You are welcomed to check out  [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=1116.0]Force analysis of a pendulum[/url]


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

  1. Adjust the length

  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:

Click show checkbox to show more information
    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.


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 \theta << 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).