Angular Impulse

Push a merry-go-round for one second and it barely speeds up. Push with the same force for ten seconds and it is really spinning. The force (as torque) is the same, but the longer push delivers more angular impulse, so it builds up more angular momentum. This is the rotational twin of ordinary impulse: force times time changes momentum, torque times time changes angular momentum. The relationship is J = tau·delta t = delta L, where L = I·omega is angular momentum.

This matters because torque alone does not tell you how much spin an object gains — you need to know how long that torque acts. A huge torque applied for a split second might change angular momentum less than a modest torque applied steadily. On a graph of torque versus time, the angular impulse is the area under the curve, exactly like area under a force-time graph gives ordinary impulse. That area equals delta L directly, no matter how the torque wiggles around in between, as long as you track the total area with sign.

Angular impulses add algebraically, with direction (sign) mattering. If a wheel at rest gets a counterclockwise angular impulse of +6 kg m²/s and then a clockwise one of -6 kg m²/s, the two cancel and the wheel ends up back at zero angular momentum, at rest again. This sets up angular momentum conservation: when the net angular impulse on a system is zero, perhaps because an external torque never acts, or equal and opposite ones trade off, angular momentum stays constant, which is exactly the idea behind spinning skaters and orbiting satellites.