Newton's second law for rotation

This is the spin-world version of F = m a — the master rule connecting the twist you apply to the response you get. Push harder on a wrench and the bolt turns faster; make an object easier to spin and the same twist produces more speed-up. It is the same logic as ordinary Newton's second law, translated into the language of rotation.

Precisely, it says the net torque on an object equals its moment of inertia times its angular acceleration: tau_net = I alpha. Line it up with F_net = m a and every piece corresponds — torque replaces force, moment of inertia replaces mass, and angular acceleration replaces acceleration. So a given torque gives a small, low-inertia object a big angular acceleration, and a large, high-inertia one only a gentle one.

You use it to work out how fast things spin up or slow down under a twist: a pulley set turning by a hanging weight, a wheel braking, a motor bringing a grindstone up to speed. Its more general and more powerful form is tau_net = dL/dt — net torque equals the rate of change of angular momentum — which still holds even when the moment of inertia is changing.