PURPOSE: To demonstrate a non-linear coupling resonance and stable fixed points.

DESCRIPTION: Start the spring oscillating vertically; the energy will then couple back and forth between pendular motion and vertical spring motion. Stationary combinations of these two oscillations (corresponding to normal modes in a linear resonance) can also be produced, by pulling the weight simultaneously down and to the side. Adding an additional weight to the spring destroys the resonance, resulting in less than total transfer of energy between the pendulum and spring motion.

This non-linear coupling resonance occurs when the spring (vertical) frequency is twice the frequency of a pendulum of the length at equilibrium. This is of interest because it is a very good mechanical analog to the v(r) = 2 v(z) resonance in the extraction region of a sector-focused cyclotron, where v(r) and v(z) are the radial and vertical betatron frequencies..


REFERENCES: (PIRA 3A70.15) See Demonstration Reference File for several good articles on the elastic pendulum.

EQUIPMENT: Elastic pendulum on mount.


Go back to Lecture-Demonstration Home Page.