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.
Topic sequence · 3 of 6
Learning objectives
Objective 1
Calculate and interpret angular momentum for both rigid fixed-axis rotation and particles moving relative to a chosen point.
Objective 2
Use Στ_ext=dL/dt and angular impulse to connect torque–time graphs with changes in angular momentum.
Objective 3
Translate among torque, angular momentum, graphs, and experiments while keeping the reference axis consistent.
- Topic 4.2 impulse and momentum change.
- Topic 5.3 torque and cross products.
- Topic 5.4 rotational inertia.
- Topic 5.6 Newton’s second law in rotational form.
Definitions first
Representations you must read and create
Representation 1
Particle diagram showing r⃗ from a chosen origin to the particle and p⃗ tangent to its path.
Representation 2
Torque-versus-time graph where signed area is angular impulse.
Representation 3
Angular-momentum-versus-time graph where slope is net external torque.
Representation 4
Rigid-body diagram labeled with axis, I, ω, and L direction/sign.
Core relationships and calculus
Variables: L kg·m²/s; I kg·m²; ω rad/s
Use: rigid system rotating about the specified fixed axis
Variables: r m from reference point; p⃗=mv⃗ kg·m/s
Use: particle angular momentum about a chosen point
Variables: φ angle between r⃗ and v⃗
Use: particle angular-momentum magnitude
Variables: τ N·m; L kg·m²/s
Use: general torque–angular-momentum relation
Variables: J_ang N·m·s = kg·m²/s
Use: angular impulse–momentum theorem
Variables: constant signed torque about one axis
Use: constant-torque special case
Start from Στ⃗_ext=dL⃗/dt. Integrate over time: ∫_{t_i}^{t_f}Στ⃗_ext dt=L⃗_f−L⃗_i. For a fixed-axis rigid body with constant I, L=Iω, so dL/dt=I dω/dt=Iα and the equation reduces to Στ=Iα. Thus Topic 5.6 is a constant-I special case of the more general angular-momentum law.
For a particle, torque about an origin is τ⃗=r⃗×F⃗. Because p⃗=mv⃗ and F⃗=dp⃗/dt, differentiate L⃗=r⃗×p⃗: dL⃗/dt=(dr⃗/dt)×p⃗+r⃗×dp⃗/dt. The first term is v⃗×mv⃗=0, leaving dL⃗/dt=r⃗×F⃗=τ⃗. Summing particles cancels internal torque pairs in the system model, leaving Στ⃗_ext=dL⃗_sys/dt.
Use one consistent reference point/axis for r, L, torque, and angular impulse. The general relation is ΔL=∫Στ_ext dt; τΔt is only the constant-torque special case. L=Iω applies cleanly to rigid rotation about a specified axis, while L⃗=r⃗×p⃗ is the particle form. Do not switch forms or axes mid-solution without re-establishing the model.
Physical examples
Example 1
A spinning disk about its axle has L=Iω; doubling ω at fixed I doubles L.
Example 2
A particle moving in a straight line can have nonzero angular momentum about an off-path origin because r⃗×p⃗ is nonzero.
Example 3
The same moving particle can have zero angular momentum about a point lying on its line of motion because the perpendicular lever arm is zero.
Example 4
A short large torque and a longer smaller torque can produce the same ΔL if the signed areas under their torque–time graphs are equal.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
First identify the reference point/axis; angular momentum cannot be quoted without it.
Reasoning pattern 2
Torque–time area gives ΔL; L–time slope gives net external torque.
Reasoning pattern 3
A particle moving in a straight line may still have angular momentum about an off-line origin.
Reasoning pattern 4
If I is constant, you may move between ΔL=IΔω and angular impulse directly.
Reasoning pattern 5
Do not mix the torque–angle graph from Topic 6.2 with the torque–time graph here.
Deeper calculus and limiting-case check
For a particle moving in a straight line with constant momentum p past an origin at perpendicular distance b, |L|=bp. Even though r changes continuously, the perpendicular lever arm b remains constant, so angular momentum about that origin is constant when no torque acts. This is a useful geometric check on the cross-product definition.
Model-selection workflow: decide what the problem is before choosing an equation
For Angular Momentum and Angular Impulse, 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 angular momentum and rigid-body angular momentum as physical quantities first and symbols second.
