Group6_4_ch11

=toc= = = = = = =

= = =LAB: STANDING WAVES=

Task A: Robert Kwark Task B: Robert Kwark Task C: Jonathan Itskovitch Task D: Jonathan Itskovitch

- What is the relationship between frequency and harmonic number? - What is the relationship between frequency and wavelength?
 * Objective:** - //What is the relationship between frequency and the tension of transverse waves traveling in a stretched spring?//

media type="file" key="Movie on 2012-05-14 at 13.30.mov" width="300" height="300"media type="file" key="Movie on 2012-05-14 at 13.39.mov" width="300" height="300" Data Table: Frequency v Tension Graph: Frequency v Tension Data Table: Frequency v Harmonic Number & Wavelength Graph: Frequency v Harmonic Number Graph: Frequency v Wavelength
 * Hypothesis:** 1. If tension increases, frequency will increase in a square root relationship. 2. Frequency will increase in a linear fashion when the harmonic number increases. 3. There is an inverse relationship between frequency and wavelength, so wavelength will decrease.
 * Methods and Materials:** We set up a configuration where there was a generator (oscillator) at one end of a string, and the other side had a pulley. Masses would be placed under the pulley. Using the oscillator, we could change the frequency of the wave in the string. First, we found the relationship between frequency and tension. To do so, we used several amounts of masses hanging from the string, which could easily give us the tension amount using Newton’s Second Law. We also used the same harmonic number each time, and measured the frequency needed by adjusting the oscillator. Then we found the relationship between the frequency and harmonic number. To do so, we kept the same mass hanging each time. The only thing that changed was the frequency. We adjusted the oscillator until we clearly saw the harmonic number in the string. We also measured the length of the string with a meterstick, and using the mass, we found the kg/m measurement. Finally, we found the relationship between frequency and wavelength. We used the same frequencies as was done in the previous trial. However, we measured the average wavelength with a meterstick, which could give us the final relationship we needed. This enabled us to measure the wave speed.
 * Videos:**
 * Data Tables & Graphs:**

CALCULATIONS ANALYSIS:

1. Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string. 2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? If this were to happen, many changes would occur. A sharp increase of tension would lead to an increase in the wave speed, due to the wave speed relative to tension equation. This would also lead to a change in the harmonic number because, since frequency remains the same and velocity increases, the wavelength must increase. This causes the harmonic number to decrease.

3. 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. The kind of string has a great effect on the hanging mass required to create a number of nodes. Each string has its own characteristics, and its kg/m ratio may change from string to string. Using the velocity to tension equation, it may take more or less tension (and therefore mass) to re-establish the same number of nodes as in the first place. This will also impact the wavelength. It is imperative to use the same string each time.

4. What is the effect of changing frequency on the number of nodes? The frequency has a direct impact on what the number of nodes will be, since there is a linear relationship. In other words, the higher the frequency, the more nodes you get. The length of the string and velocity must remain constant, however, to achieve this.

5. What factors affect the number of nodes in a standing wave? Several factors affect the number of nodes in a standing wave. Obviously, frequency plays a direct role in the number of nodes due to the linear relationship. Velocity has a role in this too, because of the equation v=lambda * frequency. If frequency is to remain the same, and velocity changes, the wavelength must change. The wavelength has a direct role in the number of nodes. Furthermore, the harmonic number of the string affects the number of nodes. The length is equal to the (harmonic number/2) times the wavelength. So if the harmonic number increases, and the length of the string stays the same, the wavelength must go down, which means the number of nodes goes down. The length of the string also plays a role, because the longer the string, the longer the wavelength, which means less nodes.

CONCLUSION:

In this lab, we addressed 3 objectives to find the relationship between certain variables. First, we had to find the relationship between tension and frequency, which we correctly guessed was a square root relationship. Then we had to find the relationship between frequency and harmonic number, which we saw was a linear, direct relationship. This was also a correct hypothesis. Finally, we had to find the relationship between frequency and wave length, which we correctly predicted was an inverse, decreasing relationship. To verify our hypotheses, we carried out 3 separate experiments, and we realized that all of out hypotheses were correct. Our data and experiments prove that indeed these relationships are true. For example, our exponent of the frequency v tension was 0.5011, which is very close to 0.5 (square root). In fact, we only got a 0.11% error! Furthermore, our exponent of the frequency v wavelength was -0.995, which is very close to -1. We only got a 0.5% error. Our thorough lab procedures gave us such great results.

Despite our great results, we could attribute our small percent error a variety of sources. For example, when adjusting the oscillator to change the frequency, it is difficult to get the exact result, and the oscillator wasn’t as cooperative as we would have liked it to be. Resolving this problem would involve the use of a more accurate or higher quality oscillator. Other errors could involve poor measurements with the meterstick. We could have used a more accurate device to solve this problem. Also, when determining the frequency of the harmonic numbers, we may not have gotten the exact frequency (the device only measured up to the nearest tenth). However, since the procedure was rather easy, there weren’t too many other places to find error.

