Angular Momentum: Rigid Body

The angular momentum of a rigid body equals the sum of angular momenta from all its tiny particles. Each particle's angular momentum is the cross product of its position vector and linear momentum. By expressing linear velocity in terms of angular velocity, the total angular momentum simplifies to Iω, where I is the moment of inertia and ω is angular velocity.

Moment of inertia represents the sum of mass times the square of distance for all particles: I = Σmiri². This quantity directly determines angular momentum through the equation L = Iω. When moment of inertia is constant, angular momentum and angular velocity remain parallel to the rotational axis, making calculations straightforward.

For rigid bodies symmetric about the rotation axis, perpendicular angular momentum components from opposite sides cancel out. Only the component along the z-axis remains, making total angular momentum parallel to the rotational axis. This symmetry simplifies the vector analysis and ensures consistent directional alignment.

When rotation is asymmetric, perpendicular angular momentum components do not cancel. The total angular momentum vector traces a cone around the rotation axis rather than aligning with it. This creates a net torque acting on the body even though angular velocity remains constant, requiring analysis of the rotation of asymmetric top behavior.

Each particle's angular momentum equals miviri, where mi is mass, vi is tangential velocity, and ri is distance from the axis. Summing all particle contributions yields total angular momentum. When particles rotate in the same plane, their angular momenta add vectorially perpendicular to that plane along the z-axis.

Axis symmetry determines whether angular momentum aligns with the rotation axis. Symmetric axes allow perpendicular components to cancel, simplifying calculations to L = Iω. Without symmetry, the calculation becomes complex because angular momentum components perpendicular to the axis must be individually tracked and combined.

Tangential velocity vi equals ωri for a particle at distance ri from the rotation axis. Substituting into the angular momentum formula miviri yields miω(ri²). Summing across all particles produces total angular momentum Iω, directly connecting rotational motion to angular momentum through the moment of inertia.