Unit 6 · Topic 09

Angular momentum

Angular momentum describes rotational motion about a chosen origin or axis.

Unit 6Angular momentumOpen navigation

Concept 01

Cross product r×p

Essential The minimum you should retain

For a particle, L=r×p \vec L=\vec r\times\vec p .

UnderstandInterpret and connect

Its magnitude is rp sinφ=pℓ and depends on origin.

DeepenFormulation and conditions

Direction follows the right-hand rule.

ExploreConnections for further study

Straight-line motion may have nonzero L about an off-line origin.

Worked example

Angular momentum of a particle

A 0.50 kg particle travels at 8.0 m/s along a line 0.30 m from the origin.

Given
  • m=0.50 kg
  • v=8.0 m/s
  • ℓ=0.30 m
Target

Find |L|.

  1. Momentum

    p=mv=4.0 kg·m/s.

  2. Lever arm

    L=pℓ.

  3. Substitution

    L=(4.0)(0.30).

  4. Result

    L=1.20 kg·m²/s.

Conclusion

|L|=1.20 kg·m²/s.

Mathematical relation

Angular momentum and torque

L→=r→×p→,τext=dL→dt,Lz=Izω \vec L=\vec r\times\vec p,\quad \vec\tau_{ext}=d\vec L/dt,\quad L_z=I_z\omega
Represents

Angular momentum and its change from external torque.

Physical interpretation

L depends on origin and is not always parallel to ω.

DeepenVariables, conditions, and checks

Variables

L
angular momentum; usual unit: kg·m²/s

Conditions of application

  • Same origin for L and τ; I_zω only for an appropriate axis.

Dimensional check

kg·m²/s.

Origin, position vector, linear momentum, perpendicular lever arm, and the axial direction of L form the geometry.prℓline of motionOparticleL ⊙

L=r×p \vec L=\vec r\times\vec p ; in the geometry shown L_z>0, so L points out of the plane, while ℓ shows dependence on the chosen origin.

Concept 02

General dynamical law

Essential The minimum you should retain

Net external torque satisfies τ_ext=dL/dt \vec\tau_{ext}=d\vec L/dt .

UnderstandInterpret and connect

Torque may change L's magnitude, direction, or both.

DeepenFormulation and conditions

Torque and L must use the same origin.

ExploreConnections for further study

Angular impulse is the time integral of torque.

Concept 03

Appropriate-axis case

Essential The minimum you should retain

For rotation about an appropriate fixed or principal axis, L_z=I_zω L_z=I_z\omega .

UnderstandInterpret and connect

I_z and ω refer to the same axis.

DeepenFormulation and conditions

The axial relation avoids summing every particle.

ExploreConnections for further study

It does not mean L is always parallel to ω in arbitrary 3D rotation.

Concept 04

Geometry of L

Essential The minimum you should retain

Lever arm ℓ is perpendicular distance from origin to the motion line.

UnderstandInterpret and connect

Moving the origin may change r, ℓ, and L.

DeepenFormulation and conditions

If the line passes through the origin, L is zero about it.

ExploreConnections for further study

Origin dependence is essential when deciding conservation.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Claiming particle angular momentum is origin-independent.

L=r×p \vec L=\vec r\times\vec p depends on r from the origin.

Claiming L is always parallel to ω in arbitrary 3D rotation.

L_z=I_zω L_z=I_z\omega is axial and conditional.