nonlin_oscillator

Experiment 7: Nonlinear Oscillator.

The Nonlinear Oscillator (Pendulum)

Introduction:

In expt.1 we have demonstrated simple harmonic motion (SHM) using the case of a simple pendulum. Now, when the angular displacement amplitude of the pendulum is large enough that the small angle approximation, as in case of SHM, no longer holds, and then the equation of motion must remain in its nonlinear form,

This differential equation does not have a closed form solution, and must be solved numerically using a computer. The small angle approximation is valid for initial angular displacements of about 20° or less. When the initial angular displacement is significantly large that the small angle approximation is no longer valid, the error between the simple harmonic solution and the actual solution becomes apparent almost immediately, and grows as time progresses.

A more detailed discussion of the above principles of the nonlinear oscillator can be found at:

http://www.kettering.edu/physics/drussell/Demos/Pendulum/Pendula.html

Further resources on the nonlinear pendulum:

http://www.pgccphy.net/ref/nonlin-pendulum.pdf

http://mathematicalgarden.wordpress.com/2009/03/29/nonlinear-pendulum/

Suggested Activities:

· For a small value of the initial angle θ, say 10⁰, find out the time period T of oscillation of the pendulum from the Amplitude-Time graph. For the particular values of g and L, calculate the T for the SHM case. Compare the two values and find the difference ΔT.

· Find T for a larger initial angle, say 80⁰. Calculate ΔT for this case. What change do you notice?

· Repeat the above for a few different θ values and make a plot of ΔT vs. θ. From this graph, what can you say about the limit of validity of the linear approximation?

· Select a small angle θ, within the limit of linear approximation, as determined above. Keeping all other parameters fixed, vary the length L of the pendulum and observe the motion of the pendulum. Make a plot of T vs. L. Does this follow the linear behavior T ~ √L? Repeat the same for a larger θ.

· Select a small angle θ, within the limit of linear approximation, as determined above. Vary the mass m of the bob and find out how T changes with m. Make plot of T vs. m.