Group4_4_ch11

Group 15toc Timothy Hwang Garrett Almeida Ryan Luo

=Mass on Spring Lab (5/4/12)= Ryan Luo - A Garrett Almeida - C Timothy Hwang - B, D

__**Objectives:**__
 * To directly determine the spring constant k of a spring by measuring the elongation of the spring for specific applied forces.
 * To indirectly determine the spring constant k from measurements of the variation of the period T of oscillation for different values of mass on the end of the spring.
 * To compare the two values of spring constant k.

__**Hypothesis:**__
 * The spring force constant that is derived by measuring the elongation of the spring will be found using the value in the equation for Hooke's Law.
 * The second spring force constant, which we will find using measurements of the variation of the period T of oscillation for different masses on the spring, will be found using calculations from slope of the trend-line of the relationship between the period and the mass of the oscillation.
 * The percent error between the two spring constants, k, will be small.

Method, Materials, and Procedure: To begin the first part of the lab, we set up a stand and attached a spring to it. The point at which the spring hung was our equilibrium point. We then added masses onto the spring; we then took measurements to determine the displacement that each mass resulted in. We used the displacement measurement we found using a meter stick that we got and found the force applied by the masses. Finally, we made an excel graph, using the data points we collected, and found the slope of the line to determine the spring constant k. As for the second part of the lab, we use the stand again to hold the bring, however we pulled the masses downwards and recorded the amount of time that it took to oscillate ten times. Once we recorded these values we divided it by ten so that we could get an individual period. Finally, we made another excel graph (period vs mass) and created a trend line to calculate the spring constant k. So, to recap, we used a stand, a spring, various masses, a stopwatch, and a meter stick.

Method 1: Method 2: __**Tables:**__ __Force vs. Displacement__ __Average Period vs. Mass on Spring__
 * __Data:__**

__**Excel Spreadsheet:**__


 * Sample Calculations:**

Method 1- k is equal to the slope of our graph k= 34.588

Method 2- solving for k percent error (compared with method 1) Examples- Top - position graph Middle - velocity graph Last - acceleration graph
 * __Discussion Questions:__**
 * 1) Do the data for the displacement of the spring versus the applied force indicate that the data for the spring constant is indeed constant for this range of forces?
 * Yes because we know that there is a linear relationship between the displacement (x) and force applied (y). Our graph shows us that as the force increases, the displacement increases as well. Each value increases at the same rate too. The slope gives us this value as our spring constant, k, which is measured in N/m. The linear slope allows us to illustrate that the k value is constant.
 * 1) Why is the time for more than one period measured?
 * The reason that we measured more than one period, specifically ten, is because it is very difficult to measure just one period. This would result in a lot of error and make our results less credible. By measuring ten oscillations and dividing by ten for the period, we are almost guaranteed more accurate period and therefore more accurate results.
 * 1) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * The k values from both of our graphs are close, in fact they show around an 8.1% error. This is a fine result. The k value determined from the first graph was probably most accurate because there was less chance of human error. In the first method, we only had to measure the displacement, this could be a source of error, although it is not a significant source. On the other hand, the timing of the oscillations in method two proved to be a more significant source of error as human reactions and instinct judged the time it took for ten oscillations. These oscillations then gave us our periods. Timing the a moving spring created more room for error, as opposed to simply measuring the distance of a spring at equilibrium at various masses. Because there was a more significant source of error in method two, the first graph of method one is more accurate.
 * 1) Generate the position with respect to time equation and the corresponding graph for
 * position with respect to time = Acos(wt) or Acos(2πft)
 * velocity with respect to time = 2πf(Acos(2πft)
 * acceleration with respect to time = 2πf^2(Acos(2πft)

Theoretically, our results should be more accurate. Our results from before were already accurate at 8.1% error. The new formula ended up increasing our percent error to over 20% and our k value to over 100. This is far more error and less accurate results. One reason for this problem could have been human error. Our reactions could have set off the error that we see in this second graph.
 * 1) A spring constant k = 8.75 N/m. If the spring is displaced -0.150 m from its equilibrium position, what is the force that the spring exerts?
 * 1) A massless spring has a spring constant of k = 7.85 N/m. A mass m = 0.425 kg is placed on the spring, and it is allowed to oscillate. What is the period T of oscillation?
 * 1) We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship m+1/3m (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using sqrt(m+1/3m), and explain these results. (mass of spring = 4.83)

Conclusion: In this lab, we found the spring force constant through two different methods and then compared the results. In our hypothesis we stated that the difference between the two spring constant values that we found would be very small if not exactly the same. For our first method, we used the Law of Conservation of Energy method that we used in a previous lab and the second method that we used was simple harmonic motion. Our first method gave of a spring constant of 34.588 N/m. We found this by looking at the slope of the trend line on a Force vs Displacement graph. We got a spring constant value of 37.39 N/m for our second method. We got a percent error of 8.1%. During the lab, there could have been many possible sources of error. One is human error. During the second method, we used the stopwatch to time 10 oscillations to figure out what the period was. We may not have started the stopwatch right when we let go of the spring so that it could oscillate or we may not have exactly stopped the stopwatch right when 10 oscillations had ended since it is hard to judge when the spring is at its crest or trough depending on where one started from. We would trial the oscillation experiment 5 times to insure the time was close to accurate. Our results were very similar to one another which gave us a small percent difference. We observed the more mass the more time the spring oscillated which proves that they are proportional to one another which is what we stated in our hypothesis. An application is a person jumping one a diving board, after the person jumps the diving board springs oscillate. The greater the weight of the person the more the diving board oscillates. We could find the k constant in this situation.

