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6.4Energy and Momentum of Rotating Systems

Conservation of Angular Momentum — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 10 of 13 1 quiz

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Lesson summary

6.4 Conservation of Angular Momentum — AP PHYSICS 1 notes: Describe the behaviour of a system using conservation of angular momentum.Conservation of angular momentum — the total angular momentum of an isolated system stays constant unless a net external torque acts: Li = Lf.Alway…

  • Always conserved
  • Non-zero external torque
  • Newton's third law
  • Change of L
  • System choice
  • Iiωi = Ifωf

Part 1 of 7The principle

1Conservation of angular momentum
  • Conservation of angular momentum the total angular momentum of an isolated system stays constant unless a net external torque acts: Li = Lf.
2Always conserved
  • Always conserved angular momentum is conserved in all interactions, even when kinetic energy is not.
3Total
  • Total the sum of the angular momenta of the parts about the same axis.
4Non-zero external torque
  • Non-zero external torque angular momentum is transferred between the system and its surroundings.

Part 2 of 7Why it holds

Why it holds
Newton's third law
  • Newton's third law the angular impulse of A on B is equal and opposite to that of B on A.
Change of L
  • Change of L any change in a system's angular momentum equals the angular impulse on it.
System choice
  • System choice a system can be chosen so its total L is constant.

Part 3 of 7Changing shape

Iiωi
Ifωf
  • a non-rigid system changes angular speed without changing L when it moves mass nearer or farther from the axis.
Mass closer
  • I decreases and ω increases.
    • (skater pulling in arms, diver tucking, tornado narrowing, person on a spinning chair)
Mass farther
  • I increases and ω decreases.
Neutron stars
  • Neutron stars spin very fast after collapse.
Planet near perihelion
  • Planet near perihelion moves faster when closer to the Sun.

Part 4 of 7Rotational collisions

Rotational collisions
Clay on a turntable or rim
  • Ltotal before = Ltotal after, including mvr⊥ of the incoming object.
Transfer within a system
  • Transfer within a system a person walking on a turntable moves angular momentum between parts while the total stays constant.
Axle forces
  • Axle forces act through the axis and have no torque about it.
Linear momentum
  • Linear momentum may not be conserved when the axle exerts an external force even though L about the axle is.

Part 5 of 7Energy

Kinetic energy
  • Kinetic energy not necessarily conserved when L is; K = L2/(2I).
Decreasing I
  • K increases, with the extra energy from internal work.
    • (skater's muscles)
Sticking collisions
  • Sticking collisions rotational kinetic energy decreases.

Part 6 of 7Applying conservation of angular momentum

1
Define the system and the axis.
2
Check that the net external torque about that axis is zero.
3
Write Li for every part (Iω for rotating bodies, mvr⊥ for moving points).
4
Write Lf using the new rotational inertia and angular speed.
5
Set Li = Lf and solve; check energy separately if asked.
  • Dart striking a tire
    • mvR sinθ = (m + M)R2ω, so ω = mv sinθ/((m + M)R).

Part 7 of 7More from the specification

Velocity
  • Rate of change of displacement.
Distance
  • Total length of path travelled, a scalar.
Angular velocity
  • Δθ/Δt in rad/s.

Key terms

TermMeaning
Conservation of angular momentumIf Στext = 0, L is constant.
I1ω1 = I2ω2Changing I changes ω.
Non-rigid systemSystem that can change shape and so change I.
Rotational collisionInteraction where objects combine with L conserved.
PerihelionClosest point to the Sun.

Avoid these mistakes

1
Applying conservation when a net external torque acts.
2
Assuming kinetic energy is conserved when L is.
3
Using different axes for L and I.
4
Forgetting the incoming object's mvr⊥ in Li.

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