Spinning, Rolling, Orbiting · 5–8% of the AP Physics 1 exam
K_rot = ½Iω²
L = Iω
J_ang = τΔt = ΔL
L_i = L_f
v_cm = rω
The basics
What's covered: Rotational kinetic energy, work by torque, angular momentum and impulse, conservation of angular momentum, rolling, and orbital satellites.
Exam weight: 5–8% — one of the lightest units.
The big trick: Units 3 and 4 had energy and momentum for linear motion. Unit 6 gives you the rotational versions.
The master idea: When no external TORQUE acts, angular momentum is conserved. Use this on every spinning-skater, collapsing-star, or "child jumps on merry-go-round" problem.
📐 Key equations
K_rot = ½Iω²
Rotational KE. The energy due to rotation. Same form as ½mv², with rotational analogs. Adds to translational KE.
K_total = ½mv² + ½Iω²
Total KE of a rolling object. Both translational AND rotational forms.
W = τΔθ
Work done by torque. Rotational analog of W = Fd. Δθ in radians. Units: joules.
P = τω
Rotational power. Rotational analog of P = Fv. Units: watts.
L = Iω
Angular momentum (rigid body). Rotational analog of p = mv. Units: kg·m²/s.
L = mvr_⊥ (point mass)
Angular momentum of a point mass. r_⊥ is the perpendicular distance from the axis to the line of motion.
J_ang = τΔt = ΔL
Angular impulse-momentum theorem. Torque applied over time changes angular momentum.
L_i = L_f (no external τ)
Conservation of angular momentum. The MOST IMPORTANT equation of Unit 6.
v_cm = rω (no slipping)
Rolling without slipping. The condition that links translational and rotational motion.
v_orbit = √(GM/r)
Orbital speed. Depends only on the central body's mass M and orbit radius r — NOT the satellite's mass.
The 6 topics at a glance
6.1 Rotational KE
K_rot = ½Iω². The rotational analog of ½mv². For rolling objects, add this to translational KE for the total.
6.2 Torque and Work
W = τΔθ; P = τω. Torque does work just like force does. Net rotational work = change in K_rot.
6.3 Angular Momentum & Impulse
L = Iω; J_ang = τΔt = ΔL. The rotational analogs of p = mv and J = Δp. Area under a τ-t graph = ΔL.
6.4 Conservation of L
L_i = L_f (no external torque). The most important principle of Unit 6. Figure skaters, neutron stars, kids jumping on merry-go-rounds.
6.5 Rolling
v_cm = rω (no slipping). Total KE = ½mv² + ½Iω². Use energy conservation for rolling-down-a-ramp problems.
6.6 Orbits
Gravity provides centripetal force: v_orbit = √(GM/r). Doesn't depend on the satellite's mass. T² ∝ r³ (Kepler).
🧠 How to solve an angular-momentum conservation problem (4 steps)
1. Identify the system — and check there's no external torque acting on it (e.g., frictionless ice, person on a merry-go-round, isolated spinning object).
2. Write L_initial and L_final. For a rigid body: L = Iω. For a point mass: L = mvr_⊥.
3. Set L_initial = L_final. If multiple objects, sum each one's L on each side.
4. Solve for the unknown — usually a final ω, a final v, or a new rotational inertia.
🛞 How to solve a rolling problem (3 steps)
1. Write the energy equation. Often mgh = ½mv² + ½Iω², for a ball rolling down a ramp.
2. Use v = rω to eliminate one variable (substitute ω = v/r into the KE equation).
3. Solve for v. The answer depends on I — different shapes give different speeds.
⚠️ Common exam traps
Conserving L requires no external TORQUE — not no force. A force directed through the rotation axis exerts zero torque and doesn't affect L.
Pulling in arms increases K_rot (NOT just ω). ω goes up while I goes down, but ω² wins. The extra energy comes from internal muscle work.
For rolling, v_cm = rω only if there's no slipping. Skidding cars, peeling out, etc. violate this.
K_rot is NOT the same as L. K = ½Iω² (scalar). L = Iω (vector). Both involve I and ω but in different combinations.
Orbital speed doesn't depend on the satellite's mass. Common surprise — but it's just like how all objects fall at the same rate under gravity.
Rolling objects have less translational KE than sliding ones for the same h. Some PE goes to rotation, leaving less for translation.