J.D. Flood and D.W. Follet
June 7, 2006
Physical Description of Vibrations
A musical sound is produced by vibration. We are going to use simple pendulums to learn some of the physical terms for describing vibrations.
The period is the time to complete a single full cycle of the vibration. For the simple pendulum this corresponds to the time for the pendulum bob to return to the point of release. (The pendulum bob is the object attached to the end of the string.)
The frequency is more commonly used to describe musical vibrations. Frequency is the number of cycles in one second. It is measured in hertz (Hz).
We will also look at how the frequency and the period depend on the length of the pendulum.
Goals
1. To get a feeling for the variation of period and frequency with pendulum length by measuring these quantities for different lengths.
2. To develop a mathematical relationship between frequency and period by considering the period and frequency data.
3. To develop a mathematical relationship between frequency and pendulum length.
4. To explore a possible origin of musical consonance by considering the simultaneous vibrations of two pendulums.
Exercise 1: Length and Frequency
Look for the stand with the two pendulums. For now work with the one that has the longer string. As you work through the exercise, fill in the table below.
Length(cm) / Time for 10 cycles
(s) / Period = time for 10 cycles/10 (s) / Number of cycles in 20 seconds / Frequency = number of cycles in one second
100
50
25
1. Adjust the length of the pendulum to be 100 cm long. This length is the distance from the point where the string is attached to the support and the middle of the pendulum bob.
2. Get the pendulum started by pulling the pendulum bob a centimeter or two to one side. Use the stopwatch to measure the time it takes to complete 10 full cycles. (We measure for 10 cycles instead of one to get a more accurate measurement. The longer the time, the less important your reaction time is). Record this data above.
3. Repeat step 2 for a 50 cm long pendulum and a 25 cm long pendulum. Record the data in the second column.
4. For each length calculate the period and the frequency using the formulas given in the table.
5. Now concentrate on the length and the frequency columns. Think about the following questions in order and write your observations in the space provided.
a. When you halve the length of the pendulum, does the frequency double?
b. If not, does it halve?
c. Does the frequency decrease as the length decreases?
d. Does the frequency increase as the length decreases?
6. Think about your experience with the monochord. As the vibrating length of the string decreases, what happens to the sound?
7. Does there seem to be any correlation between the sound and the frequency? If so, what is it?
Exercise 2: Vibrations and intervals
1. You are going to use both pendulums for this part. Adjust the length of one to be 100 cm and the length of the other to be 44.4 cm (the closer you get the better this will work!)
2. Release both pendulums from the same angle. You do this by lining up the strings before you release the pendulum bob.
3. Watch the longer pendulum complete two cycles. Now see if you can count the number of cycles completed by the shorter pendulum in the time it takes the longer pendulum to complete two cycles. If this seems difficult, try using the stopwatch to help you decide.
4. Discuss your observations with the rest of the class.
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