M058 · Rotating frames / exact coordinates

Coriolis Turntable

Orbit a three-dimensional turntable as a puck slides, a radial guide moves or a nozzle emits ballistic droplets. Compare the synchronized inertial and rotating coordinate views below, inspect forces and energy, and drag the launch point or velocity tip to set the initial state.

Reviewed time $\text{Pending}$

Apparatus scene $\text{Pending}$

Actual platform rotation $\text{Pending}$

Distance from the origin $\text{Pending}$

Inertial speed $\text{Pending}$

Relative speed $\text{Pending}$

Coriolis acceleration magnitude $\text{Pending}$

Centrifugal acceleration magnitude $\text{Pending}$

Total relative acceleration magnitude $\text{Pending}$

Real external acceleration magnitude $\text{Pending}$

Inertial kinetic energy per mass $\text{Pending}$

Relative kinetic energy per mass $\text{Pending}$

Coriolis power per mass $\text{Pending}$

External inertial power per mass $\text{Pending}$

Coordinate round-trip defect $\text{Pending}$

Velocity round-trip defect $\text{Pending}$

OBSERVE / COMPARE / VERIFY

Who sees the ball curve?

Show the straight-line guessCompare with a guess that ignores rotating coordinates.Scene: puck / guide / jet

j

0.00\,

Parent rotation rate

\Omega

0.80\,\mathrm{rad/s}

Projection latitude

\varphi

90.00\,\mathrm{{}^\circ}

Launch radius

r_0

1.30\,\mathrm{m}

Launch azimuth

\alpha_0

0.00\,\mathrm{{}^\circ}

Initial relative speed

v_0

0.90\,\mathrm{m/s}

Initial relative direction

\beta_0

180.00\,\mathrm{{}^\circ}

Record duration

T

4.00\,\mathrm{s}

Reviewed fraction of the horizon

t/T

0.00\,\%

Orange rings edit the launch point and relative velocity; sliders also support keyboard input. In the guide scene the velocity tip edits radial speed and the separate direction control is disabled. The grey straight-line guess deliberately omits frame rotation. Current velocity arrows use a display gain, and all acceleration arrows share the displayed scale. In the jet scene the leading droplet and its trail retain the first release; additional dots are separate later emissions. Blue inertial speed is constant only for a free particle; the guide does real work.

Ideal horizontal point particles, constant platform rotation and coincident origins. The two-metre rim is a reference circle on an unbounded ideal plane, with no wall collision. Guided radial motion is prescribed and its necessary external force is computed. The jet is independent ballistic droplets, without fluid interaction. Latitude projects the chosen parent rate onto the platform normal; this is an analogy, not a full Earth trajectory model. No friction, vertical motion or varying rotation.

Physics tutorial

A curve can belong to the observer

BackgroundA rotating camera can see a free particle curve even when an observer in the laboratory sees a straight line. Position and velocity must both be transformed; rotating the velocity components alone misses the platform’s transport velocity.

Why it mattersA carousel, a rotating laboratory and a geographic tangent plane share the language of rotating coordinates. Separating real forces from coordinate terms prevents the apparent sideways deflection from becoming an invented interaction.

Start with the essentials

Focus question: Which observer sees a straight path, and what real force is needed to keep a walker on a radial platform line?
One-sentence intuition: The free puck needs no horizontal force after release. A guided walker needs a real force because following a line fixed to a rotating platform is accelerated motion in the laboratory.

Core mathematical model

Position and velocity transformation

\mathbf r_I=R(\omega t)\mathbf r_R,\qquad\mathbf v_I=R(\omega t)(\mathbf v_R+\omega J\mathbf r_R)

The quarter-turn operator points counterclockwise. The added transport velocity is the velocity of the platform point under the particle.

Exact free flight

\mathbf r_R(t)=R(-\omega t)[\mathbf r_0+(\mathbf v_0+\omega J\mathbf r_0)t]

Start from an inertial straight line and transform it. This solution requires no numerical integration and retains the full centrifugal contribution.

Rotating-frame acceleration ledger

\ddot{\mathbf r}_R=\frac{\mathbf F_R}{m}-2\omega J\dot{\mathbf r}_R+\omega^2\mathbf r_R

The Coriolis term depends on relative velocity; the centrifugal term depends on position. No Euler term is present because the angular velocity is constant.

Guided radial motion

\mathbf r_R=(r_0+ut)\mathbf e_r,\qquad\frac{\mathbf F_R}{m}=2\omega u\mathbf e_\theta-\omega^2(r_0+ut)\mathbf e_r

A guide must supply both the tangential acceleration and the inward centripetal component. The relative radial acceleration itself is zero.

Energy and observer-dependent work

e_I=\frac12|\mathbf v_I|^2,\qquad\dot e_I=\frac{\mathbf F_I\cdot\mathbf v_I}{m},\qquad\mathbf a_C\cdot\mathbf v_R=0

Coriolis acceleration does no work in the rotating frame. That fact does not make relative kinetic energy constant; centrifugal and real-force terms can change it.

Latitude analogy

\omega=\Omega\sin\varphi

The latitude control projects the chosen parent rate onto the platform normal. It changes the actual simulated disk rate; it does not add Earth curvature, vertical coupling or atmospheric dynamics.

Common difficulties

A curve implies a sideways physical force

Typical misconceptionThe curved platform path means another object must push the free puck.

Better mental modelThe laboratory path is straight. The coordinate acceleration is supplied by the frame terms, while the computed real horizontal force remains zero.

The initial velocities are the same

Typical misconceptionA throw at the same relative speed has the same laboratory velocity at every launch radius.

Better mental modelThe platform’s tangential transport velocity grows with radius. Move the launch ring while keeping the relative velocity unchanged and compare the inertial speed.

Walking along a radius is free flight

Typical misconceptionThe straight path on the platform requires no force.

Better mental modelThe walker’s laboratory path curves. The guide applies an actual force and can change inertial kinetic energy.

A jet streak is one particle’s inertial path

Typical misconceptionConnect all visible droplets and interpret the result as a freely moving particle’s inertial curve.

Better mental modelLater droplets start at a later nozzle orientation and have different inertial velocities. Each separate droplet follows its own straight inertial line.

Run the experiment

01
Compare the two cameras
Use the inward throw and review the full record. Compare the rotating curve, inertial line and grey straight-line guess.
What to observe: The actual path changes coordinates, while the grey guess omitted the rotation.
02
Change the reference rate
Reverse the rotation, select zero rotation and then select the half-rate latitude projection.
What to observe: The sideways deflection reverses, vanishes or changes its rate with the actual platform rotation.
03
Drag the initial state
Move the orange launch ring and velocity tip. Compare inertial and relative speeds.
What to observe: Off-centre release adds a transport velocity before the free particle starts its laboratory line.
04
Supply the missing force
Select the radial guide, then the jet. Inspect the acceleration ledger and external power.
What to observe: The guide needs a real force; independent jet droplets need none after each release.