Pendulum

Quote from: Fu-Kwun Hwang on January 29, 2004, 05:35:38 pm

<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%">
How to change parameters?
<ol>Set the initial position
Click and drag the left mouse button
<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>
dragging the pointer (while > holding down the left button)
<ol>from the support-point (red dot) to a position that sets the length you want.</ol>
Animation starts when you release the mouse button
<li>Change gravity g</li>
Click near the tip of the red arrow,
<ol>and drag the mouse button to change it (up-down).</ol>
<li>Change the mass of the bob</li>
Click near the buttom of the black stick,
<ol>and drag the mouse button to change it (up-down).</ol>
</ol>
Information displayed:
<ul>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
<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
Try to compare velocity and the tangential component of the gravitional force!</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>You can produce a period verses angle ( T - X ) curve on the screen,just started at different positions and wait for a few second.</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 << 1 , $\sin\theta\approx \theta$
so the above equation become math_failure (math_image_error): \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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