When a tuning fork vibrates, it disturbs nearby air molecules and creates alternating regions of high and low pressure. Our ear picks up these rapid pressure changes and converts them into electrical nerve impulses. Our brain then interprets these nerve impulses as sound. In a similar fashion, microphones convert these rapid pressure changes into electrical signals. The CBL-2 can transfer these signals to your calculator to be displayed on screen. When the tone is "pure," the resulting signal appears as a sine wave.

In this activity, you will explore the characteristics of a pure sound wave produced by a tuning fork. You will then use your knowledge of sine waves to find a curve to fit the sound wave data produced by your tuning fork.

## Equipment Required

The class as a whole will need:
• 1 CBL-2 unit
• 1 Vernier Microphone/Amplifier

•
 Each group will need: 1 TI-83 Graphing Calculator Program: TUNED 1 Tuning Fork

## Setup and Data Collection

Since all groups will share the CBL-2 and Microphone, this equipment should be placed at a central location.
In turn, each group should bring its tuning fork and calculator to collect data. After connecting their calculator to the CBL-2, they should run the TUNED program and follow the directions on screen.
The data should appear to be a smooth sinusoidal curve centered about the x-axis. If it does not, simply press [CLEAR] and [ENTER] to begin another trial.
Notes:  For best results, you should strike the tuning fork sharply on the sole of your shoe. Immediately place the tuning fork near the microphone and begin sampling. To prevent damage, never strike a tuning fork against a hard surface (like wood or metal).

1.  The frequency of your tuning fork should be engraved near the handle. For example, a tuning fork that creates a 'C' note has a frequency of 261.6 cycles per second.
Frequency of your tuning fork: __________________
2.  Sketch a graph of the sine wave produced by your tuning fork in the space below.

## Analysis

3.  We will use the following equation to model the sound wave produced by the tuning fork: y = A sin(Bx+C).

The amplitude of our model is determined by the value of A. Use your calculator to find the amplitude of the data you collected. There are at least two ways to do this.

• One way is to use the [TRACE] command to find the y-coordinate of one of the high peaks of the graph.
• Another way is to use the max(L2) command to find the maximum value of the list L2, where the y-coordinate data is stored. The max( command is found on the [LIST] menu, under MATH.
•
A = __________________

4.  Recall that the period is the time between any two consecutive peaks on the graph. Use the [TRACE] command to find the period of your tuning fork, based on your data.
Period = _____________

5.  Recall that

Use the period you found above to calculate the frequency.

Frequency = _______________

6.  Compare the frequency you calculated in question 5 with the frequency engraved on the tuning fork.  Compute the relative error and record it below. (Use the value engraved on the tuning fork as the correct answer.)

Relative Error = __________________

7.  We know that.  Use this formula to solve for B.

B = ___________________

8.  To see how well your model fits the data so far:

Use your computed values for A and B to write an equation of the form y = A sin(Bx).

y = _______________________________

Then, press [Y=] on your calculator and move to an empty function register. Enter your equation and press [GRAPH] to see the results.

Does your model provide a good fit for the data? If not, describe how they differ.

9.  It is likely that your model will be shifted horizontally from the data. In that case, we say that your model is out of phase with the data. To correct for this, we compute the value of the phase shift C.

Use the [TRACE] command to find horizontal translation, or phase shift, of your data from a sine wave. (You may wish to clear the equation you entered in [Y=] to make the graph less cluttered.)

Phase Shift = ____________________

10.  Recall that

You know the Phase Shift and you know B. Use this information to solve for C.

C = ________________

11.  You now have enough information to write an equation of the form y = A sin(Bx+C) that can be used as a model for the tuning fork data.  Using your computed values for A, B, and C, write your equation below.

y = _______________________________

Check to see that your model now matches your data by graphing this equation on your calculator.

## Using What You’ve Learned

12.  We know that the amplitude of the sound wave represents the loudness of the sound: louder sounds create sound waves with greater amplitudes. Explain how you could change the amplitude, A, if you repeated this investigation.

13.  The pitch of a sound is associated with the frequency of the sound wave. Higher pitched tones produce sound waves with greater frequencies. How would your graph change if you used a higher frequency tuning fork?

14.  Use what you know about the relationship between period and frequency (see question 5) to explain how the period would change if the frequency were to increase. That is, if the frequency gets bigger, what happens to the period?

Jon Hasenbank, 2000.

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