Enough theory -- it's time to put physics to the test. Your job today is to set up a system which oscillates, predict its resonant frequency, and then check to see if you were correct.

**Calculations of uncertainty will be given a high weight in
today's experiment.**

You will have to figure out what to do for most of today's activity. Feel free to ask for help.

- Determine the spring constant
**k**of your spring. Be sure to include an uncertainty and units. - Predict the natural frequency of oscillations
for your spring. Express in two ways,
with the appropriate uncertainty and units for each.
- the natural angular frequency
**ω**_{0} - the natural frequency
**f**_{0}

- the natural angular frequency
- Attach a mass
**m = 45 g**to your spring. Measure the actual frequency of oscillations, in cycles per second, with an uncertainty. - Does your theoretical value for the natural frequency agree with the actual value, within the uncertainties?
- Set up the equipment needed to drive your spring-plus-mass
system with an oscillator -- see the figure below for tips.
Set the amplitude of the driver to the 9 o'clock position. Starting at about 10 Hertz, drive your system and measure its amplitude of oscillation. Watch for at least 5 or 6 cycles at each setting, and choose the largest amplitude you see.

Make a table of measurements from 10 Hertz down to 1 Hz. Adjust the steps in frequency to "zoom in" on interesting behavior.

- Make a graph showing amplitude as a function of frequency.
- What is the actual frequency of resonance? Provide a value, units, and an uncertainty. Does it agree with your prediction, within the uncertainties?

*
If you have time, for extra credit .....
*

A resonant system will often show behavior like that in the graph
below: the amplitude of oscillations reaches a peak at
some central frequency **ω**,
with some width **Δω** around that peak.

The "quality factor", **Q**,
of a system, is a measure of how narrow the peak
is. The narrower the peak, the higher the **Q** factor.

ω Q = ----- Δω

Systems with high **Q** values will tend to oscillate
for longer duration before the damping causes them
to decrease and stop moving.
In fact, to a rough approximation,
it is said that

Q is approx number of cycles required for the system's energy to fall to 1 / e^{2π}of the original energy

- Use your measurements and graph to determine the central frequency, width, and Q factor for your system.
- Test the relationship between Q and the time it takes for oscillations in your system to decay.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.