Energy and Momentum of Rotating Systems · Spinning, Rolling, Orbiting
Everything you need to master Unit 6 — rotational kinetic energy, work by torque, angular momentum and its conservation, rolling motion, and orbital satellites. The rotational analogs of Units 3 and 4 (plus orbits as a bonus).
5–8% of the AP exam
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Unit 6 extends Unit 5 by adding energy and momentum to rotational motion. Just as Unit 3 gave you energy conservation for linear motion and Unit 4 gave you momentum conservation, Unit 6 gives you both for rotation. There's also a bonus topic at the end — orbital motion of satellites and planets, which uses both gravity (Unit 2) and circular motion ideas.
The College Board breaks Unit 6 into 6 topics: (6.1) Rotational Kinetic Energy, (6.2) Torque and Work, (6.3) Angular Momentum and Angular Impulse, (6.4) Conservation of Angular Momentum, (6.5) Rolling, and (6.6) Motion of Orbiting Satellites. The most important topic is 6.4 — conservation of angular momentum is what makes figure skaters spin faster when they pull in their arms.
Unit 6 makes up about 5–8% of the AP exam — one of the lightest units. But it builds heavily on Units 3, 4, and 5, so make sure those are solid first.
Key terms preview
A taste of what you'll find in The Essentials and Flashcards.
Rotational Kinetic Energy
K_rot = ½Iω². The KE an object has because it's rotating. Same form as ½mv², with the rotational analogs.
Angular Momentum
L = Iω (for a rigid body). The rotational analog of linear momentum. Vector quantity, units kg·m²/s.
Angular Impulse
J_ang = τΔt = ΔL. Torque applied over time changes angular momentum.
Conservation of Angular Momentum
If no external torque acts on a system, its total angular momentum stays constant. Why figure skaters spin faster.
Rolling Without Slipping
v_cm = rω. A rolling object has both translational AND rotational KE. Where the surface meets the object, there's no slipping.
Orbital Motion
Gravity provides the centripetal force that keeps satellites in orbit. Orbital speed depends on the radius of the orbit, not the satellite's mass.
Rotating objects carry rotational KE (½Iω²) AND angular momentum (Iω) in addition to any linear KE and momentum. A rolling ball has both forms of energy at once. Torque does work just like force does, and angular impulse changes angular momentum just like linear impulse changes linear momentum.
2. Angular momentum is conserved when no external torque acts
If you can change the rotational inertia of a system without applying external torque, the angular velocity adjusts to keep L = Iω constant. That's why figure skaters spin faster when they pull in their arms, and why neutron stars spin so fast (they were once large, slow-rotating stars that collapsed).
3. Rolling objects share their energy between translation and rotation
A rolling object's total KE = ½mv² + ½Iω². When a ball rolls down a ramp from height h, mgh is split between linear and rotational KE — so the bottom speed is slower than for a sliding object. This is why a solid disk beats a hoop down a ramp: the disk has lower rotational inertia, so less of its energy goes to rotation, and more goes to translation.