Energy and Momentum of Rotating Systems
The rotational work-energy theorem, derived exactly the way Unit 104 derived the linear one — substitute \(\tau=I\,d\omega/dt\) into \(W=\int\tau\,d\theta\), use \(d\theta=\omega\,dt\), and the integral collapses to \(\tfrac{1}{2}I\omega_f^2-\tfrac{1}{2}I\omega_0^2\). Combined with rolling-without-slipping (translational and rotational kinetic energy together), this closes the loop between everything linear and everything rotational covered so far.
What you'll learn
- Derive the rotational work-energy theorem by the same substitution technique used for the linear version
- Combine translational and rotational kinetic energy for a rolling object using v=ωR
- Apply conservation of angular momentum (L=Iω constant) to systems where moment of inertia changes
1. Deriving the Rotational Work-Energy Theorem
The rotational work-energy theorem is derived exactly the way Unit 104 derived the linear one: substitute \(\tau = I\,d\omega/dt\) into \(W=\int\tau\,d\theta\), use \(d\theta=\omega\,dt\), and the integral becomes \(\int I\omega\,d\omega\), which evaluates to \(\tfrac{1}{2}I\omega_f^2-\tfrac{1}{2}I\omega_0^2\) — the same structure, one substitution mirroring the other.
2. Rolling Without Slipping: Combined Kinetic Energy
An object rolling without slipping has both translational kinetic energy, \(\tfrac{1}{2}mv^2\), and rotational kinetic energy, \(\tfrac{1}{2}I\omega^2\), simultaneously. Using \(\omega=v/R\) and writing \(I=cmR^2\) (where \(c\) is a shape factor), the two combine into \(KE_{total}=\tfrac{1}{2}mv^2(1+c)\) — a single expression covering any rolling shape once its shape factor is known.
3. Conservation of Angular Momentum with Changing I
When no external torque acts on a system, its angular momentum \(L=I\omega\) stays exactly constant — even while \(I\) itself changes. A figure skater pulling their arms in reduces \(I\), and since \(L\) can't change, \(\omega\) must increase to compensate, exactly proportionally.
Key equations
- W = ½Iω_f² − ½Iω₀² — The rotational work-energy theorem — derived from τ=Iα and W=∫τdθ by the same substitution (dθ=ωdt) used for the linear version.
- KE_total = ½mv²(1+c) — Total kinetic energy of an object rolling without slipping — translational plus rotational, combined using v=ωR.
- L₁ = L₂ (I₁ω₁ = I₂ω₂) — Conservation of angular momentum when no external torque acts — even if I itself changes (e.g., pulling arms in), the product Iω stays constant.