Angular momentum
Solid bodies

Angular momentum is also an extremely useful concept for describing rotating rigid bodies such as a gyroscope or a rocky planet.For a continuous mass distribution with density function ρ (r), a differential volume element dV with position vector r within the mass has a mass element dm = ρ (r) dV. Therefore, the infinitesimal angular momentum of this element is:

d L = r × d m v = r × ρ (r) d V v = d V r × ρ (r) v {\displaystyle d\mathbf {L} =\mathbf {r} \times dm\mathbf {v} =\mathbf {r} \times \rho (\mathbf {r})dV\mathbf {v} =dV\mathbf {r} \times \rho (\mathbf {r})\mathbf {v} }

and integrating this differential over the volume of the entire mass gives its total angular momentum:

L = ∫ V d V r × ρ (r) v {\displaystyle \mathbf {L} =\int _{V}dV\mathbf {r} \times \rho (\mathbf {r})\mathbf {v} }

In the derivation which follows, integrals similar to this can replace the sums for the case of continuous mass.
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