Predict the resonant frequency

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.

1. Determine the spring constant k of your spring. Be sure to include an uncertainty and units.
2. 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 ω0
• the natural frequency f0
3. Attach a mass m = 45 g to your spring. Measure the actual frequency of oscillations, in cycles per second, with an uncertainty.
4. Does your theoretical value for the natural frequency agree with the actual value, within the uncertainties?
5. 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.

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

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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  /  e2π

of the original energy

```

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