The Coriolis Force — Ball on a Turntable

Source

Simulation: A ball launched across a rotating turntable — one motion, two cameras: a 3D ground (inertial) view and a co-rotating view, plus a split-screen comparison. Trails in both frames.
Concept: The Coriolis force is fictitious: from the ground the ball travels dead straight; the sideways "push" exists only in the rotating frame, whose path is exactly r⃗_rot(t) = R(−θ(t))·r⃗_lab(t).
Serves: Rotating reference frames and fictitious forces: F⃗_Cor = −2mΩ⃗×v⃗_rot (|F| = 2mΩv), F⃗_cf = mΩ²r outward; deflection sense vs the sign of Ω; Earth-scale weakness (0.01458 m/s² per kg at 100 m/s) — why cyclones need days and hundreds of km.
Inquiry: Predict which way the ball curves as seen from the turntable when it spins counterclockwise; launch and check; reverse Ω.
Misconception: "The Coriolis force is a real push" — it exists only in the rotating frame; the ground-frame path is straight.

Pedagogy principles applied

ID	Principle	How it shows up here
G4	Playable from load	The turntable spins immediately; every slider and button is live; the prediction is optional and never blocks; the first launch is one click.
G1/G2	Predict → test → evaluate	Four-choice prediction sits beside the stage before the first launch; committing gives targeted feedback; each completed run appends the observed ground/rotating outcome.
H2	Misconception structural, not standing text	"It's a real push" is prediction choice A; the rebuttal appears only on selection, while the ground-frame path stays straight.
C1	Real values, honestly labeled	SI units throughout on a 1 m turntable; Ω and v in rad/s and m/s; Earth connection survives in The Physics sheet.
C2/A2	Readout discipline	Only the Ω and v badges by default; a_Cor appears as the arrow's own label; centrifugal lives behind a checkbox.
A1	One concept, one sentence	The Coriolis force exists only in the rotating frame — the ground-frame path is straight.
E1	Animation honesty	The rotating-frame path is r⃗_rot = R(−θ)·r⃗_lab — never an independently animated curve; runs pause at the rim. (Tooltip metaphor: chalk on the table.)
3D	Embodied 3D apparatus	three.js with ACESFilmic tone mapping, sRGB output, soft shadows, clamped orbit camera with hint, HTML labels projected from 3D anchors; the frame toggle re-parents only the camera.
AUD	Machine-checkable physics	window.__audit.at() returns 2Ω·v from the same helper that labels the arrow (2 N per kg at the defaults Ω = 0.5 rad/s, v = 2 m/s).

🎠 The Coriolis Force — Ball on a Turntable

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Speed

🎥 GROUND FRAME (inertial)

Legend

path in the ground frame

path in the rotating frame

initial launch position

🖱 drag to orbit · scroll to zoom

Guided Inquiry

1 · The apparatus

Let’s meet the apparatus. A turntable rotates at a constant rate Ω. A launcher bolted to its rim fires a ball at speed v across the table, aimed straight at a target painted on the table. Two cameras film the flight: one fixed to the room, and one turning with the table.

2 · The path in the room

Let’s start with the room camera. The table turns at Ω and the launcher fires, and nothing touches the ball after it leaves the muzzle. Filmed by the room-fixed camera, what do you think its path looks like?

3 · The path on the table

Now switch to the other camera. The table turns counter-clockwise at Ω = 0.50 rad/s, and the same flight is filmed by the camera turning with the table. What do you think — relative to the aim, that path

4 · Both records, one flight

Let’s see both at once. Both cameras now record the same launch, so fire the launcher, then compare the cyan room trail with the amber table trail. Which features of the flight are identical in the two records, and which aren’t?

5 · Doubling the launch speed

Now let’s put a number on it. The table keeps turning at Ω = 0.50 rad/s while the launch speed is raised from 1.0 to 2.0 m/s. What do you think — the Coriolis force on the ball in flight becomes

6 · A frame effect, and how strong

Let’s put it all in perspective. The Coriolis term has no source object and no reaction partner, and being perpendicular to the velocity it does no work — it turns the path without changing the speed measured on the table. On Earth Ω = 7.29×10⁻⁵ rad/s gives a 100 m/s wind just 0.0146 m/s²: enough to shape cyclones over hundreds of kilometres and days, far too weak to decide which way a bath drains.

