Energy and Momentum of Rotating Systems
What this unit covers
The topics below follow the published Physics C: Mech course framework for Unit 6. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.
Lessons in this unit
- Rotational Kinetic Energy & Work14 min · 3 objectivesCompute rotational kinetic energy as KE_rot = ½Iω² · Find the total kinetic energy of a rolling object as translational plus rotational · Apply the rotational work-energy relationship W = ∫τ dθ
- Angular Momentum13 min · 3 objectivesDefine angular momentum as L = Iω for a rigid body and L = r × p for a particle · Relate the net torque to the rate of change of angular momentum, τ = dL/dt · Compute the angular momentum in simple rotating situations
- Conservation of Angular Momentum13 min · 3 objectivesState conservation of angular momentum for systems with zero net external torque · Solve problems where the moment of inertia changes, using I₁ω₁ = I₂ω₂ · Explain why kinetic energy can change even when angular momentum is conserved
- Orbital Mechanics14 min · 3 objectivesApply Newton's gravitation to circular orbits, with gravity supplying the centripetal force · Derive orbital speed and relate the period to radius through Kepler's third law · Use conservation of angular momentum to relate speeds at perihelion and aphelion
Formulas in Unit 6
Every term in Unit 6
All 13 terms we publish for Energy and Momentum of Rotating Systems, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.
- Conservation of angular momentum
- Στ_ext = dL/dt, so L is conserved when no external torque acts. A collapsing star spins faster for exactly this reason.
- Angular momentum
- L = Iω for a rigid body about a fixed axis, or L = r × p for a particle about a point.
- Rotational kinetic energy
- KE = ½Iω². A rolling body has ½Mv² + ½Iω², which combine to ½(1 + I/MR²)Mv² when rolling without slipping.
- Angular momentum of a particle moving in a straight line
- Non-zero about any point not on its line of motion: L = mvb, where b is the perpendicular distance.
- Energy change when I changes
- Pulling mass inward conserves L but increases ½Iω² — the extra energy comes from the work done pulling it in.
- Rotational work and power
- W = ∫τ dθ and P = τω, the rotational analogues of ∫F dx and Fv.
- Collisions involving rotation
- Use conservation of angular momentum about the pivot; linear momentum is generally NOT conserved because the pivot exerts an external force.
- Rolling race result
- a = g sin θ/(1 + I/MR²), so a sphere (⅖) beats a disk (½) beats a hoop (1), independent of mass and radius.
- Angular momentum of a point particle
- L = mvr sin θ about a chosen origin. Non-zero even for straight-line motion, provided the line misses the origin.
- Conservation across a collision with a rod
- A projectile embedding in a pivoted rod conserves angular momentum about the pivot: mvd = (I_rod + md²)ω.
- Why kinetic energy is not conserved when I changes
- Changing I requires internal work — the skater's muscles — which adds energy while leaving L untouched.
- Precession qualitatively
- A torque perpendicular to angular momentum changes its direction rather than its magnitude, so a spinning top precesses instead of falling.
- Rotational analogue table
- x↔θ, v↔ω, a↔α, m↔I, F↔τ, p↔L, ½mv²↔½Iω², Fx↔τθ. Every linear result has a rotational twin.
What examiners penalize here
- Keep L = Iω and τ = dL/dt distinct: one is the angular momentum, the other is the rule for how torque changes it. The single most useful consequence is that zero net external torque forces angular momentum to stay constant.
- For circular orbits set gravity equal to the centripetal force (GMm/r² = mv²/r). For elliptical orbits, use conservation of angular momentum (v_p r_p = v_a r_a) at the apses and conservation of energy between them — energy and angular momentum together pin down the whole orbit.
Practice Physics C: Mech
Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.
Questions about this unit
How much of the AP Physics C: Mechanics exam is Unit 6?
Unit 6, Energy and Momentum of Rotating Systems, is worth 10–15% of the Physics C: Mech multiple-choice section according to the published course framework. Across all 7 units that makes it a substantial share — heavier than an even split would give it.
What topics are covered in Physics C: Mech Unit 6?
Energy and Momentum of Rotating Systems covers Rotational KE, Angular momentum, Conservation and Orbital mechanics. We publish 13 terms with definitions for this unit, all of them on this page.
How should I study Physics C: Mech Unit 6?
Read the 4 lessons below first — about 55 minutes — then drill the 13 terms in cram mode until you can produce each definition from memory rather than just recognize it. Recognition is what makes a unit feel finished when it is not. Finish with practice questions and read the explanation for every one you get right by elimination as well as the ones you miss.
All 7 units of AP Physics C: Mechanics
Unit names, topics and exam weights follow the published College Board course framework for AP Physics C: Mechanics. AP® is a trademark registered by the College Board, which does not endorse this site.