Group6_2_ch11

= = Lab: Transverse Standing Waves on a String toc Ali, Amanda, Jessica, Nicole What is the relationship between Frequency and the tension of transverse waves traveling in a stretched string? What is the relationship between frequency and harmonic number? What is the relationship between frequency and wavelength?
 * Objectives**

We believe that as tension increases, frequency should decrease. The relationship between frequency and harmonic number should be that as frequency increases (within the multiples of its natural resonance frequency), the number of antinodes will increase at a rate one less that that multiple. If n=5, there will be 4 antinodes. The frequency will increase because velocity is increasing. Wavelength remains constant.
 * Hypothesis**

There was a setup of a string through a pulley attached to a SHM generator, which was clamped down to the table. At the end of the string was a mass of 1.150 kg. We then turned on the generator and found its maximum amplitude. Here, we found five antinodes. Then, we used different masses to find their frequency at five antinodes. For Part B, we kept one mass constant 1.350 kg. We then tried to find different harmonic numbers, and compared them in a data chart.
 * Methods and Materials**


 * Data and Calculations**






 * Discussion Questions/ Analysis**
 * 1) Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string.
 * 2) [[image:Screen_shot_2012-05-12_at_3.37.01_PM.png]]
 * 3) What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data?
 * 4) We would see an increase in velocity if the string stretched significantly as the tension increased. Because of this, we would also see an increase in harmonic number, as the two are directly related (V=ƒ** λ **). Our results would have been different in this lab for the frequency would have been at a less voltage to get some certain harmonic value.
 * 5) What is the effect of the type of string on the amount of hanging mass needed to create a set number of nodes? Explain this.
 * 6) Any type of string will have a differing qualities than that of another type of string. Some strings may be more taut, some will be thicker, thinner, more flexible. Essentially, the natural resonance frequency will be specific to only one certain type of string. Additionally, the amount of mass added will increase tension, and therefore factor into a certain frequency of a string . For example, a string with a smaller tension will resonate at a different frequency than that of a string with a much higher tension.
 * 7) What is the effect of changing frequency on the number of nodes?
 * 8) As frequency increases, as does the number of nodes. Wavelength and frequency share an indirect relationship. As frequency increases, wavelength shortens, thus meaning that a single period shortens as well. If all is relative to a certain string, the amount of waves will therefore increase from the increase of frequency.
 * 9) What factors affect the number of nodes in a standing wave?
 * 10) Frequency, string length, type, and tension, as well as mass per unit length, all contribute to affecting the number of nodes in a standing wave.

This lab had three different objectives involving different aspects of standing waves. First we found the relationship between the tension and frequency of a transverse wave. In our hypothesis, we thought that as tension increased frequency would decrease. Our data proved that our hypothesis was incorrect. The two have proportional relationship that is when one is increasing so is the other which was the power function. For the second objective, the relationship between the frequency and the harmonic number. We found that in the experiment that as as the harmonic number was increasing as was the frequency which was what we stated in our hypothesis to some degree. The last part of the experiment was to find the relationship between the frequency and the wavelength. From our data we found that as the wave length increased the frequency decreased which was very incorrect from our hypothesis. For our graph we used a power fit trend line to give us the correct results. Our data was very precise with small percent difference from the theoretical. For each part of the lab we had under 6% percent difference. The little error that we did receive came from the frequency we used. With the mass hanging, the string was bouncing around and wasn't always staying on the wheel that could have affected our frequencies. Also our recorded frequencies were not necessarily with the maximum altitude so that could have been a source of error. We also used a lot of weight which made it very difficult to find the frequencies that were not as high up on the number scale. This lab helped our understanding between the components in standing waves and their relationship with frequency. _ = Lab: Speed of Sound - Resonance Tube =
 * Conclusion **

Nicole T, Brianna B, Jenna M, Kaila S, and Lauren K
 * Group Members**

Determine several effective lengths of the closed tube at which resonance occurs for a frequency. Determine several effective lengths of the open tube at which resonance occurs for a frequency. Determine the speed of sound from the measured wavelengths and known frequency of the sound.
 * Objectives**

