Unit 6 · Topic 10

Angular momentum conservation and precession

With zero net external torque about the chosen origin, L is constant; perpendicular torque may redirect it and produce precession.

Unit 6Conservation and precessionOpen navigation

Concept 01

External condition

Essential The minimum you should retain

If τ_ext=0, L_i=L_f about the chosen origin or axis.

UnderstandInterpret and connect

In the axial case, I_iω_i=I_fω_f I_i\omega_i=I_f\omega_f .

DeepenFormulation and conditions

Reducing I may increase ω without external torque.

ExploreConnections for further study

The system must include parts whose interactions are treated as internal.

Worked example

Angular speed after reducing I

With no external torque, I changes from 4.0 to 2.0 kg·m² and ω_i=3.0 rad/s.

Given
  • I_i=4.0 kg·m²
  • I_f=2.0 kg·m²
  • ω_i=3.0 rad/s
Target

Find ω_f and compare energies.

  1. Conservation

    I_iω_i=I_fω_f I_i\omega_i=I_f\omega_f .

  2. Final speed

    ω_f=(4·3)/2=6 rad/s.

  3. Initial energy

    K_i=18 J.

  4. Final energy

    K_f=36 J.

  5. Interpretation

    L is conserved while internal work raises K.

Conclusion

ω_f=6.0 rad/s; K_i=18 J; K_f=36 J.

Mathematical relation

Conservation and slow precession

τext=0⇒Li=Lf;Iiωi=Ifωf;Ω≈τL≈MgrIω \tau_{ext}=0\Rightarrow L_i=L_f;\ I_i\omega_i=I_f\omega_f;\ \Omega\approx\tau/L\approx Mgr/(I\omega)
Represents

Axial conservation and slow-precession approximation.

Physical interpretation

Conserving L does not require conserving ω or K.

DeepenVariables, conditions, and checks

Variables

Ω
precession angular speed; usual unit: rad/s

Conditions of application

  • Zero external torque for L conservation; symmetric gyroscope and slow precession for Ω.

Dimensional check

rad/s for Ω.

Concept 02

Internal work

Essential The minimum you should retain

Conserving L does not imply conserving K_rot.

UnderstandInterpret and connect

Changing mass distribution may require internal work.

DeepenFormulation and conditions

At fixed L, K=L²/(2I), so decreasing I increases K.

ExploreConnections for further study

Conservation laws must be checked separately.

Concept 03

Redirecting L

Essential The minimum you should retain

Torque approximately perpendicular to L mainly changes its direction.

UnderstandInterpret and connect

For a fast gyroscope, this lateral change produces axis precession.

DeepenFormulation and conditions

Gravity does exert torque about the pivot.

ExploreConnections for further study

The tip of L moves laterally in the slow-precession picture.

Concept 04

A stated approximation

Essential The minimum you should retain

For slow uniform precession, Ω≈τ/L≈Mgr/(Iω) \Omega\approx\tau/L\approx Mgr/(I\omega) .

UnderstandInterpret and connect

Increasing ω raises L and lowers Ω when other data stay fixed.

DeepenFormulation and conditions

The formula assumes a symmetric gyroscope, fast spin, and slow precession.

ExploreConnections for further study

Outside that regime corrections and nutation may appear.

Pivot, center of mass, weight, torque, angular momentum, and precession are shown conceptually.MgLaxis / r_cmpivotCMτ ⊗

Slow-precession diagram: Mg creates torque approximately perpendicular to L and mainly changes its direction.

Concept review

Common errors

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

Assuming L conservation forces constant ω.

I may change, so ω changes.

Explaining precession by zero torque.

Torque approximately perpendicular to L redirects it; Ω≈τ/L is a slow-precession approximation.