Energy and Momentum of Rotating Systems
Core ideas
Rotational kinetic energy is ½Iω², and rolling objects combine translational and rotational energy: K = ½mv² + ½Iω², with v = ωr for rolling without slipping. Angular momentum L = Iω is conserved when net external torque is zero, which explains why a spinning skater speeds up when pulling arms in (I decreases, ω increases).
Candidates must connect linear and rotational quantities (v = ωr, a = αr) and decide when to use energy conservation versus angular momentum conservation. Energy conservation applies when no nonconservative work acts; angular momentum conservation applies when external torque is zero - even if kinetic energy changes.
Rolling races are a classic: objects with smaller I/(mr²) reach the bottom first, so a solid sphere beats a hoop regardless of mass or radius.
Traps
- Forgetting rotational kinetic energy when an object rolls.
- Applying energy conservation to a skater pulling in her arms - she does work, so K increases.
- Using L = Iω without checking that the axis is fixed or through the center of mass.
- Mixing linear and angular units.
Rule of thumb
For rolling down a ramp, use mgh = ½mv²(1 + I/(mr²)). The shape factor alone decides the winner.
Worked example
A solid cylinder (I = ½mr²) rolls from height h: mgh = ½mv² + ¼mv² = ¾mv², so v = √(4gh/3). A hoop (I = mr²) gives v = √(gh), which is slower. Both results are independent of mass and radius.