represents the distance from an axis of rotation where the total mass of a body could be concentrated as a point mass, having the same moment of inertia as the original body. It is defined by the formula:
$I = Mk^2$
Where:
- $I$ is the moment of inertia of the body about the specific axis.
- $M$ is the total mass of the body.
- $k$ is the radius of gyration about that same axis.
From this formula, the radius of gyration can be calculated as:
$k = \sqrt{\frac{I}{M}}$
Therefore, the core task is to find the moment of inertia ($I$) of the sphere relative to the axis of interest.
Finding the Radius of Gyration of a Sphere: A Step-by-Step Guide
Based on the method outlined in the provided reference, finding the radius of gyration often involves considering an axis that may not pass through the sphere's center. Here are the steps:
Step 1: Determine the Moment of Inertia of the Sphere
The first step is to know the moment of inertia of the sphere. The fundamental moment of inertia for a solid sphere of mass $M$ and radius $R$ about an axis passing through its center is:
$I_{center} = \frac{2}{5}MR^2$
For a hollow sphere (spherical shell) of mass $M$ and radius $R$ about an axis through its center, the moment of inertia is:
$I_{center, hollow} = \frac{2}{3}MR^2$
The method outlined in the reference suggests dealing with cases where the axis might not be through the center, starting with determining the moment of inertia.
Step 2: Use the Parallel Axis Theorem
The reference specifically mentions using the Parallel Axis Theorem. This theorem is used when the axis of rotation is parallel to an axis passing through the center of mass, but is displaced by a distance.
If you need to find the moment of inertia ($I_{parallel}$) about an axis parallel to the central axis and at a distance $d$ from it, the theorem states:
$I{parallel} = I{center} + Md^2$
Where:
- $I_{center}$ is the moment of inertia about the axis passing through the center of mass (which we determined in Step 1).
- $M$ is the total mass of the sphere.
- $d$ is the perpendicular distance between the two parallel axes.
Applying this for a solid sphere:
$I_{parallel} = \frac{2}{5}MR^2 + Md^2$
Step 3: Simplify the Moment of Inertia
This step involves simplifying the expression obtained for the moment of inertia. For the case using the Parallel Axis Theorem, the simplified expression for the moment of inertia about the parallel axis is $I_{parallel} = \frac{2}{5}MR^2 + Md^2$.
Step 4: Relate Moment of Inertia to Radius of Gyration
The final step is to use the relationship $I = Mk^2$ to find the radius of gyration ($k$).
Using the moment of inertia $I_{parallel}$ calculated in the previous steps:
$I{parallel} = Mk{parallel}^2$
Substituting the expression for $I_{parallel}$:
$\frac{2}{5}MR^2 + Md^2 = Mk_{parallel}^2$
To find $k_{parallel}$, divide both sides by $M$:
$\frac{\frac{2}{5}MR^2 + Md^2}{M} = k_{parallel}^2$
$\frac{2}{5}R^2 + d^2 = k_{parallel}^2$
Taking the square root of both sides gives the radius of gyration about the parallel axis:
$k_{parallel} = \sqrt{\frac{2}{5}R^2 + d^2}$
This shows how the radius of gyration changes depending on the axis of rotation.
Radius of Gyration for Common Cases
Here are the formulas for the radius of gyration of a solid sphere relative to common axes:
Case 1: Axis Through the Center of the Sphere
This is the simplest case. Using $I{center} = \frac{2}{5}MR^2$ and $I{center} = Mk_{center}^2$:
$Mk_{center}^2 = \frac{2}{5}MR^2$
$k_{center}^2 = \frac{2}{5}R^2$
$k_{center} = \sqrt{\frac{2}{5}}R$
Case 2: Axis Parallel to the Center Axis
As derived using the Parallel Axis Theorem (Steps 2-4 above), for an axis parallel to the central axis at a distance $d$:
$k_{parallel} = \sqrt{\frac{2}{5}R^2 + d^2}$
Notice that if $d=0$, this formula reduces to the central axis case: $k_{parallel} = \sqrt{\frac{2}{5}R^2 + 0^2} = \sqrt{\frac{2}{5}}R$.
Summary of Radius of Gyration for a Solid Sphere
| Axis of Rotation | Moment of Inertia ($I$) | Radius of Gyration ($k$) |
|---|---|---|
| Through the Center | $\frac{2}{5}MR^2$ | $\sqrt{\frac{2}{5}}R$ |
| Parallel to Center Axis (dist d) | $\frac{2}{5}MR^2 + Md^2$ | $\sqrt{\frac{2}{5}R^2 + d^2}$ |
Note: These formulas are for a solid sphere. The values for a hollow sphere would differ based on its central moment of inertia ($\frac{2}{3}MR^2$).
Practical Insight: Why Use Radius of Gyration?
The radius of gyration simplifies rotational dynamics problems. Instead of calculating the moment of inertia for a complex shape, you can treat the object as a point mass ($M$) located at a distance ($k$) from the axis of rotation, resulting in the same rotational inertia ($I = Mk^2$). This is useful in structural engineering and mechanics to analyze how objects resist angular acceleration.
