Centrifugal Force — Ball on a Merry-Go-Round

Source

Lecture: Classical mechanics — non-inertial frames & fictitious forces
Simulation: "The Centrifugal Force — A Ball on a Merry-Go-Round" · 3D carousel with two synchronized views (ground/inertial ↔ rotating camera, plus side-by-side), trails in both frames, counterfactual "fling" mode
Concept: The centrifugal force is not a real interaction: it appears only when motion is described in a rotating frame (F_cf = mω²r, radially outward); the ground view shows straight-line or at-rest motion under real forces only.
Serves: Frame-dependence of fictitious forces · Newton's 1st law as the ground-frame explanation · reading and scaling F = mω²r (m = 1 kg, ω = 1.5 rad/s, r = 2 m → 4.5 N outward; ω = 1, r = 1 → 1 N; F ∝ ω² and ∝ r)
Inquiry: Where is the outward force in the ground view? · How does F_cf change when ω doubles? · Which facts are identical in both frames?
Misconception: "Centrifugal force is a real outward interaction acting on the ball." Nothing exerts it; the released ball's ground-frame path is a straight tangent line, never a radial fling — the outward push exists only in the rotating description.

Pedagogy principles applied

ID	Principle	How it shows up here
PRED	Optional, non-blocking prediction	3-choice ground-path prediction offered before the release; sim starts paused so the answer isn't spoiled, but everything is playable without predicting.
H1/H2	Misconception targeted structurally	The wrong idea lives only in choice A and the ⚠ mode; the rebuttal appears only when selected; the "claimed" ball/arrow is red and visibly fails.
CF	Counterfactual mode	Two labeled buttons — "✓ Reality (no push)" vs "⚠ 'Fling' claim" — both implemented exactly (tangential vs radial launch at v = ωr₀).
INV	Invariant vs frame-dependent	"Two cameras, one reality" panel: events and real forces identical in both frames; fictitious forces appear/disappear with the frame.
C1/C2	Readout discipline · real SI values	One headline badge (F = mω²r) plus contextual force labels; SI units, m = 1 kg, g = 9.81 m/s², μ = 0.40 labeled.
D6/D7	One hero scene, student-driven disclosure	Single 3D stage; frame toggle and split view are explicit, always-available controls; panels collapsed, never locked.
A3/A4	Formal layer off by default	Rotating-frame equation of motion, transformation r′ = R(−ωt)r, and equilibrium live in The Physics revision sheet (⚛︎ The Physics button).
E1/J2	Honest animation, exact physics	Inertial path is a closed-form straight line (no integrator); rotating path obtained by rotating it by −ωt; slider changes explicitly restart the run and clear trails; the episode never silently loops.
3D	3D apparatus tier	ACESFilmic + sRGB, soft shadows, clamped orbit camera with hint, HTML labels projected from 3D anchors; frame toggle re-parents the camera onto the carousel.
AUD	Machine-readable audit hook	window.__audit.at() returns centrifugalMag = ω²r (per kg) from the current sliders — identical to the displayed badge.

Teach me

Speed

drag to orbit, scroll to zoom

Guided Inquiry

1 · The apparatus

Let’s meet the apparatus. A carousel deck turns at a constant angular speed ω. A ball rests on the frictionless deck at radius r, held there by a backstop that turns with it. The same release can be filmed by a camera fixed to the ground, or by a camera bolted to the deck — and that choice will matter.

2 · The path after release

Let’s start with the ground camera. The deck turns at ω, and the backstop lets the ball go at radius r. Filmed from the ground, what path do you think the ball follows afterwards?

3 · Two cameras, one release

Now let’s bring in the second camera. Both cameras film the same release, so release the ball, then compare the cyan ground trail with the orange deck trail. Which features of the motion are identical in the two records, and which aren’t?

4 · The source of the outward push

Here’s the puzzle. On the deck camera, the released ball moves outward as though shoved. So which object do you think exerts that shove?

5 · Doubling the spin

Now let’s put a number on it. The ball is held at r = 2.0 m while the deck turns at ω = 1.00 rad/s. Then the deck speeds up to ω = 2.00 rad/s at the same radius. What do you think — the outward term in the deck frame becomes

6 · What makes a force fictitious

Let’s pull it all together. Every event and every real force is the same in both films; only the outward term appears or disappears with the choice of camera. A force with no source object and no reaction partner is called fictitious — here it’s mω²r, radially outward, in the deck frame alone.

