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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].
The problem statement, all variables and given/known data. There is the moment of inertia about an x and a y axis named, [itex]I_{x}, I_{y}[/itex]. Then there is the moment of inertia about the centroidal x and y axis named, [itex]\overline{I}_{x}, \overline{I}_{y}[/itex]. Often we can look up these values in a table (like the figure included) and apply another idea called the parallel-axis therom to find moments about different axes.