The Rotational Inertia or moment of inertia of a solid sphere rotating about a diameter is

(1)

This can be shown in many different ways, but here we have chosen integration in spherical coordinates to give the reader practice in this coordinate system. If we choose an axis such as the z axis, then we just have one moment of inertia given by

(2)

It is important to understand this distinction and the more general case about an arbitrary axis is handled by the inertia tensor. Since we have chosen z as our axis of rotation, then z in formula (2) is the distance from dm (dV) to the z axis. In the figure below this is shown as the purple line.

Then from spherical coordiantes we obtain z through

leaving us with the integral

Assuming a constant density throughout the sphere converts the infinitesimal mass dm to

and in spherical coordinates the infinitesmal volume dV is given by

giving the final function to integrate as

Integrating the r term is simply

The 𝜃 term is a little more involved and we substitude in the trigonometric relation

and the integrand for the 𝜃 term becomes

Using the technique of u substitution to solve this

so

completing the integration yields

Finally, the ϕ term integrates to 2π so

(3)

Using the simple formula for density

and we know that the volume of a sphere is

plugging these into (3) gives us our original equation in (1)

0.1 References

[1] Halliday, D., Resnick, R., Walker, J.: ”fundamentals of physics”. 5th Edition, John Wiley & Sons, New York, 1997.