Listen

Controls

▾

Camera frame

angular speed ω1.50 rad/s

hold radius r2.0 m

Formal tools — the same motion in two coordinate systems (m = 1 kg, SI units, g = 9.81 m/s²)

Ground frame · after release

A straight line at constant speed. The sim draws exactly this closed form — no integrator, no force.

Rotating frame = re-description

The orange path is the same straight line rotated into carousel coordinates — a viewpoint change, not new physics.

Newton in the rotating frame

To keep F = ma in a spinning frame you must add these two frame terms. No object exerts them; they have no reaction partner.

Centrifugal magnitude

Here: m = 1 kg, ω = 1.50 rad/s, r = 2.0 m ⇒ F_cf = 4.50 N, radially outward — matches the header badge and the reference value (4.5 N at ω = 1.5 rad/s, r = 2 m).

While held (rotating-frame equilibrium)

The backstop's real inward (centripetal) push balances the fictitious outward pull — why the held ball sits still on the carousel.

The Physics

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

At a glance

A ball sits on a spinning carousel deck, held by a backstop, and is then let go. One camera fixed to the ground and one riding on the deck film the same motion, so you can compare the two points of view.

What you'll learn

How the path of the released ball depends on which camera films it.
Which force keeps the held ball on its circle, and which way it points.
Where the outward "centrifugal force" comes from, and how to test whether a force is real.
How the size of that outward term depends on ω and r.

Seen from above: the deck turns anticlockwise at ω. The backstop holds the ball at radius r until you release it.

What's on screen

Cyan trail: the ball's path as the ground camera films it.
Orange trail: the ball's path as the deck camera films it.
N (backstop) arrow and label: the backstop's push while it holds the ball.
F_cent arrow and label: the outward term, with its size in newtons.
Coriolis arrow and label: the Coriolis term, when its chip in the rotating legend is on.
Centrifugal force readout (Controls panel): F_cent at the hold radius.

See itBoth cameras film the ball while the backstop holds it. Watch for: in the right-hand (rotating) view the deck stands still and the park turns around it.

The controls

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

Control	What it changes	Range
Camera frame	Films the carousel from the Ground, from the Rotating deck, or from both side by side.	Ground · Rotating · ⧉ Both opens on Ground
angular speed ω	How fast the deck turns.	0–2 rad/s opens at 1.50
hold radius r	How far from the axis the backstop holds the ball. The deck edge is at 3.4 m.	0.5–3 m opens at 2.0 m
Release ball	The backstop drops away and lets the ball go. Also on the stage.	once per run
Coriolis arrow	A chip in the rotating view's legend. Shows or hides the Coriolis arrow.	on / off opens off
Centrifugal force (readout)	The size of the outward term at the hold radius.	panel
Drag / scroll (stage)	Orbits and zooms the view. The motion is unchanged.	stage
▶ Play / ⏸ Pause	Runs or freezes the motion.	top bar
Speed	Plays the motion faster or slower on screen. The physics stays the same.	0.25×–4× opens at 0.5×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Moving a slider or switching camera puts the ball back in the backstop. Fixed: m = 1 kg, a frictionless deck and ground.

One release, two films

Ground camera. After release nothing pushes the ball sideways: the deck is frictionless and the backstop has dropped away. So the ball keeps the velocity it had, tangent to its circle, of size v = ωr, and moves in a straight line at constant speed (Newton's first law).
Deck camera. The same flight, re-plotted in coordinates that turn with the deck, is an outward spiral that curls back against the spin. The ball seems to be shoved outward.
Nothing about the ball differs between the two films. Only the camera does.

One release at ω = 1.50 rad/s, r = 2.0 m, seen from above. Dotted: the hold circle.

The deck film is the ground film turned back by the angle the deck has turned, ωt.

See itRelease the ball with both cameras running. Watch for: on the left the trail runs off in a straight line; on the right the same flight sweeps outward in a long curve around the carousel.

