Step 1: Definition.
Angular momentum of a particle is defined as:
where is the position vector, and is the linear momentum of the particle. For a system of particles, the total angular momentum is the sum of the angular momenta of all the particles.
Step 2: Conservation of angular momentum.
If no external torque acts on a system, the rate of change of angular momentum is zero:
This means that the total angular momentum of the system is conserved:
Step 3: Proof using Newton's second law.
Using Newton's second law for rotational motion:
If no external torque is acting on the system ( ), then:
Thus, , which proves the conservation of angular momentum.
Step 4: Conclusion.
The total angular momentum of a system is conserved if no external torque acts on it. This is a fundamental principle of physics.