Angular Momentum (L)
The rotational analog of linear momentum. For a rigid body: L = Iω. Units: kg·m²/s. A vector quantity, just like linear momentum. The central concept of Unit 6.
FoundationRolling Motion
Motion combining translation (the center of mass moves) AND rotation (the object spins). A ball rolling down a hill, a tire on a moving car. Rolling objects have both translational and rotational KE.
FoundationRotational Kinetic Energy
K_rot = ½Iω². The KE an object has because it's rotating. Same form as ½mv², just with rotational analogs. Adds to any translational KE the object also has.
6.1 Rotational KETotal Kinetic Energy
K_total = K_trans + K_rot = ½mv² + ½Iω². For a rolling object, both contribute. For a sliding object (not rotating), only the translational part counts. For a spinning top in place, only the rotational part.
6.1 Rotational KEEnergy Conservation in Rotation
Same as for linear motion (Unit 3), but include the rotational KE term. For a rolling object: K_trans_i + K_rot_i + U_i = K_trans_f + K_rot_f + U_f, when no non-conservative forces dissipate energy.
6.1 Rotational KEComparing KE of Rolling Objects
For two objects with the same m, R, and v, the one with HIGHER rotational inertia has MORE total KE (extra goes into spinning). A hoop and a disk rolling at the same speed have different total KEs.
6.1 Rotational KEWork Done by Torque
W = τ·Δθ (when torque is constant). The rotational analog of W = F·d. Units: joules (J), same as any work or energy. Torque acting over an angular displacement transfers energy.
6.2 Torque and WorkRotational Work-Energy Theorem
W_net = ΔK_rot = ½Iω_f² − ½Iω_i². The net work done by torques equals the change in rotational KE. The rotational analog of W_net = ΔK from Unit 3.
6.2 Torque and WorkRotational Power
P = τω (for constant torque and constant angular velocity). The rotational analog of P = Fv. How fast torque is doing rotational work. Units: watts (W = J/s).
6.2 Torque and WorkEnergy Transfer via Torque
When a motor applies torque to spin a wheel, it does rotational work. If torque and angular velocity are in the same direction, the motor adds energy to the wheel. Opposite directions = brake (removes energy).
6.2 Torque and WorkAngular Momentum of a Rigid Body
L = Iω. The rotational momentum of a spinning object. Larger I or larger ω = more L. Direction follows the right-hand rule (curl fingers in direction of rotation, thumb points along L).
6.3 Angular MomentumAngular Momentum of a Point Mass
L = mvr_⊥, where r_⊥ is the perpendicular distance from the rotation axis to the line of motion. A car driving in a straight line has angular momentum about any axis NOT on its line of motion.
6.3 Angular MomentumAngular Impulse
J_ang = τ·Δt. The rotational analog of linear impulse. Torque applied over time. Units: N·m·s = kg·m²/s (same as angular momentum, since J_ang = ΔL).
6.3 Angular MomentumAngular Impulse-Momentum Theorem
J_ang = ΔL. The angular impulse on a system equals its change in angular momentum. The rotational analog of J = Δp from Unit 4. Area under a torque vs. time graph equals ΔL.
6.3 Angular MomentumConservation of Angular Momentum
If no net external TORQUE acts on a system, total angular momentum stays constant: L_i = L_f. The most important principle in Unit 6. (Note: external FORCES through the axis don't matter — only external TORQUES.)
6.4 Conservation of LClosed (Isolated) System
For angular momentum: a system with no external torque acting. The internal interactions (parts pushing or pulling each other) come in equal/opposite pairs that cancel out — they can't change total L.
6.4 Conservation of LFigure Skater Effect
The classic demonstration of L conservation: a skater pulls in her arms, reducing her rotational inertia I. With no external torque, L = Iω stays constant, so ω must INCREASE. Smaller I = larger ω.
6.4 Conservation of LNeutron Star Formation
A massive star (large I, slow rotation) collapses to a neutron star (tiny I, very fast rotation). By conservation of L: as I drops dramatically, ω must rise dramatically. Neutron stars can rotate hundreds of times per second.
6.4 Conservation of LInternal vs External Torques
Internal torques (between parts of a system) cancel by Newton's third law and don't change the system's total L. Only EXTERNAL torques (from outside the system) can change L. This is why the figure skater's arm movements don't change her L.
6.4 Conservation of LRolling Without Slipping
The condition v_cm = rω, where v_cm is the center-of-mass velocity and r is the object's radius. At the point of contact, the surface and the object have ZERO relative velocity — no sliding.
6.5 RollingRolling With Slipping
When v_cm ≠ rω — the object is rolling AND sliding. Like a car peeling out: wheels spinning fast (large ω·r), car barely moving (small v_cm). The point of contact has nonzero velocity relative to the road.
6.5 RollingRolling Down an Incline
Use energy conservation: mgh = ½mv² + ½Iω². With v = rω (no slipping), solve for v. Different rotational inertias give different final speeds — solid disks beat hoops, which beat sliding (non-rotating) blocks.
6.5 RollingHoop vs Disk vs Sphere
Race down an incline: a solid sphere beats a solid disk beats a hoop (assuming same m and R). The one with the smallest I/(mr²) ratio puts the least energy into rotation, leaving more for translation — and reaches the bottom faster.
6.5 RollingTotal KE of a Rolling Object
K_total = ½mv² + ½Iω². For rolling without slipping (v = rω), this becomes K_total = ½mv² + ½I(v/r)² = ½v²(m + I/r²). Always more KE than a non-rotating object moving at the same v.
6.5 RollingOrbital Motion
A satellite orbiting Earth (or any object orbiting another) is in free fall under gravity. Gravity provides the centripetal force that keeps the satellite curving around the planet instead of flying off in a straight line.
6.6 OrbitsOrbital Speed
v_orbit = √(GM/r), where M is the mass of the central body and r is the orbital radius. Notice: it DOESN'T depend on the satellite's mass. A small and a large satellite at the same altitude orbit at the same speed.
6.6 OrbitsOrbital Period
T = 2πr/v_orbit. Time for one complete orbit. Closer orbits are faster (shorter T). The Moon orbits Earth every ~28 days; the International Space Station orbits every ~90 minutes (because it's much closer).
6.6 OrbitsKepler's Third Law (qualitative)
T² is proportional to r³. Bigger orbits take much longer. Doubling the orbital radius multiplies the period by 2√2 ≈ 2.83. This is built into v_orbit = √(GM/r) — at larger r, v is smaller AND the orbital circumference is bigger, so T grows rapidly.
6.6 Orbits