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.
Topic sequence · 4 of 6
Learning objectives
Objective 1
Choose a system and reference axis that make angular-momentum accounting transparent.
Objective 2
Apply conservation of angular momentum to shape-changing systems and rotational interactions, including I_iω_i=I_fω_f when appropriate.
Objective 3
Distinguish angular-momentum conservation from mechanical-energy conservation and use external angular impulse when the system is not isolated.
- Topic 6.3 angular momentum and angular impulse.
- Topic 5.4 rotational inertia.
- Topic 5.3 torque.
- Topic 4.3 conservation of linear momentum for system-boundary analogy.
Definitions first
Representations you must read and create
Representation 1
Before/after angular-momentum bar chart with signed components about one axis.
Representation 2
System-boundary diagram identifying which torques are internal and which are external.
Representation 3
I-versus-ω comparison showing inverse change when L is constant.
Representation 4
Torque–time graph whose external signed area gives any nonzero ΔL of the system.
Core relationships and calculus
Variables: all constituent angular momenta about same origin/axis
Use: system total
Variables: external torque only
Use: general system law
Variables: same chosen axis/reference point
Use: conservation condition
Variables: vector/component equality
Use: before–after conservation statement
Variables: fixed-axis model with L=Iω
Use: shape change or coupled rigid-body before/after states
Variables: external angular impulse
Use: non-isolated correction when external torque acts
Integrate the system law Στ_ext=dL/dt. If the external angular impulse is zero over the interaction, then L_f−L_i=∫Στ_extdt=0, giving L_i=L_f. For a one-axis rigid/shape-changing model with L=Iω at each state, I_iω_i=I_fω_f. The conservation law remains valid even when I changes; the simplified Newton form τ=Iα does not capture all changing-I dynamics by itself.
For a system, Στ_ext=dL_tot/dt. Integrate over the interaction: L_f−L_i=∫Στ_extdt. If the net external angular impulse is zero, L_f=L_i. For states that each satisfy L=Iω about the same axis, substitute to obtain I_iω_i=I_fω_f. This derivation explains why conservation follows from the absence of external angular impulse, not from the absence of internal forces or torques.
Conservation applies only to the total angular momentum of the selected system about the same chosen axis/reference point when the net external angular impulse about that axis is zero. Internal torques may redistribute L among parts. Conservation of angular momentum does not imply conservation of rotational kinetic energy.
Physical examples
Example 1
A skater pulls mass inward: I decreases and |ω| increases so Iω stays constant when external torque is negligible.
Example 2
A student on a low-friction turntable catches a spinning wheel; the final angular speed follows conservation for the combined student–wheel system, not for either subsystem alone.
Example 3
Two coaxial disks brought into contact by friction eventually share angular speed. Friction is internal to the two-disk system, so total L is conserved while mechanical energy decreases.
Example 4
If a brake pad is outside the chosen wheel system, its torque is external and angular momentum of the wheel alone is not conserved.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Draw the system boundary first, then classify torques as internal or external about the chosen axis.
Reasoning pattern 2
Use I_iω_i=I_fω_f only when both states can be represented by L=Iω about that same axis.
Reasoning pattern 3
A decrease in I at fixed L requires an increase in |ω|, but the sign of L still matters.
Reasoning pattern 4
If a known external torque acts for a known time, use L_f=L_i+J_ang,ext instead of conservation.
Reasoning pattern 5
Do not pair angular-momentum conservation with energy conservation automatically; verify the collision/coupling model separately.
Deeper calculus and limiting-case check
At fixed nonzero angular momentum L, rotational kinetic energy can be written K=L²/(2I). Therefore decreasing I while conserving L raises K. The added kinetic energy must come from internal work (for example, a skater doing muscular work while pulling arms inward). This relation is a powerful diagnostic against the misconception that conserved L implies conserved K.
Model-selection workflow: decide what the problem is before choosing an equation
For Conservation of Angular Momentum, the fastest reliable route is not equation hunting. Start by identifying the rotating system and constraint, encode it with the energy/angular-momentum representation, choose the work-energy or angular-impulse equation, and then test the rolling/orbit/isolation assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat conservation of angular momentum and system angular momentum as physical quantities first and symbols second.
