Energy and Momentum of Rotating Systems

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 6. This unit is worth 5–8% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Rotational Kinetic Energy12 min · 3 objectivesCompute rotational kinetic energy with KE = ½Iω² · Recognize a rolling object’s energy as translational plus rotational · Include rotational energy in conservation-of-energy accounting
• Angular Momentum13 min · 3 objectivesCompute angular momentum as L = Iω · Find the angular momentum of a point mass moving in a circle, L = mvr · Treat angular momentum as a vector along the rotation axis
• Conservation of Angular Momentum14 min · 3 objectivesState conservation of angular momentum for zero net external torque · Explain the spinning-skater effect in terms of I and ω · Solve I₁ω₁ = I₂ω₂ problems
• Rolling Motion13 min · 3 objectivesApply the rolling-without-slipping condition, v = rω · Explain why different shapes reach the bottom of a ramp at different speeds · Connect rolling to combined translation and rotation

Formulas in Unit 6

Every term in Unit 6

All 7 terms we publish for Energy and Momentum of Rotating Systems, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Conservation of angular momentum

Conserved when no net external torque acts. A skater pulling arms in reduces I, so ω rises and L stays constant.

Rotational kinetic energy

KE = ½Iω². A rolling object has both translational ½mv² and rotational ½Iω² kinetic energy.

Rolling race

Down the same ramp, objects with smaller I/mr² arrive first — a solid sphere beats a solid cylinder, which beats a hoop, regardless of mass or radius.

Angular momentum

L = Iω for a rigid body, or L = mvr sin θ for a particle about a point. A vector along the rotation axis.

Angular impulse

τΔt = ΔL, the rotational analogue of linear impulse.

Why ω rises when I falls

L = Iω is fixed, so halving I doubles ω. Kinetic energy ½Iω² then increases — supplied by the work done pulling the mass inward.

Rotational work and power

W = τθ and P = τω, matching the translational forms with rotational quantities substituted.

What examiners penalize here

• In a "race down the ramp" question, the object that puts *less* energy into rotation (smaller I relative to mR²) ends up moving faster. That is a direct consequence of splitting mgh between ½mv² and ½Iω².
• Angular momentum is a vector along the rotation axis. On the AP exam its direction matters most in conservation problems, where the total vector — magnitude and direction — must stay constant.
• Conservation of angular momentum needs zero *external* torque. Internal changes — pulling in arms, dropping on clay — do not violate it. Check that nothing outside the system is applying a torque before you set L constant.
• For ramp-race questions, rank shapes by how much of mgh goes into rotation: sphere (⅖MR²) beats disk (½MR²) beats hoop (MR²). Mass and radius cancel out, so the *shape* alone decides the winner.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids