Question

State and prove the principle of conservation of angular momentum.

Hint:

Angular momentum is conserved in isolated systems where no external torque acts. This principle applies to systems ranging from particles to planets.

Answer 1

Step 1: Definition.
Angular momentum \( L \) of a particle is defined as: \[ L = r \times p \] where \( r \) is the position vector, and \( p \) 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: \[ \frac{dL}{dt} = \tau_{\text{ext}} = 0 \] This means that the total angular momentum of the system is conserved: \[ L = \text{constant} \]
Step 3: Proof using Newton's second law.
Using Newton's second law for rotational motion: \[ \tau = \frac{dL}{dt} \] If no external torque is acting on the system (\( \tau = 0 \)), then: \[ \frac{dL}{dt} = 0 \] Thus, \( L = \text{constant} \), 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.