As the frequency increases, the resonance will also increase. Our graph should display a linear relationship between the length and the number of nodes. However, the closed tube should have a smaller slope on the linear graph than the open tube.
 * Hypothesis**

To do this lab, we used a resonance tube with length scale marked on the tube and moved it in and out to see when the sound was the loudest. To produce the sound, we used a frequency generator and speaker, which projected the sound through the tube. In order to find the velocity of the sound for some of our calculations, we used the class thermometer to know the temperature. This made it possible to find the velocity of sound in the room.
 * Methods and Materials**

media type="file" key="Movie on 2012-05-21 at 13.40.mov" width="300" height="300"


 * Data**





Link to Excel:

Calculations
 * Velocity, temperature of 24.3ºC**
 * Wavelength**
 * Length of tube, closed; n=3**
 * Length of tube, open; n=3**
 * Percent error, open tube; n=3**


 * Percent error, closed tube; n=3**
 * Slope of the Graphs to calculate:**
 * Wavelength, open tube**
 * Velocity, open tube**
 * Wavelength, closed tube**
 * Velocity, closed tube**

Discussion Questions**
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?
 * 2) For the open tube, the slope is equal to one half of the wavelength. The equation for length is L=n(1/2) λ, and the equation for a line is y=mx. Because L is equivalent to the y part of the line, and n is equivalent to the x part, the slope (m) must be one half of the wavelength.
 * 3) For the closed tube, the slope is equal to one fourth of the wavelength. The equation for length is L=n(1/4) λ, and the equation for a line is y=mx. Because L is equivalent to the y part of the line, and n is equivalent to the x part, the slope (m) must be one fourth of the wavelength.


 * 1) Why was the length of the tube always smaller than expected?
 * 2) The length of the tube always turned out to be smaller than expected due to the end shift of the tube, which was a result of the larger diameter. The antinode shifts and the diameter of the tube increases.

if we used 34.3 C for the temperature: V = 331.5+.6(34.5) = 352.2m/s
 * 1) Suppose that the temperature had been 10 C higher than the value measured for the room temperature. How much would that have changed the measured value of L?

V = f(wavelength) 352.2 = (450)(wavelength) wavelength = .783m

closed tube n=3 L = 1/4(3)(.783) = 0.587m This is .01025m larger compared to n=3 at 10 degrees Celsius cooler.

open tube n=3 L = 1/2(3)(.783) = 1.175m This is .02154m larger compared to n=3 at 10 degrees Celsius cooler.

This shows the resonance in a tube that is closed on the left side. n = 5 L = 5(1/4)(wavelength) L = 5/4(wavelength)
 * 1) Draw a figure showing the fifth resonance in a tube closed at one end. Show also how the length of the tube L5 is related to the wavelength, λ.

This shows the resonance in a tube that is open on both sides. n = 5 L = 5(1/2)(wavelength) L = 5/2(wavelength)
 * 1) Draw a figure showing the fifth resonance in a tube open at one end. Show also how the length of the tube L5, is related to the wavelength, λ.


 * 1) What does this have to do with making music?
 * 2) In musical instruments, specifically woodwinds, you push down on keys, buttons, or holes to change the 'length' of the tube. Although you're not actually changing the length, you're changing where the air can flow, so technically you're changing the wavelength. The musician can do this so that we can hear different sounds.

