Foundations — Angular momentum L = Iω (fixed axis), L = r × p (general)

This page assumes you know nothing. We build every letter, arrow, and symbol the parent note uses, in an order where each piece only leans on pieces already made. Read top to bottom.

1. A point, an origin, and an arrow: $O$ and $\vec{r}$

Before anything spins, we need a way to say where things are.

Why the topic needs this: angular momentum is always measured about a chosen point. Move the dot $O$, and the arrow $\vec{r}$ changes, so the answer changes. This is why the parent note keeps saying "about a chosen origin."

2. How fast, which way: velocity $\vec{v}$ and speed $v$

Why the topic needs this: angular momentum cares about motion. But — crucial hint for later — it only cares about the part of the motion that curls around $O$, not the part heading straight at or away from $O$.

3. Mass $m$ and linear momentum $\vec{p}$

See Linear Momentum for the straight-line version this whole topic is built on top of.

4. The angle $\theta$ and the lever arm $r_{\perp }$

Now we have two arrows sharing the point where the object sits: $\vec{r}$ (from $O$ to the object) and $\vec{p}$ (the object's motion). The angle between them is the star of the show.

We need to measure "how much of the motion goes around $O$." Look at the figure: drop a straight line along the direction of motion. The shortest distance from $O$ to that line is the piece that matters.

Why $\sin \theta$ and not something else? We want the piece of $\vec{r}$ that is sideways to the motion — the part that acts as a lever. In the right triangle, the side opposite to $\theta$ is exactly that sideways piece, and "opposite over hypotenuse" is the definition of $\sin \theta$. So $r\sin \theta$ pulls out precisely the lever arm.

• $\theta =90^{\circ }$: motion is fully sideways, $\sin 90^{\circ }=1$, so $r_{\perp }=r$ — maximum swirl.
• $\theta =0^{\circ }$ or $180^{\circ }$: motion is straight toward/away from $O$, $\sin \theta =0$, so $r_{\perp }=0$ — zero swirl. Nothing goes around.

5. The cross product $\vec{r}\times \vec{p}$

We have two arrows and an angle. We need one operation that:

1. spits out the swirl-magnitude $rp\sin \theta$ automatically,
2. gives zero when the motion is straight toward/away ($\theta =0$ or $180^{\circ }$),
3. tells us the axis the swirl happens around (which way it turns).

That operation is the cross product. This is why the parent note uses it and not ordinary multiplication.

Why this is exactly what angular momentum needs: the $\sin \theta$ inside the length automatically deletes the straight-in/straight-out motion (their $\sin \theta =0$), keeping only the going-around part. And the thumb-direction gives us the spin axis for free. See Cross Product for the full machinery.

6. Angular speed $\omega$ and circular motion

For a whole spinning body we stop tracking each speck's straight-line speed and switch to how fast it turns.

7. Moment of inertia $I$ and the sum symbol $\sum$

Why the topic needs this: when you add up $\vec{r}\times \vec{p}$ over every speck of a rigid body (each with $v_i=\omega r_i$), the constant $\omega$ factors out and what's left is exactly $\sum m_i{r}_i^2$. Giving that lump the name $I$ turns a messy sum into the clean $L=I\omega$. Full detail in Moment of Inertia.

8. The two headline symbols: $L$ and $\vec{L}$

Now every ingredient is defined, we can finally write the two formulas the parent note lives on.

Everything above was assembled so that these two lines mean something concrete rather than being symbols to memorise. See Rotational Kinetic Energy and Conservation of Angular Momentum for where $L$ goes next, and Torque for what changes it.

9. How the foundations feed the topic

Equipment checklist