Short on time? This page is the recap. For the derivation, diagrams and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Angular momentum is the rotational partner of linear momentum. It is always measured about a stated axis or point.
- A rigid body spinning about a fixed axis has L = Iω.
- An object moving in a straight line has angular momentum about any point off its path: L = rmv sin θ = mvd, where d is the perpendicular distance from the point to the line of motion. About a point on the path, L = 0.
- Angular impulse is torque × time. It has the same sign as the torque.
- The rotational impulse–momentum theorem says the net angular impulse equals the change in angular momentum. It follows from τ_net = Iα when I is constant.
- This course uses only a + or − sign for each sense of rotation. Three-dimensional directions of L are not tested.
Key relationships
| Relationship | Units | Use it to |
|---|---|---|
| L = Iω | kg·m²/s | angular momentum of a rigid body |
| L = rmv sin θ = mvd | kg·m²/s | angular momentum of a moving object about a point |
| angular impulse = τΔt | N·m·s (= kg·m²/s) | effect of a torque acting for a time |
| ΔL = L − L₀ | kg·m²/s | change in angular momentum, with signs |
| τ_net Δt = ΔL | — | link torque, time and change in L |
| slope of L–t graph | N·m | net torque |
| area under τ–t graph | N·m·s | angular impulse (area below the axis is negative) |
Assumptions behind the numbers
- The body is rigid and turns about a fixed axis, so I is constant while torques act.
- θ is the angle between the radial line (point to object) and the velocity, not between the path and some other line.
- Friction and drag are negligible unless stated. If they are not, include their torques in τ_net.
Mistakes to avoid
- No reference point stated. L about one point is not L about another.
- Dropping sin θ. L = rmv only when v is perpendicular to r.
- Forgetting L₀. The theorem gives the change, not the final value.
- Ignoring signs. A braking torque gives a negative angular impulse when the spin is positive.
- Comparing torques instead of angular impulses. A small torque for a long time can beat a large torque for a short time.
- Force × time instead of torque × time. Multiply by the lever arm first.
- “Straight-line motion has no angular momentum.” It does, about any point off the line.
Quick self-check
- A wheel with I = 0.30 kg·m² spins at 8.0 rad/s. What is its angular momentum? (2.4 kg·m²/s)
- A constant net torque of 0.50 N·m acts for 6.0 s. What is the change in angular momentum? (3.0 kg·m²/s, same sign as the torque)
- A 0.10 kg ball moves at 5.0 m/s in a straight line that passes 0.20 m from a post. What is its angular momentum about the post? (0.10 kg·m²/s, at every point on its path)
Next: practice questions.