Foundations — Torque = dL - dt

Before you can trust that one-line law, you must earn every symbol inside it. Below, each piece is built from nothing — plain words first, then the picture, then why the topic can't live without it. Read top to bottom: each block uses only things defined above it.

This page is the ground floor for the parent topic. Nothing here is assumed — we start from an arrow on a page.

1. An arrow: what a vector is

A plain number (like "5 kg") is a scalar — size only, no direction. But "moving 5 m/s to the east" needs a direction too, so it's a vector.

Look at the figure below. On the left, the blue arrow's length is the size and where it points is the direction — that single arrow carries two pieces of information at once. On the right, the yellow dot labelled "5 kg" is a scalar: a bare number with nowhere to point.

2. Position vector $\vec{r}$ — "where am I, measured from where?"

The key subtlety: $\vec{r}$ only means something once you name $O$. Move the origin, and $\vec{r}$ changes. That is why the parent note keeps shouting "about the same point" — every rotational quantity is measured relative to a chosen $O$.

In the figure below, the blue arrow $\vec{r}$ runs from the yellow origin $O$ to the green object. Now watch the red dashed arrow $\vec{r}{}^{\prime }$: it reaches the same object but starts from a different origin $O^{\prime }$ — so it has a different length and direction. Same object, different $\vec{r}$, purely because we moved the reference point.

3. Velocity $\vec{v}$ and mass $m$

4. Linear momentum $\vec{p}=m\vec{v}$

5. Force $\vec{F}$, the rate-of-change symbol, and Newton's second law

We need the vector version because the core law differentiates $\vec{L}$ and $\vec{p}$, which are arrows, not plain numbers.

See Newton's Second Law for the full build. The parent topic starts from this line and rotates it, so it is a hard prerequisite.

6. The cross product $\vec{a}\times \vec{b}$ — a machine for "twisting" parts

This is the one genuinely new tool, so we build it carefully. See Cross Product for more.

Why $\sin \theta$ and not, say, $\cos \theta$? Because $\sin \theta$ measures how perpendicular the two arrows are:

• If the arrows point the same way ($\theta =0°$): $\sin 0°=0$ → the cross product is zero.
• If the arrows are at right angles ($\theta =90°$): $\sin 90°=1$ → the cross product is biggest.
• If they point opposite ways ($\theta =180°$): $\sin 180°=0$ → zero again.

The figure below shows exactly this: three panels of the same two arrows at $0°$, $90°$, $180°$, with $\sin \theta$ printed under each — $0$, then $1$, then $0$ again. Watch how the "twisting power" peaks when the arrows are square to each other.

7. Angular momentum $\vec{L}=\vec{r}\times \vec{p}$

8. Torque $\vec{\tau }=\vec{r}\times \vec{F}$

Notice the perfect parallel: $\vec{L}=\vec{r}\times \vec{p}$ and $\vec{\tau }=\vec{r}\times \vec{F}$. Since $\vec{F}=d\vec{p}/dt$, torque is to $\vec{L}$ exactly what force is to $\vec{p}$. That parallel is the whole topic.

9. Radians, moment of inertia $I$, angular velocity $\vec{\omega }$, angular acceleration $\vec{\alpha }$

For a rigid body about a fixed axis these package the spin neatly as $\vec{L}=I\vec{\omega }$, which leads to $\vec{\tau }=I\vec{\alpha }$ (see tau = I alpha). The rotational dictionary:

How the foundations feed the topic

Everything funnels into the single boxed law, then branches out to the rigid-body special case and to Conservation of Angular Momentum (what happens when the twist is zero).

Equipment checklist

Cover the right side; can you answer each before moving on?