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The moment of inertia about an axis is $I=\int r_{\perp }^{2}dm$. The parallel axis theorem states $I=I_{\mathrm{CM}}+M{a}^2$, where a is the perpendicular separation from a parallel axis through the centre of mass, and M is total mass.

Since $\rho =k{r}^2$, the density coefficient has units $\mathrm{kg}{\mathrm{m}}^{-5}$. The density-power cylindrical moment of inertia follows from cylindrical shells:For an offset pivot, the physical pendulum equation is $(I_{\mathrm{CM}}+M{a}^2)\ddot{\theta }=-Mga\sin \theta$. At small amplitude,For the minimum-period offset of a physical pendulum, minimizing $T^2$ is equivalent to minimizing $a+2R^2/(3a)$. Its derivative vanishes atThe second derivative is positive. The period diverges as $a^{-1/2}$ near zero and grows as $a^{1/2}$ for large offset, giving one minimum. This assumes the stated ideal pivot can pass inside the body; an externally constrained pivot would restrict the allowable a. ^[ref]