Listen

Controls

▾

View

View ground frame trajectory View rotating frame trajectory

Rotation rate Ω

Launch speed

draw the centrifugal arrow (Ω²r, outward) in the rotating view

Ground (inertial) frame: no horizontal force after launch, so the path is a straight line at constant velocity (Newton I).

Rotating frame: two fictitious bookkeeping terms appear — Coriolis and centrifugal. Neither is produced by any real interaction.

At the defaults Ω = 0.5 rad/s, v_rot = 2 m/s, m = 1 kg, r = 1 m. The Coriolis force is perpendicular to v⃗_rot — with Ω > 0 it points to the right of the motion (F⃗ = [0, −2] N for v⃗_rot = [2, 0] m/s).

The exact view transform: the path in the rotating frame is the path in the ground frame re-plotted by a rotating camera — same motion, rotated coordinates. This sim computes that trail from this formula, point by point.

Earth: Ω⊕ = 7.29×10⁻⁵ rad/s. A 100 m/s jet-stream wind feels a feeble 0.01458 N per kg — hours to bend a path, hence cyclones spanning hundreds of km.

The Physics

The Coriolis force · what this sim shows, the physics behind it, and every control

At a glance

A launcher on a turning table fires a ball at a target painted on the table, and two cameras film the flight: one fixed to the room, one turning with the table. Comparing the two films shows where the Coriolis force comes from.

What you'll learn

What one flight looks like from the room and from the turning table, and why the two films differ.
Which extra terms Newton's law needs in a rotating frame, and why they are called fictitious.
How the direction of the Coriolis force depends on the sense of rotation.
How its size depends on Ω and on the speed measured on the table.

Top view. The launcher rides the rim and aims straight at the target painted on the table.

What's on screen

Launcher (green) and red target ✕: both turn with the table.
Path in the ground frame (cyan): the ball's track in the room.
Path in the rotating frame (amber): its track drawn on the table.
Initial launch position: a green strip on the floor.
Arrows (table camera only): green v_rot, red a_Cor, violet a_cf if switched on.
Frame badges name each camera; the status pill shows the run's state.

See itThe table turns with the launcher armed. Press 🚀 Launch when you are ready. Watch for: the launcher and the target ✕ riding round together with the table.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
View	The camera: fixed to the room, turning with the table, or both side by side.	Ground / Rotating / Both opens on Ground
Trajectory checkboxes	Show or hide the cyan ground-frame path and the amber rotating-frame path.	on / off both on
Rotation rate Ω	How fast the table turns. Positive is counter-clockwise seen from above; negative reverses it.	−2 to +2 rad/s opens at 0.50
Launch speed	The ball's speed as it leaves the muzzle, measured relative to the table.	0.5–3 m/s opens at 2.0
🚀 Launch	Fires the ball. The run pauses when the ball reaches the rim.	button
Centrifugal arrow	Adds the violet outward arrow, Ω²r, in the table camera.	on / off opens off
▶ Play / ⏸ Pause	Turns the table and flies the ball, or freezes both. After a run, Play re-arms the launcher.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Changing Ω, the launch speed, the view or the centrifugal box re-arms the launcher. Drag the stage to orbit the camera; scroll to zoom. Fixed: table radius 1 m; launcher 0.88 m and target 0.78 m from the centre; no friction or drag.

The room sees a straight line

The room is an inertial frame. Once the ball leaves the muzzle, nothing pushes it sideways, so Newton's first law applies:

Room camera: no horizontal force, so constant velocity.

The cyan path is a straight line, traced at constant speed, whatever Ω is.
The launcher rides the rim, so the ball leaves with the launch speed along the aim plus the rim's own sideways speed, Ω × 0.88 m. At the opening values that is 0.44 m/s, which tilts the line about 12° off the aim from the very start.
Meanwhile the target turns with the table, so unless Ω = 0 the ball misses it.

See itLaunch, filmed by the room camera. Watch for: the cyan trail, a straight line from the green launch strip to the rim.

The table sees a curve

One flight, two cameras (Ω > 0). The ball moves the same way in both.

The table camera records the same positions, but in coordinates that have turned by θ = Ωt since launch:

Each point of the room path, turned back by the angle the table has turned. The sim draws the amber trail this way.

Later points are turned back further than earlier ones, so the straight room line becomes a curve on the table.
For Ω > 0 the curve bends to the right of the aim. At the opening values the ball reaches the rim after 0.90 s, 49° round from the aim point: about 23° from the launcher's sideways speed, and 26° because the table turned during the flight.

