Cycloidal Pendulum

This following applet is Cycloidal Pendulum
Created by prof Hwang Modified by Ahmed
Original project 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 $T_{1/4}=\sqrt\frac{r}{g}\pi$

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

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
	Double pendulum Molecular Workbench	concord	0	11221	August 21, 2005, 09:22:30 am by concord
	compound pendulum Dynamics	qianger21	2	21827	December 20, 2018, 10:19:36 am by staytokei
	help: simple pendulum Dynamics	foofz	3	23308	November 11, 2009, 09:59:13 pm by Fu-Kwun Hwang
	Cycloidal Pendulum Kinematics	Fu-Kwun Hwang	0	18879	September 06, 2009, 05:54:01 pm by Fu-Kwun Hwang
	Motion along cycloidal curves Kinematics	Fu-Kwun Hwang	0	9285	September 09, 2009, 12:02:26 pm by Fu-Kwun Hwang

	Author	Topic: Cycloidal Pendulum (Read 9040 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).