Torque and Angular Momentum

Having established $L=I\omega$ for a rigid body's angular momentum and $\tau=I\alpha$ for the torque acting on it, these two results can be directly combined. Since $\alpha=\dfrac{d\omega}{dt}$, the torque relation can be rewritten as

(valid whenever $I$ itself stays constant, as it does for a genuinely rigid body rotating about a fixed axis), giving the compact and powerful result

This states that an external torque acting on a rigid body, about a fixed axis, produces a rate of change of the body's angular momentum about that same axis — it is exactly the rotational counterpart of the linear relation $\vec F=\dfrac{d\vec p}{dt}$, which itself is simply Newton's second law written in its most general momentum form.

Conservation of angular momentum. This relation immediately implies a conservation law: if the net external torque on a body is zero ($\tau=0$), then $\dfrac{dL}{dt}=0$, so the angular momentum $L$ must remain constant:

$\text{if }\tau=0,\text{ then }\frac{dL}{dt}=0\ \Rightarrow\ L=\text{constant}.$ …