Foundations — Rotating frames — centrifugal force, Coriolis force

This page assumes you know nothing. We build every symbol the parent note throws at you, in an order where each brick rests on the one before it.

0. What a "frame" even is

Picture two people watching the same ball: one standing on the ground, one riding a spinning platform. Same ball, two stories. The whole topic is about translating between these two stories.

Why the topic needs this: Newton's rule is only guaranteed in inertial frames. A rotating frame is non-inertial, so we must patch the rule — that patch is centrifugal + Coriolis.

1. Vectors and the little arrow $\stackrel{⃗}{}$

• $\vec{r}$ — the position vector: an arrow from the centre/origin out to where the object is. Its length is the distance from the origin.
• $\vec{v}$ — the velocity vector: how fast and in what direction the object moves.
• $\vec{a}$ — the acceleration vector: how fast the velocity is changing.
• $\vec{F}$ — the force vector: a push or pull, with strength and direction.

2. Magnitude bars $|\vec{A}|$ and unit vectors $\hat{e}$

Any vector = (its length) × (its direction signpost): $\vec{A}=|\vec{A}|\hat{A}$.

The parent note uses three special signposts on a spinning disk:

• $\hat{r}$ — points radially outward, straight away from the centre.
• $\hat{\theta }$ — points sideways, in the direction of spin (tangent to the circle).
• $\hat{z}$ — points up the axis, out of the disk.

Why the topic needs these: centrifugal force points along $\hat{r}$; Coriolis on a radially-thrown ball points along $\hat{\theta }$. Having named signposts lets us say which way each fictitious force pushes.

3. Angular velocity $\vec{\omega }$

Why the topic needs it: $\vec{\omega }$ is the input describing the rotating frame. Both fictitious forces are built out of $\vec{\omega }$.

4. The cross product $\vec{\omega }\times \vec{A}$

This is the engine of the whole topic, so we build it slowly. Full treatment: Cross Product & Right-Hand Rule.

Two facts the parent uses, now un-mysterious:

• $\hat{z}\times \hat{r}=\hat{\theta }$ — spin axis crossed with "outward" gives "sideways." (Perpendicular to both, right-hand rule.)
• $\sin \varphi =0$ when $\vec{A}\parallel \vec{B}$: if two arrows are parallel, their cross product is zero. So a velocity along the axis feels no Coriolis.

5. Newton's Second Law $\vec{F}=m\vec{a}$

Why the topic needs it: this is the rule we refuse to give up. Because it only holds in an inertial frame, the rotating observer's job is to rearrange it into $m{\vec{a}}_{\text{rot}}={\vec{F}}_{\text{real}}+(\text{fictitious terms})$ so the same familiar equation still works inside the spin.

6. Centripetal force (the real inward pull)

Here $r_{\perp }$ (read "r-perp") means the perpendicular distance from the spin axis — how far out from the pole you are, not how far from the origin. On a flat disk they're the same; on a globe they differ.

7. Derivative and $d/dt$ — "rate of change"

The parent writes ${\left(\frac{d\vec{A}}{dt}\right)}_{\text{inertial}}$ and ${\left(\frac{d\vec{A}}{dt}\right)}_{\text{rot}}$: the same arrow, its change-per-second as judged by the ground observer versus the spinning observer. These disagree precisely because the spinning observer's own signposts are turning.

How it all feeds the topic

Equipment checklist

Cover the right side; can you answer each before revealing?

Ready? Continue to 1.2.14 D2 Visual Walkthrough.