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Standing Waves on a Vibrating String
Purpose:
To verify experimentally the relationship between the length, tension, vibrating frequency, and wave speed for a vibrating string
Apparatus:
Vibrating string apparatus, power supply, function generator, frequency meter, meter stick, slotted masses, mass hanger
Introduction
A string of length L under tension F
T
has certain "natural frequencies" at which it can vibrate if it is plucked or somehow disturbed. These natural frequencies are also the resonant frequencies of the string when it is subjected to periodic driving by an external forcing agent. If the string is essentially fixed at each end, a standing wave at the driving frequency will be established. As the frequency of the external driving force is changed, resonances will occur and can be identified because the amplitude of the standing wave produced on the string reaches a maximum. At its lowest resonant frequency, the string has displacement nodes at each end and a displacement antinode at its midpoint, so the length of the string corresponds to half the wavelength of the waves travelling along the string.
By measuring the lowest resonant frequency as a function of tension for a fixed length, we can determine how the speed of the wave on the string depends on the tension.
Prediction
Predict how the fundamental frequency depends on the tension.
For a given length, the fundamental frequency will _______________ as the tension increases.
Procedure
NOTE: Certain controls have been preset and are NOT to be changed. Under no circumstances are you to molest the DC Voltage Control on the power supply, or the Offset and Amplitude controls on the function generator.
1. Adjust your string to be exactly 1.50 m from the vibrator to the center of the pulley. We'll take these to be the ends of the string.
2. Each group will be assigned a mass to hang from the string. Record your group's mass in the space below and hang that mass from the end of the string. (Remember that the hanger has mass.)
Mass = __________ kg
3. Vary the frequency of the function generator until your string is vibrating with the maximum amplitude at its three lowest resonant frequencies, which are the fundamental f
1
, 2f
1
, and 3f
1
. Since the differences between successive resonant frequencies for a vibrating string are equal to the fundamental frequency, measuring these three resonances gives us three values for the fundamental which can be averaged together. Record your resonant frequencies here:
Vibrating Strings
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Stephen Luzader

f
1
= __________ Hz f
2
= __________ Hz f
3
= __________ Hz
Now calculate frequency differences and record their values here: f
2
- f
1
= __________ Hz f
3
- f
2
= __________ Hz
Average of f
1
and the two differences = __________ Hz
4. Make a table summarizing the results of all the groups in the class.
Mass (kg) Frequency (Hz)
__________ __________
__________ __________
__________ __________
__________ __________
__________ __________
__________ __________
5. Compute the tension in each string and tabulate the tensions and the frequencies:
Tension (N) Frequency (Hz)
__________ __________
__________ __________
__________ __________
__________ __________
__________ __________
__________ __________
6.
Was your prediction about the effect of the tension correct? Explain briefly.
7. To analyze the data, we want to establish a proportionality between the tension raised to some power (call it p) and the frequency raised to some power (call it q); i. e., a graph of f q vs F
T p will be a straight line. What powers p and q will give such a straight line, and what are the slope and intercept predicted to be? Briefly explain your reasoning.
Vibrating Strings
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Stephen Luzader

8. Use Excel to plot the graph and test your prediction about the relationship between the frequency and the tension. Was your prediction correct?
9. Once you've gotten a straight line graph, use the Trendline function to plot the line on your graph and to print the equation on the graph. Read the slope and intercept from the equation and give them here with correct units : slope = __________ intercept = __________
Print a copy of the graph to include with this report.
10. We have verified the predicted relationship between frequency and tension qualitatively, but not quantitatively. We must have theoretical values of the slope and intercept to compare with the measured values. What is the predicted value of the intercept ?
Predicted intercept = __________
Does the measured intercept agree reasonably well with your theoretical prediction? Give a brief explanation.
11. Suggest a way of determining the predicted value of the slope. Then carry out the necessary measurements and calculate a theoretical slope.
Vibrating Strings
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Stephen Luzader

Calculations
12. Does the measured slope agree with the theoretical value? Give a brief explanation.
13. Please summarize the results, stating clearly how this experiment tends to support or fail to support the theoretical description of wave vibrations on a string.
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Stephen Luzader

Homework Questions
Very Important Note: You’ll need the numerical values of the tension, density, and fundamental frequency for your string to do some of these questions. Please be sure to copy that important information BEFORE you turn in your lab report!!!!!!!!
HW1. Prove that the difference between two successive resonant frequencies is equal to the fundamental frequency for a vibrating string. Please do this in general (using variables), not just as a demonstration that it works for the particular set of numbers in this experiment.
HW2. Compute the actual speed of the waves along your string.
HW3. Find the wavelength of the sound in air for the fundamental frequency of your string.
HW4. Why is the wavelength for HW3 different from twice the length of the string?
HW5. Find the average power in the wave at the fundamental for your string if the vibration amplitude is 1.00 cm.
Vibrating Strings
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Stephen Luzader