The held ball needs an inward push

Ground view. Before release the ball goes round a circle, so it is accelerating toward the axis. That takes a real inward (centripetal) force of size mω²r, and the backstop supplies it: the arrow labelled N (backstop).
Deck view. Here the ball sits still, yet N still pushes inward. For F = ma to give zero, the deck frame adds an outward term of the same size, mω²r. That is the centrifugal force, the arrow labelled F_cent.

The held ball in the deck frame, at ω = 1.50 rad/s, r = 2.0 m.

Deck-frame balance while the ball is held.

A rider on a carousel feels the same thing: the seat or rail pushing inward. The feeling of being flung out is your body's tendency to keep going straight.

See itBoth cameras film the held ball. Watch for: the N (backstop) label shows in both views, the F_cent label only in the right-hand (rotating) view.

No object exerts it

To keep F = ma true in a frame turning at ω, you must add two terms to the real forces:

Newton's second law in the deck frame.

Centrifugal: mω²r, straight out from the axis, acting wherever the ball is.
Coriolis: sideways to the ball's motion, and only when the ball moves relative to the deck. It bends the path back against the spin.
The test for a real force is to name the object that exerts it. After release the deck is frictionless and the backstop is gone, so nothing pushes outward. These terms have no source and no reaction partner, which is why they are called fictitious. Switch to the ground camera and both vanish.

See itRelease the ball on the deck camera, with the Coriolis arrow on. Watch for: the F_cent and Coriolis labels stay with the ball after the backstop has dropped away, though nothing touches it.

How big it is

The term grows as the square of ω: double the spin and it becomes four times larger. Doubling r only doubles it.
It depends on where the ball is now. After release r grows, so the term grows: at the default settings it is 4.50 N at release and 7.65 N at the deck edge (r = 3.4 m).

ω r

These sliders move the sim too.

See itHold the ball at r = 2.0 m with ω = 1.00 rad/s, then with ω = 2.00 rad/s. Watch for: the two Centrifugal force readings, side by side at the end.

See itRelease the ball on the deck camera at your settings. Watch for: the number on the F_cent label climbing as the ball moves outward.

Where it matters

The spinning Earth. We all stand on a rotating frame, turning once a day. At the equator the outward term is about 0.034 m/s², roughly 1/290 of g, so a scale reads about 0.3% less there than the same spot would without the spin. The same spin is why Earth bulges about 21 km at the equator.
Centrifuges. A lab ultracentrifuge at ω ≈ 3000 rad/s (about 29 000 rpm) and r = 10 cm gives ω²r ≈ 9×10⁵ m/s², about 90 000 g, enough to separate particles by size and density.
Spin dryers. In the ground frame, water leaves through the holes in the drum along straight tangent lines, just like the ball after release.

The outward term at Earth's equator.

Go deeper: one point in both films

Take the defaults (ω = 1.50 rad/s, r = 2.0 m), start the clock at release, and put the release point on the x axis, with y pointing the way the deck turns.
At t = 0.5 s the ground film has the ball at (2.00, 1.50) m. The deck has turned 0.75 rad (43°). Turning the point back by that angle gives (2.49, −0.27) m in the deck film.
So in the deck film the ball is already at r = 2.50 m, outside its hold circle, and behind the spoke it was released from.

Summary

In one line. The centrifugal force, mω²r outward, is a term you add to keep F = ma working in a rotating frame. No object exerts it, and in the ground frame it does not exist.

Key points

Released, the ball moves in a straight tangent line in the ground frame, and in an outward spiral in the deck frame.
While held, the backstop's real inward push keeps the ball on its circle. In the deck frame the outward term balances it.
The term is mω²r: four times larger for twice the spin, twice as large for twice the radius.

Common mix-ups

"Released, the ball flies straight out along the spoke." It leaves along the tangent. A radial path would need an outward force, and there is none.
"Centrifugal and centripetal force are an action–reaction pair." Both act on the same ball. The real pair is the backstop pushing the ball and the ball pushing back on the backstop.
"If you feel flung outward, an outward force must be acting." What you feel is the seat pushing you inward.

Things to try on your own

Set ω = 0 and release the ball. Compare the two views.
Set r = 1.0 m and then r = 3.0 m at the same ω, and compare the Centrifugal force readings.
Release the ball on the Rotating camera with the Coriolis arrow on, and note which way the Coriolis arrow points.