AP Physics C: Mechanics Unit 6: Energy and Momentum of Rotating Systems 2027
Dedicated Unit 6 chapter hub for 6 current topics. Use it to move through the lesson sequence, connect formulas and labs, and hand off cleanly to the previous or next unit without mixing topic and unit search intent.
Unit 6 map: conservation laws for rotating, rolling, and orbiting systems
Unit 6 is where the conservation tools from Units 3 and 4 are rebuilt for rotation. The recurring questions are: what is the chosen axis, what belongs inside the system, is external torque negligible, and does the constraint allow mechanical energy conservation? Rolling problems then couple translation and rotation, while satellite problems combine gravitational energy with angular momentum.
All six Unit 6 topics
Rotational Kinetic Energy
Derive K_rot from moving mass elements, compare mass distributions, and separate center-of-mass translation from rotation.
Current lesson · 6.1 →Torque and Work
Integrate torque over angular displacement and connect rotational work to changes in rotational kinetic energy.
Open 6.2 →Angular Momentum and Angular Impulse
Use L⃗=r⃗×p⃗ or Iω and integrate external torque over time to obtain ΔL⃗.
Open 6.3 →Conservation of Angular Momentum
Identify when external angular impulse is negligible and track redistribution of rotational inertia and angular speed.
Open 6.4 →Rolling
Combine translational and rotational kinetic energy with the no-slip constraint v_cm=Rω and a_cm=Rα.
Open 6.5 →Motion of Orbiting Satellites
Use gravitational force, energy, angular momentum, circular-orbit relations, and conservation reasoning for orbital motion.
Open 6.6 →Choose the right Unit 6 conservation model
| Situation | Start with | Condition to justify | Do not assume |
|---|---|---|---|
| Known torque through an angular displacement | W=∫τ dθ and W_net=ΔK_rot | Track the torque component about the chosen axis. | τ is constant unless stated or shown. |
| Torque acts over a time interval | ∫τ_ext dt=ΔL | Use external torque about the stated origin/axis. | Angular momentum is automatically conserved. |
| System changes shape while external torque is negligible | L_i=L_f | External angular impulse is negligible. | Rotational kinetic energy must also be conserved. |
| Rigid body rolls without slipping | Energy + v_cm=Rω | Pure rolling and appropriate energy-loss assumptions. | All friction necessarily dissipates energy. |
| Satellite in a central gravitational field | Energy and angular momentum | Identify whether orbit is circular or general bound motion. | Circular-orbit formulas apply at every point of an ellipse. |
Calculus and representation ladder
Work from torque
W=∫τ(θ)dθ is the angular analog of ∫F·dx. A τ-versus-θ graph therefore turns signed area into rotational work and then into ΔK_rot.
Angular impulse
τ_ext=dL/dt, so integrating a torque-time graph gives ΔL. This is the rotational version of impulse-momentum, but the chosen origin matters.
Rolling energy
For a rigid body, K=½Mv_cm²+½I_cmω². With no slip, ω=v_cm/R, so geometry enters through β=I_cm/(MR²).
Orbit energy and momentum
Central gravity gives zero torque about the central body, so angular momentum is conserved. Mechanical energy is also conserved between impulses if no nonconservative external work occurs.
Three Unit 6 synthesis problems
Rolling-body race down an incline
Two equal-mass objects with different β=I/(MR²) roll from the same height. Write Mgh=½Mv²+½Iω² and v=Rω to show v²=2gh/(1+β). The smaller β reaches the bottom faster. The result depends on mass distribution, not simply on total mass or radius.
Rotating student pulls masses inward
With negligible external torque, I_iω_i=I_fω_f. If I decreases, |ω| increases. Compare K_rot before and after: it generally changes because internal biochemical work is done while the masses move inward. Angular momentum conservation does not imply kinetic-energy conservation.
Satellite receives a brief tangential impulse
Separate the short thrust interval from the later orbit. During the thrust, external impulse and torque change the satellite’s momentum, angular momentum, and mechanical energy. After the thrust ends, central gravity conserves both orbital energy and angular momentum. Do not apply the circular-orbit speed formula unless the new state is actually circular.
Laboratory and data-analysis tasks
Rotational work-energy test
Apply a measured torque to a rotor through a known angular displacement. Integrate τ(θ), measure ω_i and ω_f, and compare work with Δ(½Iω²).
Angular impulse experiment
Record torque and angular speed versus time for a disk. Numerically integrate τ(t) and compare with IΔω for a constant-I rotor, including uncertainty.
Rolling-shape comparison
Roll objects with different rotational-inertia factors down the same incline. Test the predicted acceleration or final-speed dependence on β and evaluate rolling-without-slipping assumptions.
