Linear momentum, p⃗ = mv⃗, tracks translational motion. Angular momentum, L, is its rotational counterpart. For a rigid system rotating about a fixed axis, it's given by:
But angular momentum is more general than "how fast something spins." Any object has an angular momentum about a chosen point, whether or not it's rotating at all, given by the magnitude of a cross product:
Adjust the distance from the reference point (r), the object's momentum (p), and the angle between them (θ) — the same setup as this unit's header art — and watch the angular momentum respond.
Here's the counterintuitive part: because L = rp sinθ only needs an angle between r⃗ and p⃗ — not actual rotation — an object moving in a perfectly straight line can have nonzero, and even constant, angular momentum about a point that isn't on its path.
A 2 kg object moves in a straight line at a constant 5 m/s, passing 3 m from the origin at its closest approach (i.e., the perpendicular distance from the origin to the line of motion is 3 m). Find its angular momentum about the origin, and explain what happens to that value as the object moves further along the line.
Where work asked "how much energy does a torque transfer over an angular displacement," angular impulse asks a different question: how much does a torque change a system's motion when it acts over a time interval?
Angular impulse also has a graphical meaning: it's the area under a graph of torque plotted against time — the direct rotational analog of finding linear impulse from the area under a force-vs-time graph.
Set a constant torque and the time interval it acts over to see the angular impulse it delivers.
A motor exerts a constant 6 N·m torque on a flywheel for 4 seconds. How much angular impulse does the motor deliver?
Now connect the two ideas above. The change in a system's angular momentum, ΔL = L − L₀, is related to the angular impulse delivered to it by a rotational version of the impulse–momentum theorem you met back in Unit 4:
That relationship, τnet = dL/dt, gives you two more ways to read a graph. The slope of a graph of angular momentum vs. time equals the net torque exerted on the system at that instant — and the area under a graph of net torque vs. time equals the angular impulse delivered, which is exactly the system's change in angular momentum.
Start a system with some initial angular momentum, apply a constant net torque for a while, and see where its angular momentum ends up.
A merry-go-round starts at rest (Li = 0). A constant net torque of 8 N·m is applied for 5 seconds. Find its final angular momentum, and explain how that value could also be read off a graph.