M058 · Rotating frames / exact coordinates

Coriolis Turntable

Why does a puck sliding freely on a turntable seem to curve when nothing pushes it sideways? Because the platform turns underneath it. Seen from outside, a free puck goes in a straight line and no sideways force acts on it. The curve exists only in coordinates that turn with the platform. A walker who wants to stay on a straight line painted on the platform is a different case: that takes a real force.

Watch a puck slide across a turntable, a guide move a walker along a radius, or a nozzle spray ballistic droplets, and turn the three-dimensional view freely. Compare the inertial and rotating views below, which run in step, check the forces and energy, and drag the launch point or the tip of the velocity arrow to set the starting 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 $\text{Pending}$

Watch the motion and compare the cases

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\,\%

Drag the orange rings to set the launch point and the relative velocity; the sliders also work from the keyboard. In the guide scene, the tip of the velocity arrow sets the radial speed, and the separate direction control is turned off. The grey straight-line guess leaves out the frame's rotation on purpose. The current velocity arrows are scaled up for display, and all acceleration arrows share one scale. In the jet scene, the leading droplet and its trail come from the first release; the other dots are droplets emitted later. The blue speed in the inertial frame stays constant only for a free particle, because the guide does work.

The particles are ideal points moving horizontally, the platform turns at a constant rate, and both frames share the same origin. The two-metre rim is only a reference circle on an endless ideal plane; nothing collides with it. The guided radial motion is prescribed, and the Lab computes the external force it needs. The jet is a stream of independent ballistic droplets with no fluid interaction. Latitude projects the chosen rotation rate of the parent body onto the platform normal; this is an analogy, not a full model of trajectories on Earth. Friction, vertical motion and changes in the rotation rate are left out.

Physics tutorial

A curve can belong to the observer

BackgroundA camera turning with the platform sees a free particle curve, while an observer in the laboratory sees it move in a straight line. Switching views means transforming both position and velocity; rotating the velocity components alone leaves out the velocity of the platform under the particle.

Why it mattersA carousel, a rotating laboratory and a small flat patch of the Earth’s surface are all described with rotating coordinates. Keeping real forces apart from the extra terms that come from the coordinates stops you from mistaking the apparent sideways deflection for a real push.

Start with the essentials

Focus question: Which observer sees a straight path, and what real force keeps a walker on a line running out from the platform’s center?
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 rotates vectors counterclockwise. The extra term is the velocity of the platform point directly under the particle.

Free flight, solved exactly

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

Start from the straight line seen in the inertial frame and transform it. No numerical integration is needed, and the centrifugal effect is included in full.

Acceleration in the rotating frame

\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

The guide must supply both the tangential acceleration and the inward centripetal part. Relative to the platform, the radial acceleration 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

In the rotating frame, the Coriolis acceleration does no work. Kinetic energy measured on the platform can still change, because the centrifugal term and real forces do work.

Latitude analogy

\omega=\Omega\sin\varphi

Latitude projects the chosen rotation rate of the parent body onto the platform normal. It changes the rate at which the simulated disk actually turns; it does not add the Earth’s curvature, vertical coupling or atmospheric dynamics.

Common difficulties

A curved path does not need a sideways force

Typical misconceptionThe puck curves on the platform, so something must be pushing it sideways.

Better mental modelIn the laboratory the path is straight. The curve on the platform comes from the terms added by the rotating coordinates, and the computed real horizontal force stays zero.

Same throw, different laboratory velocity

Typical misconceptionA throw at the same speed relative to the platform has the same laboratory velocity from any launch radius.

Better mental modelThe platform moves faster the farther you are from the center, and that speed adds to the throw. Keep the relative velocity fixed, launch from a larger radius and compare the inertial speed.

Walking along a radius is not free motion

Typical misconceptionA straight path on the platform needs no force.

Better mental modelIn the laboratory the walker’s path curves. The guide pushes with a real force and can change the kinetic energy measured in the inertial frame.

A jet streak is not one particle’s path

Typical misconceptionJoining up all the visible droplets traces the inertial path of one free particle.

Better mental modelLater droplets leave when the nozzle points a different way, so they have different inertial velocities. Each droplet follows its own straight line in the inertial frame.

Run the experiment

01
Compare the two views
Choose Inward throw and press Review full record. Compare the curve in the rotating view, the line in the inertial view and the grey straight-line guess.
What to observe: The same real motion looks different in the two coordinate systems; the grey guess leaves out the rotation.
02
Change the rotation rate
Choose Reverse the rotation, then Stationary platform, then Half-rate projection.
What to observe: The sideways deflection reverses, disappears or changes in step with how the platform actually turns.
03
Drag the starting state
Move the orange launch ring and the velocity tip, and compare the inertial and relative speeds.
What to observe: A release away from the centre adds the platform’s own velocity, and the free particle then follows a straight line in the laboratory.
04
Find the force the guide supplies
Choose Guided radial walker, then Continuous ballistic jet, and look at the acceleration terms and the external power.
What to observe: The guided walker needs a real force; each jet droplet needs none once it leaves the nozzle.