There were three main objectives addressed within this lab activity, including being able to determine the effect of the length of both a closed and an open tube at which resonance occurs for a certain frequency, as well as determine the speed of sound from a measured wavelength and known frequency of sound. With these main focuses in mind, it was hypothesized that as the frequency increases, so will the resonance; this hypothesis is proven correct by the results. This is manifested by the graph, which shows the linear relationship between these two factors; as the length gets larger, so does the number of nodes. Although the graphs of both the open and the closed tube have linear graphs, the slope of the closed tube is small than that of the open tube, which was hypothesized correctly, too.
 * Conclusion**

Despite the fact that our hypotheses were accurate, there was a percent error for both the open and closed tube activities. For the open tube, there was a mere 1.94% error, which is extremely low; however, for the trials performed using a closed tube, there was a high 19.73% error. While conducting the lab, there are several possible areas of error that could have impacted the results. A main concern could be the fact that each person's perception and hearing is a little different, and no one's is perfect, therefore although we may have //thought// that we were measuring the length of the tube when the sound was at a maximum, this may not have been the actual peak. Additionally, upon measuring the length of the tube, the measurements were not as precise as they could have been; this could change the length of the tube, which also would harm the results. For future performances of this activity, all of the people in the lab group should stand close to one another, have absolute silence in the room, and spend a lot more time focusing in on the exact location of the maximum noise within the tube; this would allow for much more accurate results. Another possible solution could be to use some sort of sound detector to determine the high point of the sound in the tube.

This lab is very applicable to everyday life and real situations. For example, a person who plays a musical instrument, such as a recorder, for example, uses this type of procedure to produce different sounds. It can also be applied to playing some sort of string instrument- as the musician changes the length of the string, they can change the sound produced. Additionally, by changing the wavelength, the frequency will change, which will create music, as a result.

= Lab: Speed of Sound - Resonance Tube (Different Group) = Group Members: George Souflis, Sarah Malley, Ben Weiss

Determine several effective lengths of the closed tube at which resonance occurs for a frequency. Determine several effective lengths of the open tube at which resonance occurs for a frequency. Determine the speed of sound from the measured wavelengths and known frequency of the sound.
 * Objectives**

We think that resonance should occur at fairly regular intervals, because the wavelength is constant and antinodes occur in a pattern. This will be true for the open and closed tubes, and will show a linear relationship between length and number of nodes. However, the open tube relationship between number of nodes and length should have a steeper slope.
 * Hypotheses**

Prepare this lab by setting up a resonance tube next to a frequency generator and a speaker. Set a specific frequency to be projected through the tube. Move the inner tube (which should have a marked length scale on it) in and out to find which different positions produce the loudest sound. This should be done for both the open tube and the closed tube (which can be made by adding or removing a cap at the end that is not on the speaker). Make sure to record the length of both the open tube and the closed tube (at resonance), the frequency used, the diameter of the tube, and the temperature of the room (which can be measured by using a thermometer) on Microsoft Excel.
 * Methods and Materials**

The lab is performed with these materials in the manner shown below.


 * Data:**
 * Graphs:**

Wave Speed: Theoretical wavelength:
 * Calculations:**

Theoretical Length with Resonance (Open Tube):

Theoretical Length with Resonance (Closed Tube):

Sample Percent Error of Length of Open Tube: Sample Percent Error of Length of Open Tube: Percent Error of Slope of Open Tube Graph: Percent Error of Slope of Closed Tube Graph:

1. What is the meaning of the slope of the graph for the open tube? For the closed tube? The slope of the open tube is equal to 1/2 of the wavelength. This is because the equation L=n1/2λ is reflected on the graph as y=mx. Therefore, y is equal to L, n=x, and m=1/2 λ. The slope of the closed tube is equal to 1/4 of the wavelength. The equation L=n1/4 λ is also reflected on the graph y=mx, where L=y, n=x, and m=1/4λ.
 * Discussion Questions**

2. Why was the length of the tube always smaller than expected?

The length of the tube was always smaller than expected because of the relatively large diameter. There was an end shift of the tube. That's why, in the analysis, we used the equations L = n(1/2) λ - .8d (for an open tube) and L = n(1/4) λ -.8d (for a closed tube) - in order to remedy this.

3. Suppose that the temperature had been 10 ˚C higher than the value measured for the room temperature. How much would that have changed the measured value of L?

