M046 · Energy and geometry

Cycloid Challenge: Fastest & Isochronous

Which curve lets a bead slide down fastest and also makes beads from any height arrive together? The cycloid does both. Between two fixed points, a bead slides down a cycloid faster than along any other curve. A cycloid is also isochronous: beads released at different heights on the same cycloid reach the bottom at the same moment.

Race particles along straight, circular, polynomial and cycloid wires between the same two endpoints. Then release particles from three heights on one cycloid, and compare their raw arrival times with a circular track that has the same curvature at the bottom.

Cycloid selected station time $\text{Ready}$

Cycloid selected speed $\text{Ready}$

Cycloid arrival time $\text{Ready}$

Independent analytic cycloid time $\text{Ready}$

Maximum refinement time difference $\text{Ready}$

Acquired timing spread $\text{Ready}$

Experiment mode $\text{Ready}$

Acquired lanes $\text{Ready}$

Shape the track, then check contact and timing

Which curve lets a bead slide down fastest and also makes beads from any height arrive together?

Mode: same endpoints or same arrival

c

0.00\,

Endpoint vertical drop

H

2.00\,m

Endpoint horizontal separation

X

3.00\,m

Polynomial sag coefficient

\beta

1.00\,

Isochronous generating radius

a

1.00\,m

Largest release fraction from bottom

f

0.80\,

Uniform gravity

g

9.81\,\mathrm{m\,s^{-2}}

Integration intervals per piece

N

160.00\,

Replay time fraction

t/T_{max}

0.55\,

Sensor noise gain

\eta

0.00\,

Point masses slide without friction along wires that hold them from both sides, under uniform gravity. Ball size is schematic; rolling, friction, drag and manufacturing tolerances are left out.

In the endpoint race every track shares the same start and finish points. The cycloid may dip below the finish. The sagging polynomial is an adjustable curve for comparison; it does not search all possible paths for the fastest one.

The isochrony mode releases three beads from rest on identical cycloids. The circular reference has the same curvature at the bottom and the same highest release height. Each bead stops at its first arrival.

An acquisition records simulated raw arrival times, which are compared separately from the computed curves and the analytic reference. The noise is the noise gain times one millisecond and one tenth of a millimetre.

The particles slide without friction on wires that hold them from both sides. The fastest-descent race and the equal-arrival test are separate experiments. Numerically integrated times and analytic reference values are kept separate.

Start with the essentials

Focus question: Which curve lets a bead slide down fastest and also makes beads from any height arrive together?
One-sentence intuition: To compare speed, keep the endpoints fixed. To test isochrony, keep the bowl fixed and change only the release height.

Core mathematical model

Cycloid through both endpoints

\begin{aligned}x&=a(\theta-\sin\theta)\\y&=H-a(1-\cos\theta)\end{aligned}

Solve the endpoint ratio for the final angle, then find the radius of the generating circle. For wide spans, the cycloid can dip below the finish.

Travel time along a curve

T=\int\frac{\sqrt{dx^2+dy^2}}{\sqrt{2g(H-y)}}

The shortest path does not have to minimize this integral. The integrand blows up at a release from rest; a squared integration coordinate removes that singularity.

Exact travel time on the cycloid

T_{race}=\theta_f\sqrt{a/g}

This analytic value is independent of the numerical integration and checks its travel time for a cycloid that starts at its cusp.

Motion measured along the arc

\begin{aligned}s_b&=4a\cos(\theta/2)\\y&=\frac{s_b^2}{8a}\\\ddot s_b&=-\frac{g}{4a}s_b\end{aligned}

Measured by distance along the curve from the bottom of the bowl, the potential of the ideal sliding particle is exactly quadratic.

Same arrival time from every height

T_{first}=\pi\sqrt{a/g}

Three releases from rest, at different nonzero heights, all take the same quarter period. The circular reference has the same curvature at the bottom, but its timing depends on the amplitude.

Common difficulties

An idealized ball

Typical misconceptionThe ball on screen rolls, has suspension and behaves like real hardware.

Better mental modelThe model is a point particle that slides. The thickness of the rail and the marker is only for drawing.

Run the experiment

01
Race between the same endpoints
Change the horizontal span, the vertical drop and the polynomial sag.
What to observe: The cycloid races the other curves from the same release point to the same finish.
02
Switch to the equal-arrival test
Select Three release heights and look at the shared bottom.
What to observe: The three cycloids have the same shape and gravity, and every particle starts at rest. Only the release heights differ.
03
Test equal arrival
Compare Near-bottom releases with the circular reference.
What to observe: For small swings, the circular reference approaches the same time. From a finite height, it is slower.
04
Record and refine
Acquire arrivals with zero noise, then change the noise or the resolution.
What to observe: The raw timing is shown separately from the analytic reference. Changing the apparatus clears the old readings.