The total angular momentum of a system about a rotational axis is just the sum of the angular momenta of its individual parts about that same axis. Nothing new there — it works exactly the way total linear momentum did back in Unit 4.
What makes angular momentum powerful is what happens when the parts of a system interact with each other. Any torque one part exerts on another is matched by an equal-and-opposite torque the second part exerts back on the first — a direct consequence of Newton's third law. Add those two internal torques together and they cancel exactly.
This is exactly why choosing your system carefully matters so much in these problems: pick a system that includes everything interacting internally, and all those internal torques disappear from your bookkeeping entirely.
Here's the idea behind every spinning-skater demonstration you've ever seen: a nonrigid system can change its angular speed without any change in its angular momentum, simply by changing shape — moving mass closer to or farther from the rotation axis.
This system's angular momentum is locked at L = 24 kg·m²/s — no external torque acts on it. Slide the rotational inertia (think: a skater pulling their arms in or out) and watch ω respond so that L stays exactly the same.
A skater spins with a rotational inertia of 4 kg·m² at 3 rad/s, arms extended. She pulls her arms in, reducing her rotational inertia to 1 kg·m². No external torque acts on her. Find her new angular speed.
Put it all together and you get the master rule for this lesson. Angular momentum is conserved in every interaction, everywhere — but whether a particular system's angular momentum stays constant depends entirely on how you draw the boundary of that system:
This is exactly the scenario in this unit's header art: two flywheels, spinning independently, engage a clutch and lock together. No external torque acts on the combined two-flywheel system, so its total angular momentum before engagement exactly equals its total angular momentum after.
Two flywheels — like the ones in this unit's header art — spin independently, then engage a clutch and lock together. No external torque acts on the combined system, so their total angular momentum before matches their total angular momentum after.
A flywheel with I₁ = 3 kg·m² spins at ω₁ = 10 rad/s. It's brought into contact with a second, stationary flywheel with I₂ = 6 kg·m². They stick together and rotate as one. Find their common final angular velocity.