Additional Info

• ID Code: G2-42
• Purpose: 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 (attached to the support shaft near the bottom of the picture) 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.

  The mass required to be connected to a spring to induce this behavior can be determined as follows by noting the resonant condition:

  v(mass on spring) = 2 v(pendulum)

  or

  sqrt[ k/m ] = 2 sqrt[ g/L ]

  so

  k = 4 mg/L

  This means that you must add a weight so that the increase in length of the original spring is 1/3 of the original spring, or 1/4 of the length of the final spring (spring constant = mg/[L/4]).

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