Two particles of equal mass $m$ move in circles of radii $r_1$ and $r_2$ with the same angular velocity $\omega$. The ratio of their angular momenta $L_A/L_B$ (where $A$ has radius $r_1$ and $B$ has radius $r_2$, and $r_1 = r_2$) is:

Point mass: $\vec{L} = \vec{r}\times m\vec{v} = mr^2\omega\hat{k}$ (for circular motion). Extended body: $L = I\omega$. Rate of change: $\tau = dL/dt$ (torque = rate of change of angular momentum). Conservation: $L = $ const if $\tau_{ext} = 0$. Angular impulse: $\tau \cdot \Delta t = \Delta L$. SI unit: kg m$^2$/s = J s.

Kinematics: $\omega = d\theta/dt$; $\alpha = d\omega/dt$. Uniform angular acceleration: $\omega = \omega_0 + \alpha t$; $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$; $\omega^2 = \omega_0^2 + 2\alpha\theta$. Dynamics: $\tau = I\alpha$. Work: $W = \tau\theta$. KE: $\frac{1}{2}I\omega^2$. Power: $P = \tau\omega$.

Spinning top/gyroscope: gravity provides torque $\tau = MgR$ (where $R$ = distance of CM from pivot). This torque changes direction of $L$ rather than magnitude (perpendicular to $L$). Precession angular velocity: $\Omega = \tau/L = MgR/I\omega$. Faster spin → slower precession. Applications: gyrocompass (navigation), attitude control in spacecraft, gyroscopic stabilisers on ships.

Ice skater: $I_1\omega_1 = I_2\omega_2$. Pulls arms in → $I$ decreases → $\omega$ increases. Diver: tucks body → $I$ decreases → rotates faster. Planetary motion: $r\times mv = $ const → slower at aphelion, faster at perihelion (Kepler\'s 2nd law). Neutron star formation: stellar core collapses from $R \sim 10^5$ km to $R \sim 10$ km → $I$ decreases $\sim 10^{10}$× → $\omega$ increases $\sim 10^{10}$× → pulsars rotate hundreds of times per second.

When no external torque acts, $I\omega$ stays constant, so reducing the moment of inertia must increase the angular speed. A skater pulling their arms in, a diver tucking, and a collapsing star all do the same thing. The neutron star case is the most dramatic: a stellar core shrinking from about $10^5$ km to roughly 10 km reduces I by around $10^{10}$, so $\omega$ rises by the same factor — which is why pulsars spin hundreds of times a second.

A body rolling without slipping has angular momentum from two sources: its rotation about its own centre, $I_{cm}\omega$, and the motion of its centre of mass about the reference point, $Mvr$. The total about a point on the ground is therefore $L = I_{cm}\omega + MvR$. Omitting the second term is the standard error, and it matters because the rolling condition $v = \omega R$ links the two rather than making one redundant.

Just as force is $dp/dt$, torque is $dL/dt$. Written this way, conservation of angular momentum is not a separate law at all — it is what the equation says when $\tau = 0$. The same relation explains gyroscopic precession: a torque applied perpendicular to L cannot change its magnitude, only its direction, so the axis sweeps out a cone at angular rate $\Omega = \tau/L$ instead of simply falling over.