PlanetPhysics/Rotational Inertia of a Solid Sphere
The Rotational Inertia or moment of inertia of a solid sphere rotating about a diameter is
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
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.
\begin{figure} \includegraphics[scale=.6]{InertiaSphere.eps} \caption{Rotational inertia of a solid sphere rotating about a diameter, z} \end{figure}
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 term is a little more involved and we substitude in the trigonometric relation
and the integrand for the term becomes
Using the technique of u substitution to solve this
so
completing the integration yields
Finally, the term integrates to so
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)
References
edit[1] Halliday, D., Resnick, R., Walker, J.: "fundamentals of physics".\, 5th Edition, John Wiley \& Sons, New York, 1997.