# Proof 33: Angular momentum

If we define angular momentum as the cross-product $\vec L=\vec r \times \vec p$, its magnitude is $L=r_\perp p$ (see the Cross-product skill). For a spinning object/system, the particles it consists of rotate in a perfect circle about the axis-of-rotation, and so $r$ is always perpendicular to the momentum, $r=r_\perp$.

Each rotating particle of the object/system carries angular momentum:

$$L_\text{particle}=r p=r\underbrace{m v}_p=r m \underbrace{r\omega}_{v}=r^2m\omega$$

(The geometric bond $v=r\omega$ was used here.) All particles of the object/system rotate equally fast, if we assume a rigid body. Then $\omega$ is the same for them all. The total angular momentum $L$ for the object/system is the sum of all $L_\text{particle}$:

$$L=\sum L_\text{particle}=\sum \left(r^2m\right)\,\omega=\omega\underbrace{\sum r^2m}_I=I\omega$$

The rotational speed $\omega$ rotates about the same axis as the angular momentum $L$. Their vector versions therefore point along the same axis and we can write:

$$\vec L=I\vec \omega\qquad\blacksquare$$

## Proof 35: Kinetic energy

Linear version of kinetic energy A spaceship in outer space starts its rocket engines to propel itself forwards with a rocket force. That force does…

## Proof 26: Parallel-axis theorem

When rotating about the centre-of-mass, the moment-of-inertia $I_\text{com}$ is often easier to find. When the axis-of-rotation is somewhere else, we’ll below derive a formula that…

## Proof 25: Centre-of-mass

An object’s centre-of-mass is a point $\vec r_\text{com}=(x_\text{com},y_\text{com})$ that “represents” the object, taking into account all the “particles” it consists of as well as how…

## Proof 22: Geometric bonds

Angle $\theta$ measured in radians is ‘the number of radius-lengths along the periphery’. Multiply that number with the radius $r$ to get the total length…

## Proof 24: Newton’s 2nd law in rotation

Get some friends to help you push the stuck carousel with your daughter sitting in it. Each applies a force on her. Friction and other…