The temperature that we originally measured was 28.4º; 10º higher would make it 38.4º.



a. closed tube



This is about a .01 m difference between this value and the theoretical value at 28.4º Celsius.

b. open tube n=5 L=(1/2)(5)(.645) L=1.6125 m

There is about a .03 m difference between this value and the theoretical value at 28.4º Celsius

Because temperature is directly related to wavelength, as the temperature increases, the wavelength will also increase. This means that L, in turn, will increase. So when temperature increases, so does L.

4. Draw a figure showing the fifth resonance in a tube closed at one end. Show also how the length of the tube L5,is related to the wavelength, λ. <><><

This shows the fifth resonance in a tube that is closed at the left end but open on the right.

5. Draw a figure showing the fifth resonance in a tube open at one end. Show also how the length of the tube L5,is related to the wavelength, λ.

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This shows the fifth resonance in a tube open at both ends.

6. What does this have to do with making music? Musicians can create music from specific instruments (like woodwinds) by changing the length of the tube that is produced the music. This can be done with instruments like guitars and woodwinds. Changing the length of the tube changes the wavelength which would in turn change the frequency, leading to higher or lower notes. At the same time, some instruments can be turned into open or closed tubes, which also affects the sound of the music.

For this lab, we hypothesized that for a constant frequency, resonance should occur at regular intervals. As a result, we also hypothesized that as harmonic number increased, length (of both closed and open tubes) would increase as well to fit this relationship. Because of the fact that length of an open tube is n1/2λ and that length of a closed tube is n1/4λ, we also hypothesized that the resulting graph of length for an open tube vs. harmonic number would have a greater slope than the graph of length for a closed tube vs. harmonic number.
 * Conclusion**

To prove these hypothesis true, we recorded the lengths of closed and open tubes that provided resonance for particular harmonic numbers. To do this, we set a frequency generator to 550 Hz, allowed noise to go through a speaker, and then increased the length of a tube into which the sound was projected. When the volume of the noise reached its maximum, we then recorded the length of the tube with its corresponding harmonic number on Excel. Once this data was collected, we then graphed this data to observe the relationship.

Once doing this, we determined that resonance did occur at regular intervals, that length increased as harmonic number increased, and that the graph of length of an open tube vs. harmonic number did have a greater slope than the graph of length of a closed tube vs. harmonic number. These observations proved our hypotheses true. The next step was to determine the accuracy of our results.

To determine accuracy of our results, we used percent error in three different ways: to compare experimental lengths to theoretical lengths (which were found using our knowledge of equations for wavelength and length of tubes, as well as a measured temperature of 28.4º Celsius), to compare the slope of the open tube graph to its theoretical slope of 1/2λ (or .317), and to compare the slope of the closed tube graph to its theoretical slope of 1/4λ (.158). Percent error was used because we were comparing theoretical values to experimental values, unlike percent difference which compares one experimental value to the average experimental value. Our results were very good. For closed tube lengths, percent error for the most part ranged from about 4% to 15% (with only one value having 40% error exactly); for open tube lengths, percent error ranged from about 4% to 10%; the slope of the open tube graph had a percent error of 8.83% and the slope of the closed tube graph had a percent error of 7.85%.

Though much of our data was extremely accurate, there are several potential sources of error that could have hampered our results. For example, instead of performing this lab as individuals, we were assigned to do this lab in a large group of students. As we have learned in class, some people have better hearing than others, meaning that the lengths from which they perceive resonance might be smaller than others. Because of the fact that this group tried to find resonance at lengths that were suitable for the majority of the group, the experimental values might have been off from more ideal ones. At the same time, we were forced to estimate hundredths of meters for the lengths because the ruler on the side of the tube did not have these available. As a result, the lengths that we measured might have been somewhat off from what they actually were.

This lab has many applications beyond the classroom. For example, musicians use resonance and frequency to make music. Woodward instruments achieve different pitches and frequencies by having their lengths changed. At the same time, guitar strings can be manipulated to create different frequencies as well. By employing their knowledge of physics, musicians are able to make more complicated, sophisticated songs.