$K_{rot} = \frac{1}{2}I\omega^2$

• Energy of rotation
• Depends on moment of inertia and angular velocity
• Analogs of translational KE: $\frac{1}{2}mv^2$
• Important for rolling and spinning objects

$K_{total} = K_{trans} + K_{rot} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

• For rolling without slipping: $v = r\omega$
• $K = \frac{1}{2}mv^2 + \frac{1}{2}(mr^2)\left(\frac{v}{r}\right)^2 = \frac{1}{2}mv^2\left(1 + \frac{I}{mr^2}\right)$
• $K = \frac{1}{2}mv^2\left(1 + \beta\right)$ where $\beta = I/(mr^2)$
• Solid sphere: $\beta = 2/5$, hollow sphere: $\beta = 2/3$, cylinder: $\beta = 1/2$

• Objects with different shapes roll down incline
• Faster object reaches bottom first (has larger acceleration)
• Depends on $\beta = I/(mr^2)$
• Smaller $\beta$ -> larger acceleration
• Results independent of mass and radius

$L = I\omega$

• Rotational analog of linear momentum ($p = mv$)
• Vector quantity: direction along rotation axis
• Units: kg - m2/s
• Conserved when $\tau_{net} = 0$

$L = mvr_\perp$ (for point particle)

• Perpendicular distance from reference point to velocity line
• Depends on choice of reference point
• Conserved when net external torque = 0

$\Delta L = \tau_{net}\Delta t$

• Net torque causes change in angular momentum
• Analogous to impulse-momentum theorem
• Important for rotational collisions

• Net external torque on system = 0
• Internal torques cancel (action-reaction pairs)
• System isolated from external rotational influences
• Can hold even during collisions

• $I_i\omega_i = I_f\omega_f$
• Skater pulls arms in: $I$ decreases, $\omega$ increases
• Angular momentum conserved (no external torque)
• Used by figure skaters, divers
• Demonstrates conservation in action

• Planets sweep out equal areas in equal times (Kepler's 2nd Law)
• Conservation of angular momentum
• $L = mvr = \text{constant}$ (approximately circular orbits)
• Explains why orbits are stable