Angular momentum; rate of change equals moment

Angular momentum

The angular momentum (moment of momentum) of a particle about a fixed point O is the moment of its linear momentum about O:

H_O = r × m v

where r is the position vector of the particle from O and m v its linear momentum. Its magnitude is H_O = m v r sin φ (φ = angle between r and v), with units kg·m²/s. For motion in a plane in polar coordinates, H_O = m r² θ̇.

Proof

Differentiate H_O = r × m v with respect to time:

dH_O/dt = (dr/dt) × m v + r × m (dv/dt)

• The first term is v × m v = m (v × v) = 0.
• In the second term m (dv/dt) = m a = F, by Newton's second law, F being the resultant force.

Therefore dH_O/dt = r × F = M_O

The rate of change of the angular momentum about a point equals the moment of the resultant force about the same point. If M_O = 0 (for example for a central force), H_O is constant, which is the principle of conservation of angular momentum.