= Lab: Transverse Standing Waves on a String (5/11/12)= Ryan Luo - C Garrett Almeida - B,D Timothy Hwang - A

1. What is the relationship between frequency and tension of transverse waves traveling in a stretched string? 2. What is the relationship between frequency and harmonic number? 3. What is the relationship between frequency and wavelength?
 * Objectives**:

1. The relationship between frequency and tension of transverse waves traveling in a stretched string would resemble an increasing power fit. 2. The relationship between frequency and harmonic number should resemble a linear fit. 3. The relationship between frequency and wavelength should resemble a decreasing power fit.
 * Hypotheses**:

In the lab, we used a string and first set it up on an electronic oscillator, or the sine wave generator. This was attached to the table by a pulley and a clamp. We then added weights to the bottom a mass holder at the bottom of the string. We ran several different tests during the course of the lab. The first objective, we kept the hanging mass and the linear density of the string constant. We then changed the harmonic number, frequency, wavelength, and wave speed. We did this to determine the relationship between frequency and harmonic number and the relationship between frequency and wavelength. To determine the relationship between frequency and tension for the first objective, we kept harmonic number and linear density constant, and changed the hanging mass. After finding all of the relationships for each of the objectives, we were able to find the equations of the graphs and analyze these results.
 * Methods, Materials, and Procedure:**

__Photos__- Sine Wave Generator

System

Part 1: Part 2:
 * Data**:

__Excel Spreadsheet__-

__Sample Calculations__- 25 yds= 25*.9144= 22.86 meters .250/22.86=.0109 kg/m

Tension: F y =m*a y F y =T-w m*a y =0 T-w=0 w=mg T=mg T=.250*9.8=2.45 N

Velocity(Wave Speed)=Frequency*Wavelength V= 10.2*6.46= 65.89 m/s

Fundamental Frequency ƒ n =ƒ*n 52.6=ƒ(5) ƒ=10.52 Hz

__Percent Error and Percent Difference Calculations__- % Error Frequency vs. Tension (0.5325-0.5)/(0.5) x 100 = 6.5%

% Error Frequency vs. Wavelength (-1.012-(-1))/(-1) x 100 = 1.2%

1. Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string. ƒ=(1/(λ√(m/L)))*√(F tension ) 10.484=(1/(68.065√(0.0109)))*√(F tension )
 * Discussion Questions**:

F tension =5550 N

2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? If the string stretched significantly as the tension increased, the velocity would increase because the two are directly related. This change in results changes the harmonic # at each frequency.

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. Each type of string has its own properties unique to its type, such as the width of the string. As a result, the wavelength and the # of nodes at each frequency varies from string to string.

4. What is the effect of changing frequency on the number of nodes? As you increase the frequency, you increase the number of nodes present because frequency and the # of nodes are directly related.

5. What factors affect the number of nodes in a standing wave? Frequency of the string, length of the string, and type of string are among a variety of factors that contribute to the # of nodes in a standing wave.

Our hypotheses on the different relationships are supported by the data that we collected over the course of the lab. The relationship between frequency and tension yielded an increasing power fit, as we hypothesized. Also, the frequency vs. harmonic number graph was liner while the frequency vs. wavelength graph had a decreasing power fit. Not only were our hypotheses proven correct, but our data was strong as each graph had an r^2 value above .99. This indicates how well the values fit our trend line. Our percent error values were also very low at 6.5% and 1.2%.
 * Conclusion**:

As is the case for every lab, there will always be sources of error. One error in this lab was that it was difficult to find the exact frequency for each trial. We got a point at which it was close but it was not exactly perfect. Multiple trials could have helped to eliminate this problem. The lab helped us to understand how to calculate frequency in easy turns. With this information, we can apply it to problems we have done in class.

=Resonance in Air Columns (5/16/12)= Ryan Luo- B Garrett Almeida- A Tim Hwang- C Mike Poleway- D

__**Objective:**__
 * 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:**__ The length of the open tube will be 1/2 the wavelength times the number of nodes, so as the # of nodes increases, the length of the tube also increases, depicting a linear relationship between the length and the # of nodes. The length of the closed tube will be 1/4 the wavelength times the number of nodes, so the relationship between the length of the tube and the # of nodes is the same reason as the open tube. The speed of sound would be 331.5+.6(T) because the sound waves are traveling through air.

__**Method and Materials:**__ The lab had been set up already by the time the class arrived in the classroom. There is a speaker attached to the sine wave generator that is attached to a stand on a table. An Economy Resonance Tube was lined with the speaker. We first found the number of anti-nodes being generated by the frequency emitted from the speaker into a closed-ended tube. Mike moved the inner tube out so the class could determine the displacement of the inner tube from the outer one by listening to how loud the speaker was (loudest point is the anti-node). The same thing was also done with the open-ended tubes.

__**Picture:**__

__**Data:**__





__**Graph:**__



__**Sample Calculations:**__ __Velocity:__ __Wavelength:__ __Length of tube:__ __Percent Error:__ Closed tube (7 nodes)-

Open tube (7 nodes)-

__**Discussion Questions:**__

For the open tube, the slope is equal to one half of the wavelength. Since the equation for the length of an open tube is L=(1/2*λ)*n, and the equation of the line (with a y-intercept of 0) is y=m*x, and since L is being graphed on the y-axis on n on the x-axis, m, or the slope, is equal to 1/2*λ, or one half of the wavelength. For an open tube, this is equal to the first node, also known as the fundamental frequency.
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?

For a closed tube, the slope is equal to one quarter of the wavelength. Since the equation for the length of an open tube is L=(1/4*λ)*n, and the equation of the line (with a y-intercept of 0) is y=m*x, and since L is being graphed on the y-axis on n on the x-axis, m, or the slope, is equal to 1/4*λ, or one quarter of the wavelength. For a closed tube, this is equal to the first node, also known as the fundamental frequency. The length of the tube is always smaller than expected due to the end shift of the tube, caused by the larger diameter. Due to the diameter, the tube ends past the exact compression of the wave, so it is not the loudest it can be at the theoretical length, and is instead a little smaller. Therefore, to take that into account, the revised equation for an open tube is L=n*(1/2* λ)-0.8d, and for a closed tube it is L=n(1/4* λ)-0.8d. Temperature and velocity are directly proportional so as one increases, so does the other. We can figure this out by knowing that velocity is directly proportional to wavelength, so as it increases so does the others (V=wavelength *frequency). Because as temperture increases velocity increases, and when velocity increase, wave length increases, if the temperature was moved up 10 degrees that wavelength would increase. ( Each parenthesis is (1/4) of a wavelength. Because we know that the fifth harmonic number was 5/4 of a wavelength, we can this determine out how to draw this figure. )( In this case, each parenthesis is half of a wavelength. Because we know that the fifth harmonic number would be 5/2 wavelengths, we can use this to create the figure above. This has to do with making music because it relates to the physics behind the workings of woodwind instruments. Through different fingerings, musicians are able to change the length of their 'tube' (their instrument), and can also change it from open to closed end, allowing for different wavelengths and in turn different frequencies. This lets distinct 'notes' to be played, creating music. __**Conclusion:**__
 * 1) Why was the length of the tube always smaller than expected?
 * 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?
 * 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.
 * 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?

For this lab, we wanted to find the different lengths at which an opened or closed tube would produce a sound at a given frequency. We also wanted to find the speed of sound. We believed that the open tube would have a length that is 1/2 the wavelength times the number of nodes. The length would keep on increasing, and in turn, create a linear relationship. The closed tube would have a length that is 1/4 the wavelength times the number of nodes, also with a linear relationship. By looking at our graphs of the open and closed tubes, we know that our hypothesis was right. In addition, we believed the speed of sound would be 331.5+.6(T) in air because it depends on temperature and increases as temperature does. This was correct as well, and through this lab, we were able to prove everyone of our hypotheses. We got good results for this lab according to our percent error. Whenever you have below 10%, you're results are considered good, but we definitely could've got a better percent error. We got 5.31%, which isn't bad, but there were several possible errors throughout each trial that could've messed us up. The R^2 values for both graphs were excellent as well, so we know that we gathered great results all around. The most common error in this lab was definitely trying to pinpoint exactly where the sound is the loudest in the experiment. We had to find the loudest sound, but there were many points where the sound sounded exactly the same. Poor hearing by us could've hurt our results pretty badly. There isn't really an easy way to fix this though, and we had to try and make a best estimate. Furthermore, there was one person trying to make exact measurements. It is very possible that these measurements could've been slightly off or the person could've make a mistake. For example, I was doing the measurements in this experiment and I had to fix my measurement a few times because I had messed up. If there were another person checking the measurements with me, then it would've been easier to get exact answers. To get the best possible results, we would need to eliminate these sources of error, and if we were to do the lab again, we would hope to get better ones. All in all, we did a good job in this lab though as our percent error shows. In the future, a great real-world application for this lab would be with musical instruments. The length of the pipe in the instruments will determine several factors that we found in this lab.