### Author Topic: Pendulum  (Read 466347 times)

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### Pendulum
« on: January 29, 2004, 06:35:38 pm »
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You are welcomed to check out Force analysis of a pendulum

How to change parameters?
Set the initial position
Click and drag the left mouse button
The horizontal position of the pendulum will follow the mouse Animation starts when you release the mouse button

2.  dragging the pointer (while > holding down the left button)
from the support-point (red dot) to a position that sets the length you want.

Animation starts when you release the mouse button
3. Change gravity g

4.  Click near the tip of the red arrow,
and drag the mouse button to change it (up-down).

5. Change the mass of the bob

6.  Click near the buttom of the black stick,
and drag the mouse button to change it (up-down).

Information displayed:
1. red dots: kinetic energy K = m v*v /2 of the bob 2. blue dots: potential energy U = m g hof the bob
Try ro find out the relation between kinetic energy and pontential energy! 3.black dots (pair) represent the peroid T of the pendulum
move the mouse to the dot :
will display information for that dot in the textfield

blue arrow(1): gravity green arrows(2): components of gravity red arrow
(1): velocity of the bob
Try to compare velocity and the tangential component of the gravitional force!

The calculation is in real time (use Runge-Kutta 4th order method). The period(T) is calculated when the velocity change direction.
You can produce a period verses angle ( T - X ) curve on the screen,just started at different positions and wait for a few second.

Therotically, the period of a pendulum \$T=sqrt{g/L}\$.
Purpose for this applet:
1. The period of the pendulum mostly depends on the length of the pendulum and the gravity (which is normally a constant)
2. The period of the pendulum is independent of the mass.
3. The variation of the pendulum due to initial angle is very small.

The equation of motion for a pendulum is \$ frac{d^2   heta}{dt^2}=-frac{g}{L}, sin   heta\$
when the angle is small \$   heta << 1\$ ,\$sin   hetaapprox   heta\$
so the above equation become \$frac{d^2   heta}{dt^2}approx-frac{g}{L},   heta\$
which imply it is approximately a simple harmonic motion with period \$T=2pi sqrt{frac{L}{g}}\$

What is the error introduced in the above approximation?
From Tayler's expansion \$sin   heta=   heta-frac{   heta^3}{3!}+frac{   heta^5}{5!}-frac{   heta^7}{7!}+frac{   heta^9}{9!}-frac{   heta^11}{11!}+...\$
To get first order approximation, the error is \$frac{   heta^3}{3!}=frac{   heta^3}{6}\$
So the relative error (error in percentage)= \$frac{   heta^3/6}{   heta}=frac{   heta^2}{6}\$
If the angle is 5 degree, which mean \$   heta=5*pi/180approx=5/60=1/12\$
So the relative error is \$ frac{   heta^2}{6}=1/(12^2*6)=1/(144*6)=1/864approx 0.00116\$

For angle=5 degree , the relative error is less than \$0.116%\$
For angle=10 degree , the relative error is less than \$0.463%\$
For angle=20 degree , the relative error is less than \$1.85%\$

So the period of the pendulum is almost independent of the initial angle (the error is relatively small unless the angle is much larger than 20 degree- for more than 2% error).

Registed user can get files related to this applet for offline access.
Problem viewing java?Add http://www.phy.ntnu.edu.tw/ to exception site list

• Guest
##### topic11
« Reply #1 on: January 30, 2004, 12:24:21 pm »
Subject: Thanks
Date: Wed, 9 Dec 1998 16:07:30 -0500
From: louise heaven <gw_heaven@compuserve.com>
To: Fu-Kwan Hwang <hwang@phy03.phy.ntnu.edu.tw>
I was very pleasently surprized to find you had done so. Thankyou again.
I would also like to say that you have a very good web page and i shall
look there first when i am researching physics.

Joseph Heaven

• Guest
##### topic11
« Reply #2 on: January 30, 2004, 05:39:57 pm »
From: Bill Kinsella <wkinsella@csi.com>
To: "'hwang@phy03.phy.ntnu.edu.tw'" <hwang@phy03.phy.ntnu.edu.tw>
Subject: Java Applets
Date: Sat, 6 Nov 1999 21:04:16 -0000

Dear Sir,

I came across your site when I was searching for material for my son who is
studying science and in particular the pendulum. I was facinated by the
immediacy and effficacy of the applets. Surely this must represent a major
advancement in the teaching of physics as well as being great fun.

Unlike you I spent most of may life as a software developer although I know
nothing of Java type languages and now work as a power company nework
controller an can think of many interactive applications for our intranet.

