Lab
Relationship Between Tension in a String and Wave Speed Along the String
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We have observed that an increase in the tension of a string causes an increase in the velocity that waves travel on the string. In this activity we will examine the precise relationship between tension (T) the force applied to the string, the wave speed (v
_{w}
) and the linear mass density of the string (µ = m/L which is measured in kg/m).
We will stretch a string across two “bridges”, creating two fixed ends, and then allow the remaining string to hang over a supporting bar with different increments of mass generating its tension. This will allow us to increase tension in the string by the addition of mass, while keeping a constant wavelength. This will cause the velocity to change with the frequency of the string like a guitar with its tuning pegs.
A microphone will be placed next to the string and when plucked the frequency of the note will be displayed on a scale using a frequency analyzer. Notice that several frequencies are observed, the must discernible and lowest frequency represents the fundamental - seen as a dark red line with green/yellow highlights. The other frequencies represent higher harmonics (or overtones). You can notice that they are evenly spaced in frequency as predicted by our model of standing waves.
Measuring the length of the vibrating string allows us to calculate the wavelength. Then by focusing on the fundamental frequency (which has only one loop) and using our model for fixed-fixed standing waves we can determine the wave speed along the string.
The experiment will be performed on two fishing lines having different pound-tests or linear density.
Samples of each of these fishing lines will now be provided to you so that you can measure their mass and length. This information will allow you to determine the
linear mass density
for each type of string used in the experiment.
Refer to the following information for the next three questions.
String #1 (Heavy String)
length of sample in meters
mass of sample in kilograms
linear mass density (µ = m/l) in kg/m
Refer to the following information for the next three questions.
String #2 (Light String)
length of sample in meters
mass of sample in kilograms
linear mass density (µ = m/l) in kg/m
Refer to the following information for the next two questions.
Part A: Data Collection
Complete the following tables for String #1 and String #2.
Record below the length of the
vibrating string segment
for both strings.
Based on the length recorded above, what is the wavelength for both vibrating strings?
Refer to the following information for the next five questions.
Part B: EXCEL
Using EXCEL, graph
Frequency vs Tension
for both data sets and then
Frequency
^{2}
vs Tension
for both data sets.
What is the filename of your group's EXCEL workbook?
On the graph of
Frequency vs Tension
, what is the
exponent
on
x
for String #1?
On the graph of
Frequency vs Tension
, what is the
exponent
on
x
for String #2?
When rectified, what is the
slope
of your group's graph for
Frequency
^{2}
vs Tension
for String #1?
When rectified, what is the
slope
of your group's graph for
Frequency
^{2}
vs Tension
for String #2?
Refer to the following information for the next nine questions.
Part C: Conclusions
What is the reciprocal of String #1's slope? (give your answer to three significant digits)
What is the linear mass density, µ, of String #1? (give your answer to three significant digits)
Based on your measured values calculated using the sample's mass and length what is your group's percent difference for String #1's linear mass density?
What is the reciprocal of String #2's slope? (give your answer to three significant digits)
What is the linear mass density, µ, of String #2 (give your answer to three significant digits)
Based on your measured values calculated using the sample's mass and length what is your group's percent difference for String #2's linear mass density?
Conceptual Questions
1. Waves are created on two ropes, a thick rope and a thin rope. If the tension on each of the ropes is the same, what is true about the wave speeds?
a. The thick rope (greater µ) has a higher waves speed.
b. The thin rope (smaller µ) has a higher wave speed.
c. The two ropes have the same wave speed because they are under the same tension.
2. What would be true about the frequency for the two ropes in Question #1 above if the wavelength was to be kept constant?
a. The thicker rope (greater µ) will have a greater frequency
b. The thinner rope (smaller µ) will have a greater frequency
c. The two ropes will have the same frequencies
3. Which of the following statement(s) are true about waves traveling along strings?
a. wave speed is proportional to the tension along a string
b. wave speed is proportional to the square of the tension along a string
c. The square of the wave speed is inversely proportional to the linear mass density, µ, along a string
d. The square of the wave speed is proportional to the linear mass density, µ, of a string
e. The square of the wave speed is proportional to the tension along a string
f. The wave speed is inversely proportional to the tension along a string
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