Motion along cycloidal curves

The following simulation try to illustrate the properties of cyclodial curves.

$\theta=\omega t$ , $x=r(\theta-\sin\theta), y=r(1+\cos\theta)$
so $x'=r(1-\cos\theta), y'=r\sin\theta$
$x'^2+y'^2=2r^2(1-\cos\theta)$
From conservation of energy: $\tfrac{1}{2}mv^2=mgy$ , $v=\tfrac{ds}{dt}=\sqrt{2gy}$
$dt=\frac{ds}{v}=\frac{\sqrt{dx^2+dy^2}d\theta}{\sqrt{2gy}}=\frac{\sqrt{2r^2(1-\cos\theta)}}{\sqrt{2gr(1-\cos\theta)}}=\sqrt{\frac{r}{g}}d\theta$ , so $T_{1/4}=\sqrt{\frac{r}{g}}\pi$

However, from an intermediate point $\theta_0$ , $v=\frac{ds}{dt}=\sqrt{2g(y-y_0)}$
So $T=\int_{\theta_0}^{\pi}\sqrt{\frac{2r^2(1-\cos\theta)}{2gr(\cos\theta_0-\cos\theta)}}d\theta$

with $\sin\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}}, \cos\frac{x}{2}=\sqrt{\frac{1-\cos x}{2}}$

$T=\sqrt{r}{g}\int {\theta_0}^{\pi} \sqrt{\frac{\sin\tfrac{\theta}{2}d\theta}{\cos^2\tfrac{\theta_0}{2}-\cos^2\tfrac{\theta}{2}}}$
define $u=\frac{\cos\tfrac{\theta}{2}}{\cos\tfrac{\theta_0}{2}}, du=\frac{\sin\frac{\theta}{2}d\theta}{2\cos\tfrac{\theta_0}{2}}$

$T=\sqrt{r}{g}\int_1^0 \frac{du}{\sqrt{1-u^2}}=2\sqrt{r}{g}\sin^{-1}u|_0^1=\sqrt{r}{g} \pi$
So the amount of time is the same from any point.

1. 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.
2. Cycloidal curves is the curve of fastest descent under gravity
3. the period of a ball rolling back and forth inside this curve does not depend on the ball’s starting position.

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
	simple harmonic motion with angular motion Wave	inayet ali	1	23654	May 15, 2009, 08:39:26 pm by daryy
	relations between simple harmonic motion and circular motion Kinematics	Fu-Kwun Hwang	2	39700	April 15, 2016, 11:21:50 pm by lookang
	Blackbody radiation curves for different temperatures Modern Physics	Fu-Kwun Hwang	23	70498	April 14, 2009, 10:08:50 pm by lookang
	Cycloidal Pendulum Kinematics	Fu-Kwun Hwang	0	19494	September 06, 2009, 05:54:01 pm by Fu-Kwun Hwang
	Cycloidal Pendulum kinematics	ahmedelshfie	2	9482	June 25, 2010, 08:21:50 pm by ahmedelshfie

	Author	Topic: Motion along cycloidal curves (Read 9470 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).