In the two previous labs you gained experience with free and damped mass-spring oscillators. In this third and final adventure with the mass-spring system, you will apply a small, periodic force to the system, very much like the periodic nudge you would give to a child on a swing. If the forcing frequency matches the undamped frequency, the system will resonate. Its amplitude will grow with each nudge until limited by inherent damping. If a system such as our mass-spring system has very little damping, the magic of resonance can turn a mouse into a moose. In our setup, the amplitude of the vibrator motion is quite small, about a millimeter, yet it is able to stimulate amplitudes in the mass-spring system over 50 times greater!
Mount the vibrator to the bench post upside down as shown in the figure (if you mount it upside up, the tongue will not move), and connect the sine wave generator to the vibrator. Place 15g on the hanger, attach the hanger to the spring and hang the assembly from the spring system. Center the position sensor directly beneath the hanger. Adjust the height of the hanger so that its bottom is more than 20 cm above the position sensor. Open the Logger Pro File, stabilize the hanger and zero the position sensor. Please also take note of Walter Fendt's Applet.
Turn off the sine wave generator, induce small vertical oscillations in the mass-spring system (small enough to avoid "clicking" of the vibrator tongue). Collect data and fit a sine curve to the position plot. From the fit parameters, read the frequency ω0. You should get a frequency of f = ω0 / 2π ~ 1.90-2.00 ± 0.04 Hz. If necessary, adjust the mass to achieve this frequency.
Set the generator amplitude to minimum, turn on the generator, set the generator frequency to 1.90-2.00 Hz, start collection, then set the generator amplitude to maximum. The mass-spring system will begin to oscillate, and the oscillations will quickly grow to a maximum.
Vary the generator frequency, selecting two values above and two values below the resonance frequency which are close to the "free oscillation frequency". Force the system as before and collect data. Allow the system to oscillate for a while and observe its behavior carefully.
Even though our intent is to employ the hanging mass-spring system as a one dimensional vertical oscillator, the system is capable of moving in three dimensions and sustaining three independent oscillation modes simultaneously.
Place 15g mass on the hanger, induce oscillations and obtain the free frequency ω0. The frequency should be f = ω0 / 2π ~ 1.90-2.00 ± 0.04 Hz. Force the system and obtain the forced frequency ωf and maximum amplitude Af. Remove 15g and place the 20cm diameter, 15g damping disk on the hanger, induce oscillations and obtain the damped frequency ωd and damping factor (the hanger and damping disk plus any additional mass should total 65gm). Force the system to obtain the forced/damped frequency and maximum amplitude.