PHYS-302, Physics of Waves (Summer 2007)

Examples of Oscillating Systems

• Phase diagrams (momentum versus position) for oscillatory motion (with JAVA applet)
• Oscillation in an LC circuit: derived the equation of motion, found the natural frequency, and looked at a JAVA applet: www.falstad.com/circuit/)
• Simple Pendulum: hands-on activity (see below) to explore how period depends on length, and how period depends on maximum angle. Here's a spreadsheet with the student results and with Dr. Russell's summary.
• Simple Pendulum: derived equation of motion, discussed small angle solution, discussed nonlinear solution and the fact that for a nonlinear restoring force the period now depends on the maximum amplitude of the motion. Looked at a JAVA physlet comparing the linear and nonlinear solutions.
• Acoustic oscillator (Helmholtz resonator). Derived the restoring force and equation of motion for an acoustic oscillator, found how the frequency depends on neck length, neck area, and cavity volume. Dr. Russell demonstrated a collection of bottles: how changing the volume (by filling with fluid) alters the frequency, and how two bottles with very different shapes can have the same frequency.
• Physical demonstration and interaction with a JAVA applet of an interesting mechanical oscillator for which the restoring force is due to friction.

Two groups of three students each investigated how the period of a pendulum depends on the length. Using a mass hanging from a string, they cut the string shorter and shorter, measuring the period and plotting the data. Both groups found that the period increased as a function of string length, but the trend was not linear. After plotting on a log-log plot and extracting the power fit, one group found T = B L0.48 and the other group found T = B L0.501. After working through the theory in class we learned that the correct result is T = B L0.5.

The other two groups (one with three and one with four students) investigated how the period of a pendulum depends on the maximum amplitude. They measured the period for initial angles over the range of 10o to 80o. Both groups found that the period was fairly constant (and close to the simple theoretical prediction) as long as the angle was below 30o. But for larger angles the period increased noticeably as the initial angle increased. When we looked at the theory for the "realistic pendulum" we found out why the period increased with increasing amplitude.