# How does torque affect angular momentum?

Apr 25, 2014

Let's start with Newton's 2nd law for rotational motion:

$\sum \setminus \vec{\tau} = I \setminus \vec{\alpha}$

An unbalance torque ($\tau$) must cause an angular acceleration ($\alpha$) in the object. This will cause a change in the angular speed ($\omega$) of the object.

$\setminus \vec{\tau} = I \left(\frac{d}{\mathrm{dt}} \setminus \vec{\omega}\right)$

Angular momentum ($L$) is defined as product of the moment of inertia ($I$) and the angular velocity $\setminus \vec{\setminus} \omega$

$\setminus \vec{L} = I \setminus \vec{\omega}$

Any change in the angular velocity will cause a change in the angular momentum.

Therefore applying an unbalanced torque to an object will change its angular momentum. The amount of time the torque is applied for will determine the magnitude of the change in momentum.

$\Delta \setminus \vec{L} = \setminus \vec{\tau} \Delta t$

This is the rotational analog of the momentum-impulse theorem. Rearranging it, and taking the limit $\setminus \Delta t \setminus \to 0$ we can see that just as instantaneous force is the instantaneous time rate of change of linear momentum, torque is the instantaneous time rate of change of angular momentum:

\vec F = \frac{d \vec p}{dt}; \vec \tau = \frac{d \vec L}{dt}