Energy and Momentum of Rotating Systems

Notes

Rotational energy and work

• $\displaystyle K_{rot} = \frac{1}{2}I\omega^2$. Rolling: $\displaystyle K = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I_{cm}\omega^2$ with $v_{cm} = R\omega$ (no slipping).
• Work by torque: $\displaystyle W = \int\tau\,d\theta$. Power: $P = \tau\omega$.
• For a rolling object released from height $h$: $\displaystyle v = \sqrt{\frac{2gh}{1 + I/(MR^2)}}$.

Angular momentum

• Particle: $\vec L = \vec r \times \vec p$, magnitude $mvr\sin\theta$. Rigid body: $L = I\omega$.
• $\displaystyle \vec\tau_{net} = \frac{d\vec L}{dt}$. Angular impulse: $\displaystyle \int\tau\,dt = \Delta L$.
• Conservation: with no net external torque, $L$ is constant.
• A particle moving in a straight line still has angular momentum about a point not on its path.

Rolling with friction

• Rolling without slipping: static friction does no work (the contact point isn't sliding).
• A ball that starts sliding and spinning wrong will have kinetic friction act until $v = R\omega$; angular momentum about the contact point is conserved during that process.

Orbits

• Angular momentum is conserved in orbits (gravity is a central force). Kepler's second law (equal areas in equal times) follows.
• Circular orbit: $\displaystyle v = \sqrt{\frac{GM}{r}}$; total energy $\displaystyle E = -\frac{GMm}{2r}$.