Energy and Momentum of Rotating Systems

Six topics that carry rotation into energy, momentum, and orbits. Rotational kinetic energy $K_{rot} = \tfrac{1}{2}I\omega^2$ stores motion in a spinning body the way $\tfrac{1}{2}mv^2$ does for a sliding one; torque does work through $W = \int \tau\,d\theta$ and delivers power $P = \tau\omega$; angular momentum $\vec{L} = I\vec{\omega} = \vec{r}\times\vec{p}$ obeys the impulse law $\int \vec{\tau}\,dt = \Delta\vec{L}$ and is conserved whenever the net external torque is zero; rolling without slipping locks $v_{cm} = R\omega$ so a rolling body splits its energy between translation and spin; and orbiting satellites leave the constant-$g$ world for real inverse-square gravity, where $v_{orb} = \sqrt{GM/r}$ and the energy of a circular orbit is $E = -\dfrac{GMm}{2r}$ (with the semimajor axis $a$ in place of $r$ for a general ellipse).