A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).

Background:
The standard pendulum period of $2\pi \sqrt{L/g}$ or frequency $\sqrt{g/L}$ holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.

In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by $(x,y)=R(\theta -\sin \theta ,-1+\cos \theta )$, where $\theta =0$ corresponds to the cusp. Consider a pendulum of length $L=4R$ hanging from the cusp, and let $\alpha$ be the angle the string makes with the vertical, as shown (in the proof).

Do you know how to calculate the area under a cycloid curve?

[ Beau Janzen/reason4math]
#Cycloid #CycloidCurve #CycloidArea #AreaCycloid #Geometry #Area

Forgive the recent apparent obsession (I’d call it a fascination) with the #cycloid but I’ve just discovered something I’d not heard of before. It is also called a #TautochroneCurve or #Isochrone curve, which means that a particle starting from any location on the curve will get to the #MinimumPoint at precisely the same time as a particle starting at any other point.

A couple of weeks ago, I posted an #animation of a point on a circle generating a #cycloid.

If you turn the curve "upside down", you get the #BrachistochroneCurve. This curve provides the shortest travel time starting from one cusp to any other point on the curve for a ball rolling under uniform #gravity. It is always faster than the straight-line travel time.

Imagine a circular wheel rolling, without skidding, on a flat, horizontal surface. The #locus of any given point on its #circumference is called a #cycloid. It is a #periodic #curve with #period over the #circle's circumference and has #cusps whenever the point is in contact with the surface (the two sides of the curve are tangentially vertical at that point).

Cycloid (2013) [3 min] by Tomoki Kurogi | #Japan

https://www.youtube.com/watch?v=udxCW25nEfg

I had been wondering about a problem of "kissing" circles bounded by a #cycloid. The central circle would be the circle that generates the cycloid, and the circles on either side would fill the remaining space.

I'd only got part way with the problem, but once the #maths were done I'm sure it will look cool...

The hotel I'm staying at: you mean this?

Cycloid Drawing Machine Uses Sneaky Stepper Hack - Stepper motors are great for projects that require accurate control of motion. 3D printers, CNC mach... more: https://hackaday.com/2019/06/15/cycloid-drawing-machine-uses-sneaky-stepper-hack/ #classichacks #steppermotor #pantograph #spirograph #cycloidal #cycloid