Energy and Momentum of Rotating Systems
Units 4 and 5 built energy and momentum conservation for translating objects; this unit is what happens when those objects also rotate. Rolling without slipping is the linchpin lesson -- it's the constraint that links a rolling object's translational and rotational motion, and it's why two objects of equal mass and radius can roll down the same incline at different rates depending on how their mass is distributed, which the final lesson compares directly.
What you'll learn
- Calculate rotational kinetic energy for a rotating object.
- Apply conservation of energy to systems with both translational and rotational motion (rolling objects).
- Analyze rolling without slipping, relating linear and rotational quantities.
- Calculate angular momentum for a rotating object or point particle.
- Apply conservation of angular momentum where net external torque is zero.
- Compare different rolling shapes down an incline using energy methods.
- Analyze rotational speed changes due to changing moment of inertia (e.g. a skater pulling in their arms).
1. Rotational Kinetic Energy
A rotating object has kinetic energy due to its rotation, \(KE_{rot} = \tfrac{1}{2}I\omega^2\) — directly analogous to \(\tfrac{1}{2}mv^2\), with \(I\) replacing \(m\) and \(\omega\) replacing \(v\).
2. Rolling Without Slipping
An object rolling without slipping satisfies the constraint \(v = \omega r\), linking its translational speed to its spin rate. Its total kinetic energy is the sum of both contributions: \(KE_{total} = \tfrac{1}{2}mv^2 + \tfrac{1}{2}I\omega^2\).
Using \(\omega = v/r\) and writing \(I = cmr^2\) (where \(c\) is a shape factor — \(\tfrac{1}{2}\) for a solid disk/cylinder, \(\tfrac{2}{5}\) for a solid sphere, \(1\) for a thin hoop), energy conservation for an object rolling from rest down a height \(h\) gives \(v^2 = \dfrac{2gh}{1+c}\). A smaller shape factor means more of the available energy goes into translation rather than spin, producing a higher final speed — and an earlier arrival at the bottom of an incline.
3. Angular Momentum
Angular momentum for a rigid body rotating about a fixed axis is \(L = I\omega\); for a point particle, \(L = mvr\sin\theta\) about a chosen axis. Angular momentum is directly analogous to linear momentum, with \(I\) replacing \(m\) and \(\omega\) replacing \(v\).
4. Conservation of Angular Momentum
When no net external torque acts on a system, its angular momentum is conserved: \(L_{before} = L_{after}\), i.e., \(I_1\omega_1 = I_2\omega_2\). Because \(L\) depends on the *product* of \(I\) and \(\omega\), a system can change its rotational speed dramatically just by changing its moment of inertia — a spinning skater speeds up by pulling in her arms (reducing \(I\)), a collapsing star spins faster as it shrinks, and a diver spins faster by tucking into a ball.
5. Comparing Rolling Objects
Using the shape factor \(c\) from Section 2, objects with smaller \(c\) always win a rolling race down an incline, regardless of their mass or radius (both cancel out of the result). This makes for a common exam ranking task: order several shapes released simultaneously by which reaches the bottom first.
Key equations
- KErot = 1/2 Iω^2 — Rotational kinetic energy
- KEtotal = 1/2 mv^2 + 1/2 Iω^2 — Rolling without slipping, combined KE
- L = Iω (rigid body); L = mvr sinθ (point particle) — Angular momentum calculations
- ΣL(before) = ΣL(after) — Conservation of angular momentum
- v = ωr — Rolling-without-slipping condition