Short on time? This page is the recap. For diagrams, the graph test and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- The total angular momentum of a system about an axis is the sum of the signed angular momenta of its parts about that axis.
- Angular momentum is conserved in every interaction. It can move between objects but is never created or destroyed.
- For a chosen system: if the net external torque is zero, its L is constant. If not, its change in L equals the angular impulse from outside.
- Parts of a system exert equal and opposite angular impulses on each other (Newton’s third law), so internal torques cannot change the total.
- A non-rigid system can change ω by moving mass towards or away from the axis, with L unchanged.
Key relationships
| Relationship | When it applies | Use it to |
|---|---|---|
| L_total = ΣIω + Σrmv sin θ | always, all about one axis | add up the parts of a system |
| L_initial = L_final | net external torque on the system is zero | collisions, landings, shape changes |
| I₁ω₁ = I₂ω₂ | one system changing shape | predict the new ω |
| ΔL_system = τ_ext Δt | external torque acts | find the transfer to or from the surroundings |
| K = ½Iω² = L² / (2I) | any rotating rigid body | track energy when L is fixed |
| ω against 1/I is a straight line through the origin | L constant | test conservation from data |
Assumptions behind the numbers
- Axles are frictionless unless stated; otherwise the axle exerts an external torque.
- Forces that act through the axis, or parallel to it, exert no torque about it.
- Small objects far from the axis are treated as points, with I = mr².
Mistakes to avoid
- Averaging angular speeds instead of using I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω.
- Assuming kinetic energy is conserved. Sticking loses K; pulling mass in gains K.
- Using linear momentum when a pivot acts. The pivot’s force is external; its torque about the pivot is zero.
- Dropping signs for parts spinning in opposite senses.
- Leaving out mr² for an object that sticks to a rotating body.
- Choosing a system with an unknown external torque, then claiming L is constant.
Quick self-check
- A spinning system with no external torque halves its rotational inertia. What happens to ω and to K? (Both double.)
- A disk (I = 0.20 kg·m²) at 6.0 rad/s has a second disk (I = 0.10 kg·m²), at rest, dropped onto it on the same axle. What is the common angular speed? (4.0 rad/s)
- Is a wheel’s angular momentum constant while a brake pad rubs on it? (Not for the wheel alone: the pad exerts an external torque. For wheel + brake + frame + Earth, yes.)
Next: practice questions.