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6.3.2Energy and Momentum of Rotating Systems

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

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Lesson summary

6.3.2 Angular Momentum of a System — AP PHYSICS 1 notes: Find the total angular momentum of a system of several bodies and particles and use it in collisions about a pivot.Ltotal = ΣLi = Σmiviri — the sum of the angular momenta of the parts about the same axis.Rotating parts — L …

  • Particles moving past the axis
  • Single axis
  • Iiωi = Ifωf
  • Mass moved inward
  • Mass moved outward
  • Person walking outward on a platform

Part 1 of 7Total angular momentum

Total angular momentum
Ltotal
ΣLi = Σmiviri
  • the sum of the angular momenta of the parts about the same axis.
Rotating parts
  • L = Iω.
Particles moving past the axis
  • L = r⊥mv.
Signs
  • Signs counterclockwise positive; equal spins in opposite senses total zero.
Single axis
  • Single axis all angular momenta must be calculated about one common axis; shift points if needed.

Part 2 of 7Nonrigid systems changing shape

Nonrigid systems changing shape
Iiωi
Ifωf
  • with no external torque, L stays constant as shape changes.
Mass moved inward
  • I decreases so ω increases.
    • (skaters, spinning chair, tornadoes)
Mass moved outward
  • I increases so ω decreases.
Person walking outward on a platform
  • I of the system rises so ω falls.

Part 3 of 7Internal and external torques

Internal torques
  • Internal torques act in equal and opposite pairs and cancel in the total.
External torques
  • External torques only torques from outside the system change Ltotal.
Benefit
  • Benefit complex interactions can be handled without knowing the forces, e.g. two discs engaging by friction need only initial and final L.

Part 4 of 7Particle striking a rotating body

Particle striking a rotating body
Before
  • L = r⊥mv for the particle plus Ibodyω for the body, about the pivot.
    • (zero if at rest)
After
  • L = (Ibody + mr2)ωf with r the stuck particle's distance from the pivot.
Linear momentum
  • Linear momentum not conserved because the pivot supplies an external force, which has no torque about the pivot.

Part 5 of 7Spin and orbital parts

Spin and orbital parts
L
Lorbital + Lspin
Mvcmr⊥ + Icmω.
Rolling wheel
  • Rolling wheel both parts point the same way about a point on the ground and add.
Two skaters grasping hands
  • Two skaters grasping hands total L before is the sum of two orbital terms, afterwards Itotalω;
    • the interaction is internal so totals match.

Part 6 of 7Checking the answer

Direction
  • Direction the final rotation should agree with the sign of total L.
Energy
  • Energy kinetic energy must be lower after a sticking collision.
Opposite spins
  • Opposite spins the final ω can be zero if total L is zero.

Part 7 of 7More from the specification

More from the specification
Angular speed
  • Magnitude of the angular velocity.
Isolated system
  • System with no net external force.
Rigid system
  • System that holds its shape while rotating.
Inertia
  • Resistance of an object to changes in its motion.
Velocity
  • Rate of change of displacement.
Impulse
  • J = FΔt = Δp.
    What the symbols mean: F = force (N) · Δt = time the force acts (s) · Δp = change in momentum (kg m s−1); FΔt is the impulse (N s)
Newton
  • Unit of force, kg m s-2.

Key terms

TermMeaning
SystemCollection of objects whose total angular momentum is considered.
Total angular momentumSum of the angular momenta of all parts about the same axis.
PivotFixed point about which the body rotates.
Perfectly inelastic collisionObjects stick and share a common angular velocity.
TurntableRotating platform used to study changes in I and ω.

Avoid these mistakes

1
Using linear momentum conservation for a collision with a pivoted rod.
2
Leaving the stuck particle's mr2 out of the final moment of inertia.
3
Using the particle's distance from the wrong point in L = r⊥mv.
4
Adding magnitudes of angular momentum for objects rotating in opposite senses.
5
Assuming kinetic energy is conserved in a sticking collision.

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