PhysicsLAB Lab
Relationship Between Tension in a String and Wave Speed

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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 (vw) 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.
Refer to the following information for the next question.

Part A: Data Collection
The experiment will be performed on two fishing lines having different pound-tests or linear density.
 
Record below the length of the first vibrating string segment - the heavy string.
 
L1 = ______________ m 

Complete the following table by measuring calculating the required values for string one.
 
 
Record below the length of the second vibrating string segment - the light string.
 
L2 = ______________ m 

Complete the following table by measuring calculating the required values for string two.
 
 
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 five questions.

Part B: EXCEL
Using EXCEL, graph Wave Speed vs Tension for both data sets and then Wave Speed Squared vs Tension for both data sets.
What is the filename of your group's EXCEL workbook?
 

On the graph of Wave Speed vs Tension, what is the exponent on x for your group's heavy string?
 

On the graph of Wave Speed vs Tension, what is the exponent on x for your group's light string?
 

When rectified, what is the slope of your group's graph for Wave Speed Squared vs Tension for your group's heavy string?
 

When rectified, what is the slope of your graph for Wave Speed Squared vs Tension for your group's light string?
 

Refer to the following information for the next seven questions.

Part C: Conclusions
The equation relating wave speed and tension in a string is given as
 
 
To make this equation fit our lines, we must square both sides
 
 
This solution tells us that the slopes of your lines for Wave Speed Squared vs Tension represent the reciprocal of each string's linear mass density, µ.
 
What is the reciprocal of your heavy line's slope? (give your answer to three significant digits) 

Based on your measured values calculated using the sample's mass and length, in Part B above, what was your group's percent difference for the heavy string's linear mass density? 

What is the reciprocal of your light line's slope? (give your answer to three significant digits) 

Based on your measured values calculated using the sample's mass and length, in Part B above, what was your group's percent difference for the light string's linear mass density? 

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?


 
2. What would be true about the frequency for the two ropes in question #1, if the wavelength was kept constant?


 
Which of the following statements are true about waves traveling on strings?





 



 
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William A. Hilburn
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