<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

.

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

when the angle is small

**math_failure (math_unknown_error): \theta << 1**
,

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

**math_failure (math_image_error): T=2\pi \sqrt{\frac{L}{g}}**
What is the error introduced in the above approximation?

From Tayler's expansion

**math_failure (math_image_error): \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

**math_failure (math_image_error): \frac{\theta^3}{3!}=\frac{\theta^3}{6}**
So the relative error (error in percentage)=

If the angle is 5 degree, which mean

So the relative error is

For angle=5 degree , the relative error is less than

For angle=10 degree , the relative error is less than

**math_failure (math_image_error): 0.463%**
For angle=20 degree , the relative error is less than

**math_failure (math_image_error): 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.