Orbit-model simulation
Use position/velocity data from a numerical central-force orbit. Check constancy of total energy and angular momentum, then identify where speed is largest and why.
Unit 6 traps that cost points
Wrong axis for L or τ
Angular momentum and torque depend on the origin or axis. A conservation claim can change when the reference point changes.
Conserving K with L
If I changes while L is conserved, rotational kinetic energy can change because internal or external work may be involved.
Using Iω for every object
L=Iω is natural for rigid-body rotation about a fixed axis. A particle generally uses L⃗=r⃗×p⃗.
Rolling means friction loss
Static friction can enforce no slip without dissipating mechanical energy at the contact point. Check the actual model instead of labeling all friction nonconservative.
Mixing circular and elliptical orbit formulas
v=√(GM/r) and E=−GMm/(2r) are circular-orbit relations when r is the circular radius. General orbit reasoning needs conservation laws.
Ignoring translation
A rolling object has both center-of-mass translational kinetic energy and rotational kinetic energy.
Where Unit 6 goes next
Unit 7 · Oscillations
Rotational energy and torque reappear in physical and torsion pendulums. Small-angle restoring torque becomes an angular SHM equation.
Unit 3 · Energy revisited
Unit 6 is not a separate bookkeeping language; it extends the same work-energy and conservation logic to rotation and gravity.
Unit 4 · Momentum analogy
Force impulse changes linear momentum; torque impulse changes angular momentum. System boundaries and external interactions remain the controlling ideas.
E&M bridge
Later magnetic forces can create torques and rotational responses, but the mechanical energy/angular-momentum analysis still uses the Unit 6 framework.
Unit 6 mastery checklist
Unit 6 lesson sequence
This dedicated unit hub owns the chapter intent. Every lesson keeps its own canonical topic URL and returns here for unit context.
Topic 6.1 · Rotational Kinetic Energy
A rigid system rotating about an axis has rotational kinetic energy K_rot=½Iω². The equation follows by summing the translational kinetic energy of every moving mass element. When a rigid body both translates and rotates, its total kinetic energy separates into center-of-mass translation plus rotation about the center of mass: K_tot=½Mv_cm²+½I_cmω².
Open Topic 6.1 →Topic 6.2 · Torque and Work
A torque transfers energy when it acts through angular displacement. The general rotational-work relationship is W=∫τ dθ; for constant torque it becomes W=τΔθ, and the signed area under a torque-versus-angle graph gives the work. Net rotational work changes rotational kinetic energy, providing the rotational analogue of the translational work–energy theorem.
Open Topic 6.2 →Topic 6.3 · Angular Momentum and Angular Impulse
Angular momentum is the rotational analogue of linear momentum, but it is always defined about a chosen point or axis. For a rigid system about a fixed axis, L=Iω; for a particle, L⃗=r⃗×p⃗. Net external torque changes angular momentum according to Στ⃗_ext=dL⃗/dt, so angular impulse ∫Στ⃗_ext dt equals ΔL⃗ and is the signed area under a torque–time graph.
Open Topic 6.3 →Topic 6.4 · Conservation of Angular Momentum
Angular momentum of a selected system remains constant about a chosen axis when the net external torque about that axis is zero. For a rigid or shape-changing system that can be modeled with L=Iω about the same axis, conservation becomes I_iω_i=I_fω_f. The decisive AP Physics C skill is system and axis selection: internal torques redistribute angular momentum, while external angular impulse changes the system total.
Open Topic 6.4 →Topic 6.5 · Rolling
Rolling combines center-of-mass translation with rotation about the center of mass. For rolling without slipping on a stationary surface, v_cm=Rω, a_cm=Rα along the rolling direction, and the contact point is instantaneously at rest relative to the surface. Total kinetic energy is ½Mv_cm²+½I_cmω². If the body slips, those kinematic constraints fail and kinetic friction dissipates mechanical energy.
Open Topic 6.5 →Topic 6.6 · Motion of Orbiting Satellites
Orbital motion is governed by gravity together with conservation of energy and angular momentum. For a satellite of negligible mass m orbiting a much more massive body M, a circular orbit satisfies GMm/r²=mv²/r, so v=√(GM/r), K=GMm/(2r), U=-GMm/r, and E=-GMm/(2r). Elliptical orbits conserve total mechanical energy and angular momentum while K and U vary; escape occurs when total mechanical energy reaches zero. A higher circular orbit has lower circular speed but greater (less negative) total energy, so raising an orbit requires net energy input rather than simply 'slowing down.'
Open Topic 6.6 →Unit 6 resource network
Explanations, examples, interactive tools and study organization are original Salar Cafe content. AP® is a registered trademark of College Board, which is not affiliated with or endorsing this site.