M046 · Energy and geometry

Cycloid Challenge: Fastest & Isochronous

Race straight, circular, polynomial and cycloidal wires between the same endpoints. Then release particles from three heights on the same cycloid, with a curvature-matched circular reference and raw arrival timing.

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 $\text{Ready}$

GEOMETRY / CONTACT / TIMING

Which curve wins—and which arrives 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\,

Frictionless sliding point masses on bilateral wires under uniform gravity. Ball size is schematic; rolling, friction, drag and manufacturing tolerances are omitted.

The endpoint race uses equal release and finish coordinates. The cycloid may dip below its finish; polynomial sag is an editable comparison, not an unrestricted path optimizer.

Isochrony compares three rest releases on identical cycloids. The circular reference matches bottom curvature and the highest release height. Each marker stops at its first arrival.

Acquire arrivals to compare simulated raw timing separately from computed curves and the analytic reference. Noise gain scales one millisecond and one tenth of a millimetre.

Frictionless bilateral sliding particles. Fastest descent and equal arrival times are separate protocols; computed quadrature and analytic references remain distinct.

Start with the essentials

Focus question: Which curve wins—and which arrives together?
One-sentence intuition: Keep endpoints fixed for speed, and keep the bowl fixed for isochrony.

Core mathematical model

Equal-endpoint cycloid

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

Solve the endpoint ratio for the final angle, then determine the generating radius. Wide spans can descend below the finish.

Time functional

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

The shortest path need not minimize this integral. Rest-release singularities are removed by a squared integration coordinate.

Cycloid analytic clock

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

An independent analytic value checks the numerical travel-time quadrature for the cusp-start cycloid.

Arc-coordinate oscillator

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

Distance from the bowl bottom gives an exactly quadratic potential for the ideal sliding particle.

First arrival is isochronous

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

Three distinct nonzero rest releases take the same quarter-period. The circular reference shares bottom curvature but has amplitude-dependent timing.

Common difficulties

Idealized apparatus

Typical misconceptionA rendered ball includes rolling, suspension and real hardware dynamics.

Better mental modelThis is a sliding point-particle model. Rail and marker thickness are schematic.

Run the experiment

01
Race the same endpoints
Change horizontal span, vertical drop and polynomial sag.
What to observe: The cycloid is compared under the same release and finish conditions.
02
Separate the two protocols
Select Three release heights and inspect the common bottom.
What to observe: The three cycloids share shape, gravity and zero initial speed; release heights differ.
03
Challenge equal arrival
Compare Near-bottom releases with the circular reference.
What to observe: The circular reference approaches the same small-amplitude clock but slows at finite height.
04
Acquire and refine
Acquire arrivals with zero noise, then change noise or resolution.
What to observe: Raw timing is separate from the analytic reference; changing the apparatus clears old readings.