🚀
AP Physics C: Mechanics · Unit 6 of 7
Energy and Momentum of Rotating Systems
10–15% of the exam7 lessons · 96 min34 terms
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.
Rotational KEAngular momentumConservationOrbital mechanics
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
- Angular Momentum of a Point Particle14 min · 3 objectivesCompute angular momentum as L = r × p for a particle · Explain why a particle moving in a straight line can have constant nonzero angular momentum · Relate torque to the rate of change of angular momentum
- Rolling Down an Incline: Why Shape Beats Mass14 min · 3 objectivesApply energy conservation to a rolling object including rotational kinetic energy · Derive the acceleration of a rolling object in terms of I/MR² · Explain why mass and radius cancel while shape does not
- Kepler's Laws from Angular Momentum & Energy14 min · 3 objectivesDerive Kepler's second law from conservation of angular momentum · Relate Kepler's third law to Newtonian gravitation for a circular orbit · Analyze an elliptical orbit using conservation of energy and angular momentum together
Formulas in Unit 6
Rotational and total kinetic energy
KE_rot = ½Iω² KE_total = ½mv_cm² + ½Iω²
The rotational twin of ½mv². For rolling, add the translational and rotational parts, linked by v_cm = Rω.
Rotational work and power
W = ∫ τ dθ P = τω
Torque integrated over angle gives work; torque times angular velocity gives power. Both mirror their translational forms exactly.
Angular momentum
L = Iω (particle: L = rmv sin θ)
Rigid body on the left, single particle on the right. Units: kg·m²/s. A vector directed along the rotation axis.
Torque as the rate of change of angular momentum
τ_net = dL/dt
The rotational twin of F = dp/dt. When τ_net = 0, L is conserved.
Conservation of angular momentum
L = Iω = constant I₁ω₁ = I₂ω₂
Holds when the net external torque is zero. Reducing I (pulling mass inward) speeds up the spin; increasing I slows it.
Circular orbit
GMm/r² = mv²/r → v = √(GM/r) T² = (4π²/GM) r³
Orbital speed falls as 1/√r. The period relation T² ∝ r³ is Kepler's third law, obtained by combining v = √(GM/r) with T = 2πr/v.
Angular momentum at the apses
L = mvr = constant → v_p r_p = v_a r_a
At perihelion and aphelion the velocity is perpendicular to the radius, so L = mvr directly. The closer approach forces the faster speed.
Angular momentum and its rate of change
L = r × p |L| = rp sin θ = mvb Στ = dL/dt
For a rigid body rotating about a fixed axis this reduces to the familiar L = Iω. The point-particle form is the more general one and the two must agree wherever both apply.
Rolling without slipping
v = ωR Mgh = ½Mv² + ½Iω² a = g sin θ / (1 + I/MR²)
The rolling constraint v = ωR is what couples the two motions and lets one equation determine both. It holds only while the object rolls without slipping.
The three laws in Newtonian form
ellipse with sun at a focus dA/dt = L/2m = constant T² = (4π²/GM)a³
In the third law a is the semi-major axis of the ellipse, which reduces to the radius for a circular orbit. The constant depends only on the mass of the central body, so it is the same for every planet in a system.
Every term in Unit 6
All 34 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 and L = r × p = mvr sin θ for a particle. Since dL/dt = Στ, L is conserved when the net external torque is zero.
- 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.
- Rotational kinetic energy as the analogue of ½mv²
- K_rot = ½Iω², the exact analogue of ½mv² with I for m and ω for v. It is energy of rotation about the axis and adds to any translational kinetic energy.
- Total kinetic energy of a rolling object
- K = ½Mv² + ½Iω², with the rolling constraint v = ωR linking them. This split is why a rolling object accelerates more slowly than a sliding one.
- Ratio of rotational to translational energy
- K_rot/K_trans = I/MR², a pure number set by shape: 2/5 for a solid sphere, 1/2 for a disk, 1 for a hoop. It determines the rolling race order.
- Rotational work
- W = ∫τ dθ, the angular counterpart of ∫F dx. A constant torque through an angle does W = τΔθ.
- Rotational power
- P = τω, matching P = Fv. This is why an engine is specified by torque and rotational speed together rather than by either alone.
- Work–energy theorem for rotation
- W_net = ΔK_rot = ½Iω² − ½Iω₀². For an object both translating and rotating, the total work equals the change in total kinetic energy.
- Angular momentum of a rigid body
- L = Iω about a fixed axis. For a point particle the more general L = r × p applies, and the two must agree where both are valid.
- Direction of the angular momentum vector
- Along the rotation axis by the right-hand rule — curl the fingers with the rotation and the thumb gives L. This is why a spinning wheel resists having its axis tilted.
- Torque as the rate of change of angular momentum
- Στ = dL/dt, the rotational counterpart of ΣF = dp/dt. It is the more general statement, valid even when I changes.
- When angular momentum is conserved
- Whenever the net external TORQUE about the chosen point is zero. This can hold even when momentum is not conserved — a rod struck and pivoting about a fixed hinge is the standard example.
- Skater pulling in her arms
- I falls, and with L = Iω constant, ω rises. Kinetic energy ½Iω² = L²/2I therefore INCREASES, supplied by the muscular work done pulling the arms inward.
- Angular impulse
- ∫τ dt = ΔL, the rotational counterpart of the impulse–momentum theorem. Useful for sudden collisions involving rotation.
- Particle striking a pivoted rod
- Angular momentum about the pivot is conserved because the hinge force acts at the pivot and exerts no torque there. Linear momentum is NOT conserved, because the hinge exerts an external force.
- Choosing the point for angular momentum
- L and τ are defined relative to a chosen point, and a conservation argument valid about one point can fail about another. Always state the point.
- Rolling constraint and energy
- v = ωR converts the two-variable energy equation into one variable. It holds only while rolling without slipping, which is exactly when static friction does no work.
- Rolling race is independent of mass and radius
- a = g sin θ/(1 + I/MR²) contains only the shape factor, so a marble and a bowling ball tie and a hoop loses to both.
- Gyroscopic precession
- A torque perpendicular to L changes its direction rather than its magnitude, so a spinning top precesses about the vertical instead of falling. It is the clearest demonstration that angular momentum is a vector.
- Kepler's second law as angular momentum
- dA/dt = L/2m, so equal areas in equal times is exactly the statement that L is conserved under a central force.
- Angular momentum in an elliptical orbit
- L = mv_p r_p = mv_a r_a at perihelion and aphelion, where velocity is perpendicular to the radius. This gives the speed ratio directly from the distance ratio.
- Orbital angular momentum is conserved but energy determines the shape
- L fixes the areal speed and E fixes the semi-major axis; together they determine the eccentricity. Free-response orbit questions almost always require both.
- Why rotational kinetic energy is not conserved when I changes
- K = L²/2I, so at fixed L a decrease in I raises K. The extra energy is supplied by whatever did the work to change the mass distribution.
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.
- When a rolling problem appears, immediately identify k = I/MR² and reach for a = g sin θ/(1 + k). Deriving it from scratch each time is a valid but slow route, and the derivation is worth doing once carefully so the formula can be used with confidence thereafter.
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 34 terms with definitions for this unit, all of them on this page.
How should I study Physics C: Mech Unit 6?
Read the 7 lessons below first — about 95 minutes — then drill the 34 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.