See itOne launch, both cameras side by side. Watch for: the amber trail curving away from the target ✕, while the cyan trail stays straight.

Two terms the turning frame needs

To use F = ma in the table's coordinates, two extra terms must be added to the real forces:

Newton's second law in a frame turning at a steady rate Ω.

Coriolis, −2mΩ × v_rot: acts only on a body that is moving relative to the table.
Centrifugal, −mΩ × (Ω × r): points straight out from the axis with size mΩ²r. It acts even on a body sitting still on the table.
Neither term comes from another object, and neither has a reaction partner. That is why they are called fictitious (or inertial) forces. Inside the turning frame they predict the motion exactly; in the room they do not exist.

See itTable camera at Ω = 2 rad/s, with the centrifugal arrow on. Watch for: when the ball stops at the rim, the violet a_cf arrow pointing straight out from the centre and the red a_Cor arrow pointing across the motion.

The sign of Ω sets the side

Ω × v_rot is always perpendicular to v_rot, so the Coriolis force does no work: it turns the path without changing the speed.
Ω > 0 (counter-clockwise): it points to the right of the motion. Ω < 0: to the left.
The ball's speed on the table can still change during a flight, by a lot at high Ω. That is the centrifugal term's work, not the Coriolis term's.

The Coriolis force is always at 90° to v_rot.

See itTable camera, Ω = +1 rad/s, paused mid-flight. Watch for: the red a_Cor arrow pointing across the green v_rot arrow, to its right.

See itThe same moment with Ω = −1 rad/s. Watch for: the red arrow now to the left of the green one, and the amber trail bending left.

How strong it is

The Coriolis acceleration (force per kilogram), 2Ωv, is proportional to Ω and to the speed v measured on the table. Double either and it doubles; if either is zero it vanishes.
The red arrow's label shows 2Ω times the ball's speed on the table at that moment, so it changes during a flight.

Ω v

These sliders move the sim too.

See itLaunch at 1.0 m/s, then at 2.0 m/s, with Ω = 0.5 rad/s, pausing each just after launch. Watch for: the card listing the Coriolis acceleration for both speeds.

Where it matters

Earth is a turntable with Ω_⊕ = 7.29 × 10⁻⁵ rad/s, about 6900 times slower than the opening table.

The largest value, reached at the poles. For horizontal motion the part that matters is 2Ω_⊕v sin(latitude).

Weather. Air flowing toward a low is turned right in the northern hemisphere and left in the southern, so storms turn counter-clockwise in the north and clockwise in the south. Hurricanes almost never form within about 5° of the equator, where sin(latitude) is small.
Foucault's pendulum. Its swing plane turns by 360° × sin(latitude) per sidereal day: about 11° an hour in Paris.
Not your sink. For water moving a few cm/s, the Coriolis acceleration is less than a millionth of g. Leftover swirl from filling the basin decides the direction.

Inflow toward a low, as seen from above each hemisphere.

Go deeper: what sets the shape of the bend

Measure time in units of 1/Ω and every flight in this sim depends only on the ratio v/(ΩR), with R the table radius. Runs with the same ratio trace the same amber curve, just at different speeds.
Meteorologists use the same kind of ratio, the Rossby number U/(2ΩL sin latitude). When it is small, as for weather systems hundreds of kilometres across, the Coriolis effect dominates.

Summary

In one line. The Coriolis force is one of two terms a rotating frame must add for F = ma to work in its turning coordinates. It acts at right angles to the motion, with size 2mΩv for motion across the table.

Key points

In the room the ball moves in a straight line; on the table the same motion curves, right for Ω > 0 and left for Ω < 0.
The Coriolis term acts only on bodies moving relative to the table, and does no work.
The centrifugal term, mΩ²r outward, acts on everything off the axis.

Common mix-ups

"The table pushes the ball sideways." Nothing touches the ball in flight.
"Everything on a turntable feels a Coriolis force." Only things moving relative to it.
"Earth's rotation decides which way a sink drains." At that scale it is far too weak.

Things to try on your own

Set Ω to 0 and launch. Compare the two trails.
Launch with Ω = 0.5 rad/s and v = 1.0 m/s, then with Ω = 1.0 rad/s and v = 2.0 m/s, and compare where the amber trail meets the rim.
Turn on the centrifugal arrow, set Ω = 2 rad/s, and follow its label as the ball crosses the table.