Pendulum in 3D

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3085

1 Pendulum in 3D

Embed this message

on: September 05, 2009, 11:25:17 pm » posted from:Taipei,T\'ai-pei,Taiwan

This is a pendulum in 3D.
Let the angle between the pendulum and the vertical line is $\theta$ and the angular velocity $\omega=\frac{d\theta}{dt}$
And the angle of the pendulum (projected to x-y plane) with the x-axis is $\phi$ , and it's angular velocity $\dot\phi=\frac{d\phi}{dt}$

The lagrange equation for the system is $L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta)$

The equation of the motion is

$\ddot\theta=\sin\theta\cos\theta\dot{\phi}^2-\frac{g}{L}\sin\theta$ ...... from $\frac{d}{dt}(\frac{\partial L}{\partial \dot{\theta}})-\frac{\partial L}{\partial \theta}=0$
and
$m L^2 \sin\theta^2 \dot{\phi}=const$ Angular momentum is conserved. ...... from $\frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})-\frac{\partial L}{\partial \phi}=0$

And the following is the simulation of such a system:
When the checkbox (circular loop) is checked, $\omega=0$ . and $\dot{\phi}= \sqrt{\frac{g}{L\cos\theta}}$ It is a circular motion.
The vertical component tangential of the string balanced with the mass m, and the horizontal component tangential provide the centripetal force for circular motion.

You can uncheck it and change the period $T=\frac{2\pi}{\dot{\phi}}$ ,
and you will find out the z-coordinate of the pendulum will change with time when
$\omega\neq 0$ or $\dot{\phi}\neq \sqrt{\frac{g}{L\cos\theta}}$
You can also drag the blue dot to change the length of the pendulum.

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!

pendulum3d.jpg (21.07 KB, 580x499 - viewed 997 times.)
	Logged

msntito

Newbie

Offline

Posts: 2

2 Re: Pendulum in 3D

Embed this message

Reply #1 on: September 28, 2009, 05:50:10 pm » posted from:Kanpur,Uttar Pradesh,India

Hi,
I was trying to derive the same equations of motion,
Why have you considered the RHS of second equation = constant ?

I got:- m(L^2) * [ (sin theta)^2 * phi_dot_dot + sin(2*theta) * theta_dot * phi_dot ] = 0
as the second eq of Motion.

After this, how to solve?

regards,
tito


	Logged

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3085

3 Re: Pendulum in 3D

Embed this message

Reply #2 on: September 28, 2009, 09:21:41 pm » posted from:Taipei,T\'ai-pei,Taiwan

The lagrange equation for the system is $L=T-V = \tfrac{1}{2}m (L\dot\theta)^2+\tfrac{1}{2}m (L\sin\theta \dot{\phi})^2- (-mgL\cos\theta)$

from $\frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})-\frac{\partial L}{\partial \phi}=0$
you will find $m L^2 \sin\theta^2 \dot{\phi}=const$

The reason is:
since there is no $\phi$ in lagrange equation so $\frac{\partial L}{\partial \phi}=0$
which mean $\frac{d}{dt}(\frac{\partial L}{\partial \dot{\phi}})=0$ ,
so $(\frac{\partial L}{\partial \dot{\phi}})$ must be a constant. i.e. $m L^2 \sin\theta^2 \dot{\phi}=const$


	Logged

msntito

Newbie

Offline

Posts: 2

4 Re: Pendulum in 3D

Embed this message

Reply #3 on: September 29, 2009, 11:16:37 am » posted from:Kanpur,Uttar Pradesh,India

`thanks a lot.
I was wondering if this problem can be solved using Euler angles instead of phi and theta (like in rigid body dynamics).How should I begin, Do you have any idea?


	Logged

Fu-Kwun Hwang

Administrator
Hero Member

Offline

Posts: 3085

5 Re: Pendulum in 3D

Embed this message

Reply #4 on: September 29, 2009, 01:34:11 pm » posted from:Taipei,T'ai-pei,Taiwan

You will need to use Euler's angle if the pendulum is not a sphere,
for example: when it is a cylinder, then, you can use the following rules to draw it:
1. rotate around z axis by $\phi$
2. rotate around y axis by $-\theta$
3. rotate around z axis by $\phi$

You can check out the following applet.

Embed a running copy of this simulation

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

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!


	Logged

diinxcom

watchlist
Jr. Member

Offline

Posts: 25

Designer

6 Re: Pendulum in 3D

Embed this message

Reply #5 on: December 14, 2014, 05:54:03 pm » posted from:,,Satellite Provider

-*-
Thanks alot!

I enjoy this content


	Logged

lookang

Hero Member

Offline

Posts: 1794

http://weelookang.blogspot.com

7 Re: Pendulum in 3D

Embed this message

Reply #6 on: November 19, 2015, 02:19:33 pm » posted from:SINGAPORE,SINGAPORE,SINGAPORE

made this JavaScript version!
enjoy
http://iwant2study.org/ospsg/index.php/interactive-resources/physics/02-newtonian-mechanics/09-oscillations/274-pendulum3d


	Logged

Pages: [1] Go Up

An interdisciplinary approach. ...Wisdom

« previous next »

Related Topics
	Subject	Started by	Replies	Views	Last post
	Pendulum « 1 2 » Dynamics	Fu-Kwun Hwang	33	456702	December 14, 2014, 05:51:29 pm by diinxcom
	Pendulum in 3D dynamics	ahmedelshfie	1	4679	June 08, 2010, 07:25:03 pm by ahmedelshfie
	Pendulum Examples for beginner who want to learn EJS	Fu-Kwun Hwang	1	5702	January 05, 2019, 02:23:23 pm by vivozi
	Pendulum dynamics	ahmedelshfie	1	4026	September 24, 2010, 06:58:52 pm by ahmedelshfie
	3D rotational pendulum also known as Furuta's pendulum by Francisco Esquembre Dynamics	lookang	9	17393	February 04, 2020, 01:27:35 pm by Fannieiley

	Author	Topic: Pendulum in 3D (Read 27240 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).