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Author Topic: Pendulum  (Read 373335 times)
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Fu-Kwun Hwang
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Reply #30 on: June 05, 2010, 11:33:39 pm » posted from:Taipei,T\'ai-pei,Taiwan

The source code is available as attached file under the first message (Please read the topic message carefully and you should have found it).
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TaraLaster
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Reply #31 on: December 18, 2013, 10:50:39 am » posted from:Makati,Manila,Philippines

<center><applet code="pendulumSystem.class" width=500 height=350 codebase="/java/Pendulum/"><param name="Reset" value="Reset"><param name="Pause" value="Pause"><param name="Show" value="Show"><param name="Resume" value="Resume"><param name="" value=""></applet></center>

You are welcomed to check out Force analysis of a pendulum
<hr ALIGN=LEFT WIDTH="100%">
<font size=+1>How to change parameters?</font>
<ol>Set the initial position
<font color="#0000FF">Click and drag the left mouse button</font>
<ol>The horizontal position of the pendulum will follow the mouse Animation starts when you release the mouse button</ol>
<li>Adjust the length</li>
 <font color="#0000FF">dragging the pointer (while > holding down the left button)</font>
<ol><font color="#0000FF">from the support-point </font>(red dot) to a position that sets the length you want.</ol>
Animation starts when you release the mouse button
<li>Change gravity <font color="#FF0000">g</font></li>
 <font color="#0000FF">Click near the tip of the red arrow</font>,
<ol>and drag the mouse button to change it (up-down).</ol>
<li>Change the mass of the bob</li>
 <font color="#0000FF">Click near the buttom of the black stick,</font>
<ol>and drag the mouse button to change it (up-down).</ol>
</ol>
Information displayed:
<ul>1. red dots: kinetic energy <font color="#0000FF">K = m v*v /2 </font>of the bob 2. blue dots: potential energy <font color="#0000FF">U = m g h</font>of the bob
<font color="#0000FF">Try ro find out the relation between kinetic energy and pontential energy!</font> 3.black dots (pair) represent the peroid T of the pendulum
<ul>move the mouse to the dot :
<ul>will display information for that dot in the textfield</ul>
</ul></ul>
Click show checkbox to show more information
<ol>blue arrow(1): gravity green arrows(2): components of gravity red arrow
(1): velocity of the bob
<font color="#0000FF">Try to compare velocity and the tangential component of the gravitional force!</font></ol>
<hr WIDTH="100%">The calculation is in real time (use Runge-Kutta 4th order method). The period(T) is calculated when the velocity change direction.
<ul><font color="#0000FF">You can produce a period verses angle ( T - X ) curve on the screen,just started at different positions and wait for a few second.</font></ul>

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\theta}{dt^2}=-\frac{g}{L}\, \sin\theta
when the angle is small math_failure (math_unknown_error): \theta &lt;&lt; 1 ,\sin\theta\approx \theta
so the above equation become \frac{d^2\theta}{dt^2}\approx-\frac{g}{L}\, \theta
which imply it is approximately a simple harmonic motion with period T=2\pi \sqrt{\frac{L}{g}}

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


So sad.. still can't find resource code. Sad Do you have any other way?
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"Progress, therfore, is not and accident,¡K" ..."Herbert Spencer(1820-1903, British philosopher)"
 
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