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to derive the moment of inertia of a solid sphere of uniform density and radius R. [tex] dI = r_{\perp}^{2} dm = r_{\perp}^{2} \rho dV [/tex] â. where [itex] r_{\perp} [/itex] is the perpendicular distance to a point at r from the axis of rotation. Therefore, [itex] r_{\perp} = |\mathbf{r}|\sin{\theta} [/itex].
A table of some of these moments of inertia can be found on page 223 in the textbook. Note that for an object where the mass is all concentrated at the same distance from the axis of rotation, such as a small ball being swung in a circle on a string, the moment of inertia is simply MR2 .