Cycloidal Pendulum

A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line. It is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid is the solution to the brachistochrone problem (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e. the period of a ball rolling back and forth inside this curve does not depend on the ball’s starting position).

The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y), with

$x = r(\omega t - \sin \omega t)$
$y = r(1 - \cos \omega t)$

If its length is equal to that of half the cycloid, the bob of a pendulum suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, also traces a cycloid path. Such a cycloidal pendulum is isochronous, regardless of amplitude. This is because the path of the pendulum bob traces out a cycloidal path (presuming the bob is suspended from a supple rope or chain); a cycloid is its own involute curve, and the cusp of an inverted cycloid forces the pendulum bob to move in a cycloidal path.

$v_x=\frac{dx}{dt}=r\omega*(1+\cos\omega t)$
$v_y=\frac{dy}{dt}=r\omega \sin \omega t$
Combined the above two equations: $(v_x-r\omega)^2+v_y^2)=(r\omega)^2$
So $v_x=\frac{v_x^2+v_y^2}{2r\omega}=\frac{2g\Delta y}{2r\omega}=\frac{g\Delta y}{r\omega}$
and $\frac{dy}{dx}=\frac{v_y}{v_x}=\frac{\sin\omega t}{1+\cos\omega t}$

The time required to travel from the top of the cycloid to the bottom is $T_{1/4}=\sqrt\frac{r}{g}\pi$

$a_x=\frac{d^2x}{dt^2}=-r\omega^2\sin\omega t$
$a_y=\frac{d^2y}{dt^2}= r\omega^2\cos\omega t$

Please check out http://mathworld.wolfram.com/TautochroneProblem.html or http://en.wikipedia.org/wiki/Cycloid for more information.

Embed a running copy of this simulation

Embed a running copy link(show simulation in a popuped window)

Full screen applet or Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list
Press the Alt key and the left mouse button to drag the applet off the browser and onto the desktop. This work is licensed under a Creative Commons Attribution 2.5 Taiwan License

Please feel free to post your ideas about how to use the simulation for better teaching and learning.
Post questions to be asked to help students to think, to explore.
Upload worksheets as attached files to share with more users.

Let's work together. We can help more users understand physics conceptually and enjoy the fun of learning physics!

Related Topics
	Subject	Started by	Replies	Views	Last post
	compound pendulum Dynamics	qianger21	2	22332	December 20, 2018, 10:19:36 am by staytokei
	help: simple pendulum Dynamics	foofz	11	25331	April 04, 2021, 12:16:22 am by vinyas
	Pendulum in an accelerated car Kinematics	Fu-Kwun Hwang	2	29932	June 09, 2010, 07:28:55 pm by ahmedelshfie
	Motion along cycloidal curves Kinematics	Fu-Kwun Hwang	0	9528	September 09, 2009, 12:02:26 pm by Fu-Kwun Hwang
	Cycloidal Pendulum kinematics	ahmedelshfie	2	9734	June 25, 2010, 08:21:50 pm by ahmedelshfie

	Author	Topic: Cycloidal Pendulum (Read 19777 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).