I would like to see an applet developed illustrating the principles of
simple roof truss design.

Thanks for the enjoyment your work provided,

Bill Kinsella

• Guest
##### topic11
« Reply #3 on: March 22, 2004, 02:55:17 pm »
From what I learned in physics. The equation for the period of a simple pendulum is T=2(pi)(L/G)^1/2. Which means constant length should result in constant period. However I change the angle of release on the pendulum and the period changes!!??.

#### ratznium

• Newbie
• Posts: 1
##### topic11
« Reply #4 on: February 07, 2005, 11:49:45 pm »
There's must be an energy leak somewhere in the system. Instead of it's simulated perpetual motion, the bob eventually increases speed so that it ends up going right around a full circle, above the top of the java applet.

It's happened twice in a row now as I've left the applet running in the background while going through physics questions.

Try it out yourself if you're interested. Leave the applet running for at least an hour, and it ought to go wild.

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### topic11
« Reply #5 on: February 12, 2005, 02:57:17 pm »
For the computer simulation, there is always some error due to calculation.
Yes. it will happened when running the simulation for a long time.

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### topic11
« Reply #6 on: February 12, 2005, 03:07:43 pm »
[quote:0748e969ef="Anonymous"]From what I learned in physics. The equation for the period of a simple pendulum is T=2(pi)(L/G)^1/2. Which means constant length should result in constant period. However I change the angle of release on the pendulum and the period changes!!??.[/quote:0748e969ef]

The period of the pendulum is almost constant if the amplitude if small (small angle vibration)
However, the period will change very small amount when the angle increase.
It only increase less than 2% for 20 degree (relative to vertical line).

#### rhipple

• Jr. Member
• Posts: 22
• Relativity, Electromagnetism, Open Source Physics
##### topic11
« Reply #7 on: April 24, 2006, 08:18:06 am »
I would like to execute this applet offline. This feature appears to be disabled at the current time. This post will serve as my notification when to try again.

#### rhipple

• Jr. Member
• Posts: 22
• Relativity, Electromagnetism, Open Source Physics
##### topic11
« Reply #8 on: April 25, 2006, 10:22:51 am »
Great! I have a local version of the applet. Now may I see the source? I would like to tinker with it.

#### maryyoung

• Newbie
• Posts: 1
##### Re: Pendulum
« Reply #9 on: March 09, 2007, 08:28:55 pm »
Hi there, I was very excited to find the pendulum simulation, but I am trying to measure differences with different lengths, then with the same length and different masses at the end of the pendulum and I cant seem to change the mass without it disappearing off the end of my screen! I am obviously doing something wrong.  I would like to simulate a length of 30cm with masses of 100,200,300,400,500 grams, is this realistic?  I want the angle to be 45 degrees - would be grateful if you could help me to do this.  Thanks and regards Mary

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### Re: Pendulum
« Reply #10 on: March 10, 2007, 12:31:09 am »
If you want to set the length and angle of the pendulum, move your mouse to the red dot at the center on the top of the simulation, click down the mouse and drag the mouse away. The textfield on the top will display length and angle of the pendulum. When you are done just release the mouse.
If you want to change mass of the object, drag the vertical line (label with mass) up and down to change  mass.

#### DKMFan

• Newbie
• Posts: 1
##### Re: Pendulum
« Reply #11 on: June 03, 2007, 09:12:26 pm »
Wow. I like. A lot.

Do you mind if I use that for my coursework? It involves making a pendulum have the time period to be used for a Grandfather Clock. I'm asking in case something shows up in the mark scheme which means I'll have to eventually.

From what I learned in physics. The equation for the period of a simple pendulum is T=2(pi)(L/G)^1/2. Which means constant length should result in constant period. However I change the angle of release on the pendulum and the period changes!!??.

And thank you for making it a lot easier to find the equation. I think I needed that for my homework. Hmmm.

#### Fu-Kwun Hwang

• Hero Member
• Posts: 3062
##### Re: Pendulum
« Reply #12 on: June 04, 2007, 11:12:53 pm »
You are welcomed to use it for your coursework.

The equation T=2(pi)(L/G)^1/2 is good only for small angle.
(The sin? was replaced by ? when derive the equation)
It will be a little different when the angle is larger.
However, the difference is usually very small.
So it is still a very good approximation unless you need very high resolution results.

#### green

• Newbie
• Posts: 1
##### Re: Pendulum
« Reply #13 on: February 25, 2008, 11:52:20 am »
i have download it, but i still can not find the source code. can u help me??
how can i get all of your source code from this site??