Pendulum in 3D

This following applet is Pendulum in 3D
Created by prof Hwang Modified by Ahmed
Original project Pendulum in 3D

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!

Related Topics
	Subject	Started by	Replies	Views	Last post
	Pendulum « 1 2 » Dynamics	Fu-Kwun Hwang	33	455238	December 14, 2014, 05:51:29 pm by diinxcom
	Pendulum wave Dynamics	Fu-Kwun Hwang	6	25712	December 15, 2015, 06:29:30 pm by Normaz
	Pendulum in 3D Dynamics	Fu-Kwun Hwang	6	26837	November 19, 2015, 02:19:33 pm by lookang
	3D rotational pendulum also known as Furuta's pendulum by Francisco Esquembre Dynamics	lookang	7	17047	May 17, 2019, 11:41:14 am by angelinajolie
	Coffee cup pendulum (demonstration and simulation) Dynamics	Fu-Kwun Hwang	0	8042	June 11, 2013, 10:28:57 am by Fu-Kwun Hwang

	Author	Topic: Pendulum in 3D (Read 4568 times)
0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).