Angular Impulse

Angular momentum changes only when a net external torque acts, and how much it changes depends on both how hard you twist and how long you twist for. That combination is angular impulse: the integral of torque with respect to time. For a constant torque it is just tau times delta t. For a torque that varies with time, you have to integrate tau(t) dt over the interval. Either way, the result equals the change in angular momentum, L_f minus L_i. This is the rotational twin of linear impulse, where force times time changes linear momentum.

A steady push on a merry-go-round beats a quick shove because the steady push keeps adding angular impulse every instant it acts, while the shove delivers its torque for only a blink. Same idea as pushing a stalled car: a gentle sustained push can get it rolling just as well as one hard shove, because what matters is the accumulated effect over time, not the instantaneous torque alone.

This sits right next to conservation of angular momentum: when net external torque is zero, angular impulse is zero, so L stays constant, which is exactly the rule behind spinning skaters and orbiting planets changing speed as their radius changes. It also connects back to rotational kinematics and energy: once you find the angular momentum delivered, dividing by the moment of inertia gives the angular velocity, which then feeds into rotational kinetic energy calculations. Graphing angular momentum versus time makes the connection visual, since the slope of that graph is just the net torque at that instant.