Choose a representation
Build Before/after angular-momentum bar chart with signed components about one axis.. If the prompt supplies System-boundary diagram identifying which torques are internal and which are external., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is L⃗_tot = ΣL⃗_i. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Conservation applies only to the total angular momentum of the selected system about the same chosen axis/reference point when the net external angular impulse about that axis is zero. Internal torques may redistribute L among parts. Conservation of angular momentum does not imply conservation of rotational kinetic energy. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | Before/after angular-momentum bar chart with signed components about one axis. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | L⃗_tot = ΣL⃗_i | system total. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | Στ⃗_ext = dL⃗_tot/dt | general system law. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Integrate the system law Στ_ext=dL/dt. If the external angular impulse is zero over the interaction, then L_f−L_i=∫Στ_extdt=0, giving L_i=L_f. For a one-axis rigid/shape-changing model with L=Iω at each state, I_iω_i=I_fω_f. The conservation law remains valid even when I changes; the simplified Newton form τ=Iα does not capture all changing-I dynamics by itself. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
A student and turntable have rotational inertia I_s=3.0 kg·m² and rotate at +2.0 rad/s. The student holds a wheel of inertia I_w=0.50 kg·m² spinning at −8.0 rad/s about the same vertical axis. The wheel is then stopped relative to the student, so all parts share final angular speed ω_f. With negligible external torque, initial L=(3.0)(2.0)+(0.50)(−8.0)=2.0 kg·m²/s. Final inertia is 3.5 kg·m², so ω_f=2.0/3.5=0.571 rad/s. Mechanical rotational kinetic energy is not generally conserved in this internal interaction, so do not impose both angular-momentum and kinetic-energy conservation unless the physics justifies it. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 6.4
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship L⃗_tot = ΣL⃗_i should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Before/after angular-momentum bar chart with signed components about one axis. and System-boundary diagram identifying which torques are internal and which are external.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a low-friction rotating platform with movable masses. Measure I and ω before and after sliding equal masses radially inward while external torque is minimized. Compare I_iω_i and I_fω_f with uncertainty. In a second trial, apply a known external torque pulse and show that the measured change in L matches the external angular impulse rather than zero.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Conserving angular momentum without first checking the net external torque about the chosen axis. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Changing the reference axis between the initial and final states. | Return to the work-energy or angular-impulse equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming angular speed is conserved; it may change when I changes even though L is constant. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Assuming rotational kinetic energy is conserved whenever angular momentum is conserved. Internal work can change K while L stays constant. | Return to the work-energy or angular-impulse equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Conserving angular momentum for one subsystem when the interaction torque from another subsystem is external to that chosen boundary. | Return to the energy/angular-momentum representation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 4.3 uses the same system-boundary logic for linear momentum. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 6.3 supplies the torque–angular-impulse law that produces the conservation condition. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 5.4 explains how redistributing mass changes I. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write L⃗_tot = ΣL⃗_i and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 6.4 mastery ladder
Interactive angular-momentum conservation check
Assume negligible external angular impulse about one axis. Enter initial and final rotational inertias to solve Iᵢωᵢ=I_fω_f and compare rotational kinetic energy.
Experimental connection
Use a low-friction rotating platform with movable masses. Measure I and ω before and after sliding equal masses radially inward while external torque is minimized. Compare I_iω_i and I_fω_f with uncertainty. In a second trial, apply a known external torque pulse and show that the measured change in L matches the external angular impulse rather than zero.
Connections across Physics C
Connection 1
Topic 4.3 uses the same system-boundary logic for linear momentum.
Connection 2
Topic 6.3 supplies the torque–angular-impulse law that produces the conservation condition.
Connection 3
Topic 5.4 explains how redistributing mass changes I.
Connection 4
Topic 6.5 rolling problems may use angular momentum about a contact point only when the torque/reference-point model is justified.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Conserving angular momentum without first checking the net external torque about the chosen axis.
Avoid this
Changing the reference axis between the initial and final states.
Avoid this
Assuming angular speed is conserved; it may change when I changes even though L is constant.
Avoid this
Assuming rotational kinetic energy is conserved whenever angular momentum is conserved. Internal work can change K while L stays constant.
Avoid this
Conserving angular momentum for one subsystem when the interaction torque from another subsystem is external to that chosen boundary.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Does zero external force guarantee angular momentum is conserved?
No. The relevant condition is zero net external torque about the chosen reference axis.
Why does a skater spin faster when pulling arms in?
I decreases. With negligible external torque, L=Iω remains constant, so |ω| increases.
Is kinetic energy conserved when angular momentum is conserved?
Not necessarily. Internal work or dissipative coupling can change kinetic energy while total angular momentum remains constant.
Can I choose any axis?
You may choose a useful reference axis, but you must calculate all angular momenta and external torques about that same axis and keep it consistent.