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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

- Adjust the length

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

- Change gravity g

Click near the tip of the red arrow,

and drag the mouse button to change it (up-down).

- Change the mass of the bob

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

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.

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).