A practical application of the oscillator would be with sound. AutoTune, for example, that many artists use to correct their voices when singing, adjusts the frequency of the sound waves to make it more in line with the background music

= Lab: Resonance Tube =

Task A: Robert Kwark Task B: Jonathan Itskovitch Task C: Jonathan Itskovitch Task D: Matt Ordover

1. The length of the closed tube is 0.25 of the wavelength times the harmonic number. This depicts a linear relationship. Thus, as harmonic number increases, effective length increases. It should increase at the same rate each time. 2. The length of the open tube is 0.5 of the wavelength times the harmonic number. This depicts a linear relationship. Thus, as harmonic number increases, effective length increases. It should increase at the same rate each time. 3. The equation for wave speed is v=f*lambda. With the given frequencies and measured wavelengths, we multiply the two to get v. However, lambda is measured, and is not perfectly accurate. The best way to calculate the speed of sound in air is v=331.5+0.6T, with v being velocity and T being temperature in centigrade. Thus, if we know temperature, we can easily calculate v this way. An economy resonance tube is lying across a lab table. On one end, a generator producing sine waves is attached to a ring stand. This is connected to a speaker that amplifies the noise produced by the wave (so we can hear it). There is 2 parts to the lab: the closed tube and opened tube. For the closed tube, we put a cap on the same end as the speaker and generator. Then, we slowly pulled the tube farther and farther out until the noise was loudest – this is the antinode. Each time the noise got loudest, this was the antinode for the next harmonic number (though for closed tube it was every odd harmonic number). We did 3 trials for this. The same thing occurred for the open tube, except we pulled off the cap. media type="file" key="Movie on 2012-05-18 at 10.48.mov" width="300" height="300" Data Table: Closed Tube Resonance Data Table: Open Tube Resonance Graph: Open and Closed Resonance, n vs L
 * Objectives: **
 * 1) Determine several effective lengths of the closed tube at which resonance occurs for a frequency.
 * 2) Determine several effective lengths of the open tube at which resonance occurs for a frequency.
 * 3) Determine the speed of sound from the measured wavelengths and known frequency of the sound.
 * Hypothesis:**
 * Methods and Materials:**
 * Video:**
 * Data Tables:**
 * Graphs:**
 * Calculations:**
 * Analysis:**
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?

The slope of the graph of the closed tube is equal to the length of the pipe at the first harmonic. The slope for the closed graph equals 0.1159. If we take the length at the first harmonic, we say L=n1/4lambda, which is 1*0.25*0.4612. This is equal to 0.1153. This demonstrates that this is true. Similarly, the slope of the graph of the open tube is equal to the length of the pipe at the first harmonic. The slope for the open graph equals 0.2319. If we take the length at the first harmonic, we say L=n1/2lambda, which is 1*0.5*0.4612. This is equal to 0.2319. This demonstrates that this is true.


 * 1) Why was the length of the tube always smaller than expected?

The length of the tube is smaller than expected because of the end correction issue. When there is resonance, there should be 0 displacement at the node, the node is not always perfectly in phase. The node goes beyond where it is expected. Because the wave travels farther, the length of the tube is smaller. Furthermore, the end correction is equal to the extra distance the wave travels, and we can get rid of this problem with a simple calculation. This will give us the true acoustic length.


 * 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?

It will increase the measured value of L. The velocity is 331.5+0.6T, so if T increases, velocity increases. Then, If frequency is to stay the same, the wavelength will increase, because v=f*lambda, so lambda is v/f. If v is higher, and f is the same, lambda increases. Then, the L increases, because L=n(1/2 - open)or(1/2-closed)lambda. If n is the same, and lambda increases, L must increase.

L=n1/4lambda L=(5)(1/4)lambda L=5/4lambda
 * 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.

L=n1/2lambda L=(5)(1/2)lambda L=5/2lambda
 * 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.

Brass instruments and woodwinds make noise due to the vibrations produced by blowing into a mouthpiece. The length of the air column helps to determine the frequency of the vibration, which in turn makes a noise of a certain note. Even for pianos, when the hammer hits the string, the length of the string determines the sound.
 * 1) What does this have to do with making music?

The purpose of the lab was to accurately determine the lengths of an open or closed pipe that produced a sound at a given frequency. In doing so, we were proving the equations L=n1/2lambda (open) and L=n1/4lambda (closed) true. We hypothesized that, as the harmonic number increases, the L must increase in a linear and constant fashion. As it turns out, the slopes of the lines of the graphs are n1/2lambda and n¼ lambda, with n being 1 (the slope is of the first harmonic). The graphs confirmed that these equations were true. We also hypothesized that the speed of the sound in air is determined solely by the temperature, and increases as the temperature increases. This was also correct. This accurate velocity helped us to get excellent results for the lab.
 * Conclusion:**

Our results for the lab were really excellent, and these results confirmed what we were trying to prove in the first place. What we wanted to get in the end was a measured length of the pipe that was similar to the length of the pipe theoretically. For example, we had only a 0.22% error between the length of the measured and theoretical pipes for the closed tube at one harmonic number. This comes to demonstrate that our accurate measurements in this lab and our good procedure produced good results. Our percent error never exceeded 5% for the closed tube, again showing just how accurate our results are. Also, for the open tube, our percent error for the length of the pipe never exceeded 3%, again showing just how great these results are. One of our harmonic numbers had a percent error as low as 0.32%!

Despite these great results, certain factors may have prevented perfect numbers. First of all, the end shift issue. Because the waves go beyond where they are expected to, the L is usually lower than expected. This produced some error for us as well. Fixing this error includes using the end shift equation. Another source of error is poor hearing judgment. It is hard to determine exactly where the sound was loudest, so our measurements are really just estimates. Another source of error is that we don’t know what the wave speed is for sure. We calculated it but we never verified that speed with a device. Using such a device would have made our results more accurate. Finally, pulling out the pipe too fast may have skipped certain antinodes, skewing results. To remedy this, one must pull the pipe slowly.

One good application of this lab is with music. The length of the air-pipe of the instrument determines the frequency, or pitch, of the note it produces.