3.11: Torque and Rate of Change of Angular Momentum
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The rate of change of the total angular momentum of a system of particles is equal to the sum of the external torques on the system.
Thus:
L=∑iri×pi
∴˙L=∑i˙ri×˙pi
But the first term is zero, because ˙r and pi are parallel.
Also
˙ri=Fi+∑Fij
˙Li=∑iri×(ri+∑jFij)=∑iri×Fi+∑iri×∑jFii
∴∑iri×Fi+∑iri×∑jFii
But ∑i∑jFij=0 by Newton’s third law of motion, and so ∑i∑jri×Fij=0.
Also ∑iri×Fi=τ, and so we arrive at
˙L=τ
which was to be demonstrated.
If the sum of the external torques on a system is zero, the angular momentum is constant.