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6.1Energy and Momentum of Rotating Systems

Rotational Kinetic Energy — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 2 of 13 1 quiz

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Lesson summary

6.1 Rotational Kinetic Energy — AP PHYSICS 1 notes: Describe the rotational kinetic energy of a rigid system in terms of its rotational inertia and angular velocity.Krot = ½Iω2 — K in J, I in kg·m2, ω in rad/s.Scalar — rotational kinetic energy is a scalar, a type of kinetic ener…

  • Mass distribution
  • Ktotal = Kr + Kt
  • Centre of mass at rest
  • System energy accounts
  • Rotational work–energy theorem
  • Total kinetic energy (rolling)

Part 1 of 7Rotational kinetic energy

Krot
½Iω2
  • K in J, I in kg·m2, ω in rad/s.
Scalar
  • Scalar rotational kinetic energy is a scalar, a type of kinetic energy, not a new kind of energy.
Origin
  • Origin it is the sum of ½mv2 of every part of the body using v = rω; for one mass ½(mr2)(v/r)2 = ½mv2.
Mass distribution
  • I = mr2 describes how mass is spread about the axis, and ω is the same at all points.

Part 2 of 7Total kinetic energy of a rigid system

Total kinetic energy of a rigid system
Ktotal
Kr + Kt
½Iω2 + ½Mvcm2.
Condition
  • Condition rotational laws hold if the axis passes through the centre of mass and the direction of rotation does not change.
Centre of mass at rest
  • Centre of mass at rest a system can still have Krot because its points move.
    • (flywheels, turntables, merry-go-rounds)
System energy accounts
  • System energy accounts include Krot for spinning pulleys and flywheels.

Part 3 of 7Rolling without slipping

Relation
  • Relation vcm = Rω, so K = ½(M + I/R2)vcm2.
Friction
  • Friction static friction at the instantaneously stationary contact point does no work, so mechanical energy is conserved.

Part 4 of 7Rolling down an incline

Energy
  • Mgh = ½Mv2 + ½Iω2 with ω = v/R.
Result
  • Result smaller I/(MR2) gives greater speed, independent of mass and radius.
ObjectIv at bottomTranslational share of K
Frictionless sliding blockNot applicable√(2gh)All
Solid sphere(2/5)MR2√(10gh/7)5/7
Solid cylinder½MR2√(4gh/3)2/3
HoopMR2√(gh)1/2

Part 5 of 7Work by torque

1W = τΔθ
  • W = τΔθ constant torque through an angle in radians.
2Rotational work–energy theorem
  • Wnet = ΔKrot = ½Iωf2 − ½Iωi2.

Part 6 of 7Energy of a rolling object on a slope: steps

1
Define the object–Earth system and check the object rolls without slipping.
2
Write the initial energy Mgh (starting from rest).
3
Write the final energy ½Mv2 + ½Iω2.
4
Substitute I for the shape and ω = v/R.
5
Solve for v and check it is smaller than the sliding value √(2gh).

Part 7 of 7More from the specification

Average power
  • Energy transferred divided by time taken.
Displacement
  • Change in position.
Linear
  • A graph that is a straight line through the origin, so resistance is constant.
Acceleration
  • Rate of change of velocity.

Key terms

TermMeaning
Rotational kinetic energyKrot = ½Iω2.
Total kinetic energy (rolling)½Mv2 + ½Iω2.
Energy conservationMgh = ½Mv2 + ½Iω2.
Shape factorLarger I/(MR2) means more energy in rotation and lower speed.
FlywheelDevice storing energy as rotational kinetic energy.

Avoid these mistakes

1
Omitting the translational term for a rolling object.
2
Using rpm instead of rad/s in ½Iω2.
3
Thinking a heavier object rolls faster — mass cancels.
4
Saying static friction dissipates energy in rolling without slipping.

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