Tuesday, February 2, 2010

Pendulum

Nonlinear Pendulum:
A simple pendulum consists of a point mass m attached to a rod of negligible weight. The torque [Graphics:Images/PendulumMod_gr_1.gif] is

(1) [Graphics:Images/PendulumMod_gr_2.gif],

where [Graphics:Images/PendulumMod_gr_3.gif] denotes the the angle of the rod measured downward from a vertical axis. The moment of inertia for the point mass is [Graphics:Images/PendulumMod_gr_4.gif] where l is the length of the rod. The torque can also be expressed as [Graphics:Images/PendulumMod_gr_5.gif], where [Graphics:Images/PendulumMod_gr_6.gif] is the angular acceleration, using Newton's second law, and the second derivative, this can be written as

(2) [Graphics:Images/PendulumMod_gr_7.gif].

Equating (1) and (2) results in the nonlinear D. E.

(3) [Graphics:Images/PendulumMod_gr_8.gif].

Linear Pendulum:
Introductory courses discuss the pendulum with small oscillations as an example of a simple harmonic oscillator. If the angle of oscillation [Graphics:Images/PendulumMod_gr_9.gif] is small, use the approximation [Graphics:Images/PendulumMod_gr_10.gif] in equation (3) and obtain the familiar linear D. E. for simple harmonic motion:

(4) [Graphics:Images/PendulumMod_gr_11.gif],

Using the substitution [Graphics:Images/PendulumMod_gr_12.gif], the solution to (4) is known to be

(5) [Graphics:Images/PendulumMod_gr_13.gif],

which has period [Graphics:Images/PendulumMod_gr_14.gif]. When the solution (5) is written with a phase shift, it becomes

(6) [Graphics:Images/PendulumMod_gr_15.gif].

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Introductory Methods of Numerical Analysis