Choose a representation
Build Particle diagram showing r⃗ from a chosen origin to the particle and p⃗ tangent to its path.. If the prompt supplies Torque-versus-time graph where signed area is angular impulse., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is L = Iω. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
Use one consistent reference point/axis for r, L, torque, and angular impulse. The general relation is ΔL=∫Στ_ext dt; τΔt is only the constant-torque special case. L=Iω applies cleanly to rigid rotation about a specified axis, while L⃗=r⃗×p⃗ is the particle form. Do not switch forms or axes mid-solution without re-establishing the model. 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 | Particle diagram showing r⃗ from a chosen origin to the particle and p⃗ tangent to its path. | 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 | L = Iω | rigid system rotating about the specified fixed axis. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | L⃗ = r⃗ × p⃗ | particle angular momentum about a chosen point. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Start from Στ⃗_ext=dL⃗/dt. Integrate over time: ∫_{t_i}^{t_f}Στ⃗_ext dt=L⃗_f−L⃗_i. For a fixed-axis rigid body with constant I, L=Iω, so dL/dt=I dω/dt=Iα and the equation reduces to Στ=Iα. Thus Topic 5.6 is a constant-I special case of the more general angular-momentum law. | 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 0.20 kg puck moves at 6.0 m/s along the +x direction. At one instant it is 0.50 m above the origin, so the perpendicular distance from the origin to its line of motion is 0.50 m. Its momentum magnitude is p=mv=1.20 kg·m/s. Therefore |L|=rp sin90°=(0.50)(1.20)=0.60 kg·m²/s. The direction follows r⃗×p⃗. If a net external torque of opposite sign averages 0.15 N·m for 2.0 s, the angular impulse is −0.30 N·m·s and the final angular momentum component becomes 0.30 kg·m²/s. This problem shows that angular momentum is reference-point dependent while angular impulse changes the value about that same reference. 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 6.3
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 L = Iω should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Particle diagram showing r⃗ from a chosen origin to the particle and p⃗ tangent to its path. and Torque-versus-time graph where signed area is angular impulse.. 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
Mount a disk or wheel on a low-friction rotary sensor. Apply a measured torque pulse with a force sensor at a known radius and record τ(t) and ω(t). Numerically integrate the measured torque–time graph to obtain angular impulse, then compare it with I(ω_f−ω_i). Repeat with different pulse shapes but similar area to test whether equal angular impulse gives equal ΔL within uncertainty.
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 |
|---|---|---|
| Using L=Iω for a translating particle instead of L⃗=r⃗×p⃗ about the specified point. | 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? |
| Forgetting that changing the origin or axis can change the angular momentum of the same motion. | 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? |
| Taking absolute area under a torque–time graph; angular impulse is signed/vector-valued. | 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? |
| Using applied torque rather than net external torque when relating torque to ΔL. | 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? |
| Confusing torque–time area with rotational work; torque–time area gives angular impulse, whereas torque–angle area gives work. | 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? |
| Using τΔt for a torque that varies with time instead of integrating the signed torque-time graph. | 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? |
| Computing particle angular momentum about one point and torque about another; both quantities must use the same reference point or 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? |
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 4.2 is the translational analogue J=∫Fdt=Δp. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 5.6 follows from Στ=dL/dt when I is constant. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 6.2 integrates torque over angle for work; Topic 6.3 integrates torque over time for angular impulse. 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 L = Iω 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 6.3 mastery ladder
Interactive angular-impulse integral
For signed net torque τ(t)=at+b, integrate from t₁ to t₂ and apply J_ang=ΔL.
Experimental connection
Mount a disk or wheel on a low-friction rotary sensor. Apply a measured torque pulse with a force sensor at a known radius and record τ(t) and ω(t). Numerically integrate the measured torque–time graph to obtain angular impulse, then compare it with I(ω_f−ω_i). Repeat with different pulse shapes but similar area to test whether equal angular impulse gives equal ΔL within uncertainty.
Connections across Physics C
Connection 1
Topic 4.2 is the translational analogue J=∫Fdt=Δp.
Connection 2
Topic 5.6 follows from Στ=dL/dt when I is constant.
Connection 3
Topic 6.2 integrates torque over angle for work; Topic 6.3 integrates torque over time for angular impulse.
Connection 4
Topic 6.4 uses zero external torque to conserve total angular momentum.
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
Using L=Iω for a translating particle instead of L⃗=r⃗×p⃗ about the specified point.
Avoid this
Forgetting that changing the origin or axis can change the angular momentum of the same motion.
Avoid this
Taking absolute area under a torque–time graph; angular impulse is signed/vector-valued.
Avoid this
Using applied torque rather than net external torque when relating torque to ΔL.
Avoid this
Confusing torque–time area with rotational work; torque–time area gives angular impulse, whereas torque–angle area gives work.
Avoid this
Using τΔt for a torque that varies with time instead of integrating the signed torque-time graph.
Avoid this
Computing particle angular momentum about one point and torque about another; both quantities must use the same reference point or axis.
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
What is the difference between L=Iω and L⃗=r⃗×p⃗?
L=Iω is the convenient rigid-body fixed-axis form. r⃗×p⃗ defines particle angular momentum about a chosen point.
What does area under a torque–time graph mean?
The signed area is angular impulse, equal to the change in angular momentum.
Can a particle moving in a straight line have angular momentum?
Yes. About an origin not on its line of motion, r⃗×p⃗ can be nonzero.
Why does the reference point matter?
Both r⃗ and torque are defined relative to that point, so changing it can change the angular momentum value and the torque used to predict its change.