Angular momentum (L) is the rotational analog of linear momentum. Just as linear momentum p = mv measures how difficult it is to stop an object moving in a straight line, angular momentum measures how difficult it is to stop — or change — an object spinning about an axis.
A gyroscope with large I or large ω has large L and is very hard to tip over or knock off its axis. This is not because of mass alone — it is because angular momentum resists changes to rotation the same way linear momentum resists changes to translation.
For a rigid system rotating about a fixed axis with rotational inertia I and angular velocity ω:
Units: kg·m²/s (equivalent to N·m·s). This is the rotational counterpart of p = mv — I replaces m, ω replaces v. Both I and ω contribute linearly: doubling either one doubles L.
A solid disk (I = 0.6 kg·m²) spins at 8 rad/s CCW. A hoop (I = 1.2 kg·m²) spins at 4 rad/s CW. Which has more angular momentum magnitude and what are their signs?
The angular impulse-momentum theorem is the rotational version of the linear impulse-momentum theorem from Lesson 4.2:
A net torque applied over a time interval produces a change in angular momentum equal to the angular impulse. The structure is identical to J = FΔt = Δp — just with rotational quantities throughout.
Set rotational inertia, initial angular velocity, applied torque, and contact time. Watch angular impulse change L — and see the new omega. The angular impulse-momentum theorem always holds: J_ang = ΔL.
Set tau negative — the torque opposes rotation and L decreases. Set tau = 0 — L never changes regardless of time. This is conservation of angular momentum (Lesson 6.4) showing up when Στ = 0.
A wheel (I = 2 kg·m²) spins at 10 rad/s CCW. A brake applies a CW torque of 4 N·m for 3 seconds. Find the final angular velocity.
Two graphical skills from Unit 4 carry over directly to angular momentum — with tau and L replacing F and p:
The area under a torque vs. time graph equals the angular impulse delivered, which equals ΔL. For constant torque: rectangle = τ × Δt. For variable torque: calculate the geometric area.
The slope of an angular momentum vs. time graph equals the net torque at that instant. Steep positive slope = large CCW torque. Horizontal line = zero net torque (L is conserved).
The most general statement of Newton's Second Law in rotational form involves angular momentum:
Net torque equals the rate of change of angular momentum. For systems where rotational inertia I is constant, this reduces to the familiar: