Group1_2_ch11

Julia, Amanda, Caroline, John Lab: Speed of Sound- Resonance Tube Objectives: Hypothesis: As frequency increases, resonance will as well. Our graph should show a linear relationship between length and the number of nodes. Though, the closed tube should have a smaller slope than the open tube, and therefore should be lower on the graph. Methods and Materials: Using the resonance tubes, we will find the frequencies at which resonance occurs. We pulled out the inner white tube to find different lengths at which resonance occurred. The frequency generator made these waves while the speaker produced the sounds. Because the tubes are marked, we were able to record the different places at which resonance occurred. Data:
 * 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.

Calculations:

Discussion Questions: Conclusion:
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?
 * 2) open tube: The slope is equal to one half the wavelength. The equation for length is L= n1/2(wavelength), the equation of a line is y=mx. L is equivalent to the y part, n is equivalent to the x part, so 1/2 wavelength is the slope.
 * 3) closed tube: The slope for a closed tube is one quarter the wavelength. The equation for length is L= n 1/4 (wavelength) and the equation of a line is y=mx. L is the y axis, n is the x axis, so 1/4 wavelength is the slope.
 * 4) Why was the length of the tube always smaller than expected?
 * 5) Because of the large diameter of the tube, an end shift of the tube occured. The tube ends past the antinode of the wave because of the diameter, so it has to be accounted for in calculations.
 * 6) 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?
 * 7) Because of the direct relationship between temperature and velocity, if one is increased, the other increases as well. If the temperature were to increase, the velocity would have increased and so would the wavelength.
 * 8) 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.
 * 9) The parentheses each represent 1/4 of a wavelength for a closed tube. The 5th resonance has a harmonic number of 5 and therefore has 5/4 wavelengths.
 * 10) 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.
 * 11) The equation for length in an open tube is L= n 1/2 wavelength, from this equation we know that each parentheses is 1/2 of a wavelength so 5/2 wavelengths are needed for the 5th resonance.
 * 12) What does this have to do with making music?
 * 13) This is how woodwinds work, the different finger positions cause the tube to become a different length, changing the wavelengths and frequencies.
 * 1) What does this have to do with making music?
 * 2) This is how woodwinds work, the different finger positions cause the tube to become a different length, changing the wavelengths and frequencies.

For this lab, all data collection was done as a class using large resonance tubes and a frequency generator. We were trying to find the wavelength at which resonance occurs for a frequency. We also were trying to determine the speed of sound from our wavelength and frequency results. We hypothesized that resonance would increase as frequency increased, and therefore a linear, increasing graph would be produced for number of antinodes and length. Our hypothesis proved correct since our experimental graphs did produce this linear, increasing relationship. In addition, our percent errors were all pretty low, with the exception (above 10%) being the error between the closed tube at 1 node at 28.89%. Though, our slopes of the graphs, which were indicative of wavelength had very small percent errors at 3.47% and 5.345 and for speed of sound it was also small at 3.48% and 5.32% which further suggests that our data was lacking much error. Though, of course, there were multiple sources of error in this lab. The largest source of error would probably be in our method for determining length of the maximums. Each person has a different hearing level and therefore may perceive one sound as a maximum while another may determine a different one to be a maximum. In addition, sound is heard different at every position, which could also mess up the person's perception of a maximum. The different perception on a maximum would directly affect the lengths and could throw off our data. To fix this problem, we could have used a different method for determining the maximums, maybe a sensor that can definitely and accurately detect them.

=Lab: Transverse Standing Waves On a String = toc Sarah and Caroline Objectives:
 * 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?

hypothesis: The relationship between frequency and tension of transverse waves traveling in a stretched string would be a power fit, so it increases by a power. the relationship between frequency and harmonic number is linear. And the relationship between frequency and wavelength is also power fit.

Methods and Materials A string was attached to an electric driven oscillator on one end and a pulley system on the other end with known masses hanging on the end. The assembly was held onto the table with a clamp the prevent moving. The frequency of the oscillator was changed to vary the number of nodes in the standing wave with the mass kept constant. Then the frequency was changed to keep the number of nodes in the standing wave the same as the mass was changed.

Data/Sample Calculations //solving for mass per unit length://  //solving to find V (one way):// //solving for wavelength (first way) // //solving for wavelength (second way) //



Graphs/Analysis <span style="font-family: 'Comic Sans MS',cursive;">

//<span style="font-family: 'Comic Sans MS',cursive;">percent difference (velocity) //



//<span style="font-family: 'Comic Sans MS',cursive;">percent difference (fundamental frequency) //



//<span style="font-family: 'Comic Sans MS',cursive;">percent error (exponent in frequency vs. tension graph): //



//<span style="font-family: 'Comic Sans MS',cursive;">percent difference (between wavelengths solved for) //



//<span style="font-family: 'Comic Sans MS',cursive;">percent error (exponent in frequency vs. wavelength graph) //

<span style="font-family: 'Comic Sans MS',cursive;">Discussion Questions: <span style="font-family: Arial,Helvetica,sans-serif;">- <span style="font-family: Arial,Helvetica,sans-serif;">- The string’s velocity would increase as the tension increased. This would have influenced the harmonic number at each given frequency, so the results would have been significantly varied. <span style="font-family: Arial,Helvetica,sans-serif;">-The type of string can change this lab significantly. If a string is stretchy versus rigid, it would affect the results of the lab because the tension would be different for different masses, resulting in different velocities. If the velocities were changed, the number of nodes at frequencies would be changed as well. <span style="font-family: Arial,Helvetica,sans-serif;">- As the frequency increases, the string is vibrated more which increases the number of nodes because there are more waves being sent through the string. <span style="font-family: Arial,Helvetica,sans-serif;">- The frequency, string length, tension, elasticity, and mass per unit length all affect the number of nodes.
 * <span style="font-family: Arial,Helvetica,sans-serif;">1. Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string. **
 * <span style="font-family: Arial,Helvetica,sans-serif;">2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? **
 * <span style="font-family: Arial,Helvetica,sans-serif;">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. **
 * <span style="font-family: Arial,Helvetica,sans-serif;">4. What is the effect of changing frequency on the number of nodes? **
 * <span style="font-family: Arial,Helvetica,sans-serif;">5. What factors affect the number of nodes in a standing wave? **

<span style="font-family: 'Comic Sans MS',cursive;">Conclusion : Our hypotheses proved to be essentially correct. We found the relationship of frequency and tension on a graph to be a power fit, as we had originally predicted. The relationship between frequency and harmonic number on the graph was linear, also as we originally predicted. And the fit between frequency and wavelength was a power fit, further supporting our original hypothesis. The graphs we compiled with the data we collected had R values that seemed to show how precise/accurate our data was - two with about .99 and one with about .98. In this experiment, we solved to find velocity in two different ways: one by finding the relationship on a frequency vs. wavelength graph (where velocity was the slope) and the other by calculating it in a formula based on our experimental data. The velocities were 76.5 and 69.1 m/s, respectively. The percent difference between these two was 9.9%. We can, as always, attribute this to human error. The frequency, used to calculate this, may not have always been 100% correct. Though it was measured on a machine, the machine was dialed by us, and we may not have not stopped it always at the exact frequency was the amplitude was highest. If this frequency was off, the velocity we obtained from the graph would be slightly off. The exponent in the equation in the frequency vs. tension graph also proved close to what it should be - .58 as opposed to .5. Though the percent error for this is large (16%) that is only because we were working with such small numbers. The difference between the two is only .08 - fairly small as far as this experiment is concerned.The percent difference in the fundamental frequency was a mere 0.83%. This could likely be attributed to rounding errors when doing calculations. The percent error in the exponent in the frequency vs. wavelength graph was even smaller - .5%. This, again, can be attributed to the smallness of the numbers we are working with; in reality, the number we obtained through experimental means was only .005 different than the actual number. The percent difference between the wavelengths we solved for was also very small - 6.73%. This could be because of inaccuracies in the measurement of the string (we used a tape measure). It could also be due to rounding and calculation related issues.

=Lab: What is the relationship between a mass on a spring and its period of oscillation?=


 * 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**: Both of the spring constants that we find should be the same. If we use the same spring, the spring constant k should not change - hence //constant//.

Next, we set up a procedure to find the constant based on the variation of the period T of oscillation. We added started by adding one mass. We stretched out the mass to start the oscillation. We then recorded the period of the oscillation of the spring moving up and down five times. We repeated this procedure three times. We recorded our data in the Excel spreadsheet, and then we repeated the entire procedure four more times, each time adding more masses. media type="file" key="Movie on 2012-05-02 at 08.37.mov" width="300" height="300"
 * Methods and Materials:** To experiment with the first part of the objective, we set up a stand with a spring, a mass holder, and a meterstick. We added one mass and then recorded the distance of elongation of the spring. We repeated this several times, each time adding progressively more masses. We recorded this data in an Excel spreadsheet and were able to make a graph from the results.


 * Data/Calculations and Graphs**:




 * 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?**
 * 1) Yes because its always the same (or fairly close to the same), indicating that its constant


 * 2. Why is the time for more than one period measured?**
 * 1) To be more accurate and account for human error and reaction time. We timed for 5 periods and divided the time by 5 to get the time for one period. Also, we did three trials of 5 periods to make our times even more accurate.

The graphs are both very similar in that the spring force constant is constant. They are different because of human error. The first graph (force vs. distance) graph is more accurate because it left less room for error. The second graph required calculating the period by using a stopwatch which left room for error because of human reaction time.
 * 3. Discuss the agreement between the k values derived from the two graphs. Which is more accurate?**
 * 4. Generate the corresponding graphs for a) position with respect to time, b) velocity with respect to time, and c) acceleration with respect to time.**
 * 5. 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?**
 * 6. 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?**
 * 7. We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship [[image:Screen_shot_2012-05-02_at_8.42.23_AM.png]] (where //m// is the hanging mass and //m2// is the mass of the spring)? Redo graph #2 using the equation [[image:Screen_shot_2012-05-02_at_8.42.30_AM.png]] and explain these results.**

The point of this lab was to find the spring force constant of the spring that we were using through two different methods and comparing them. Through our hypothesis, we had hoped that these two values of the spring constants would come out to be the same thing. In order to find these two values, it was necessary to use two separate methods, which used different variables to find the same value. The methods we used were conservation of energy and harmonic motion. The first value, which came from conservation of energy, was 3.33 N/m, while the second value, which came from harmonic motion, was 3.27 N/m. When graphing the second values of oscillation, we used the power fit for the trend line so that we could limit error because without using this fit, the slope value would have been off and changed our values. We found that the percent difference between these values is 1.88%, so therefore, we got a very low percent error, although we still did get some. Error throughout this experiment could have derived from our lack of exact instruments. We had to eyeball some of our measurements due to the constant motion of the springs, which could have slightly skewed our results. We also were forced to use the stopwatch, which could have been slightly off considering that it didn’t have a motion detector attached that could send signals to stop it, and therefore make results more accurate. We did try to avoid error here though by making our period 5 oscillations, although some error should still be accounted for. If we used a motion detector, this would have greatly helped us in eliminating a great source of error throughout the experiment. An application of this to real life situations would be in a gymnastics board. It has springs underneath and a person runs onto it. Therefore, the mass of the person causes the system to oscillate, and therefore corresponds mass with the k value.
 * Conclusion**