Group2_4_ch11

Maddie Margulies, Nicole Tomasofsky, Lauren Kostman, Lauren Barinsky toc

Lab: Spring Force Constant
What is the relationship between the 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

The same spring will be used for both methods, and the spring force constant wont change, the spring constant value of k should be similar. As the mass increases the period should increase and as it decreases, the period should decrease. Although, the spring constant should remain the about same.
 * Hypothesis:**

A mass was added to a spring to set it at equilibrium. Then we added more mass and took the displacement of the spring. This data was used to find the Spring Constant value of K in excel.
 * Methods and Materials:**
 * Method 1:**

Masses were placed on the spring and it was pulled down. Then we timed 5 oscillations to get the period by dividing by 5. Then we used the formula for the period of a spring to calculate k. Then excel was used to make a mass versus period graph to see the relationship between the period and the mass. media type="file" key="Movie on 2012-05-04 at 11.07.mov" width="300" height="300" media type="file" key="Movie on 2012-05-04 at 11.19.mov" width="300" height="300"
 * Method 2:**
 * Video of Method 1:**
 * Video of Method 2:**
 * Data:**
 * Method 1: Force versus Distance**
 * Method 2: Period versus Mass**
 * Graphs:**
 * Method 1: Force versus Distance**
 * Method 2: Period versus Mass**
 * Link to Excel spreadsheet:**
 * Sample Calculations:**
 * Method 1:**


 * Method 2:**


 * 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?
 * 2) Yes. The data shows a linear relationship between displacement and force. As force increases, displacement increases. The equation for the line has k as the slope, showing that k is constant.
 * 3) Why is the time for more than one period measured?
 * 4) Since the period is so short, it would be difficult to accurately record the time with a stopwatch. By getting a time for 10 periods and then dividing by 10, we get a period with less error so that our calculations for k are more accurate. Error due to human reaction time has more of an impact on such a small period than it does on a larger one.
 * 5) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * 6) We got a k value of 32.67 N/m with the first method and 32.7 N/m with the second method. I believe that the first method was more accurate because the required data was easy to obtain by simple measurements. The R^2 value for that graph is 0.99 with a good amount of data points. The second method, however, called for the more difficult measurement of the period. Since a stopwatch was used for this and there was more room for error, we can assume this data was more inaccurate. The R^2 was lower, 0.96, and there are only a few data points.
 * 7) Generate the position with respect to time equation and the corresponding graph for
 * 8) 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)
 * [[image:x_v_a_sin.GIF width="420" height="272"]]
 * 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?
 * 2) [[image:number_5.png width="145" height="55"]]
 * 3) 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?
 * 4) [[image:number_6.png width="123" height="157"]]
 * 5) We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship m + 1/3 ms (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using the square root of m + 1/3 ms, and explain these results.
 * 6) [[image:Screen_shot_2012-05-10_at_8.45.37_AM.png]]


 * Conclusion**:

The objective of this lab was to find the spring force constant using two different techniques, and then use this experimental value of k to determine the relationship between a mass on a spring and its period of oscillation. Our hypothesis states that even with two different techniques, our experimental values of k should be very close. It also states that there is a direct relationship between the mass on a spring and the oscillation period, meaning as the mass increases, the period should also increase and vice versa.

Our lab results were relatively good. Using the force vs. distance method, we graphed a k value of 32.67 N/m, and using the period vs. mass method we graphed a k value of 32.7 N/m. However, to check for accuracy, using the method 2, we can compare the exponent on our graph's equation, which should ideally be .5. We got a 6.6% error between our exponent (.276) and the ideal exponent (.5). This verifies that our methods were conducted well, due to the fact that method 2 was proven to be accurate based on exponent, which proves that method 1 is relatively accurate based on the small difference between the two.

There are reasons to our sources of error though. For starters, our spring was very tight, so we knew it had a relatively high k value for a spring of that sort. However, as a result, it became difficult to record oscillation time for method 2. If we did not wait until the spring reached the very top, which was difficult because it was very quick, the results of the oscillation period would be altered. Even as we added more mass, which made the oscillation period longer, the period was still relatively short making it difficult to measure at many masses. In addition reaction time difference also made this method more of a problem because we used a basic stopwatch. We could have fixed this problem with a different spring entirely that had a smaller k value so the oscillation period was naturally longer making it easier to measure and more clear at which points to measure. Or, we could have used a force sensor to more accurately give us times. Problems occurred with the first method as well. Although we made sure to mark our equilibrium point and measure from that point each time, since the spring moved so little between addition of masses because they were relatively small, measuring could have been off. For example, if the ruler was not completely straight results could have been off which could have altered our k value a lot. However, as discussed before method 2 was most likely more accurate because of the comparison of exponent values. If we used a motion detector our results would have been more accurate. Moreover, a general problem could have been the spring itself, because it was a little overused. However, all in all, there this lab proved to successfully state our hypothesis, with relatively good results.

=Transverse Standing Waves on a String= May 11, 2012

__**Objectives:**__ __**Hypotheses:**__ __**Procedure (video):**__ media type="file" key="Movie on 2012-05-11 at 11.19.mov" width="300" height="300"
 * Determine the relationship between frequency and the tension of transverse waves traveling in a stretched string.
 * Determine the relationship between frequency and harmonic number.
 * Determine the relationship between frequency and wavelength.
 * Frequency and tension are indirectly related, therefore as the frequency increases, the tension should decrease, and vice-versa.
 * The frequency of a wave and its harmonic number, or the number of antinodes, are directly proportional; therefore, as the frequency increases, the harmonic number should, too. Due to this, the graph should be a linear function.
 * As the frequency increases, the wavelength should decrease. The relationship between the two is a power-fit graph.

__**Data Tables:**__

__**Graphs:**__



__**Link to Excel Spreadsheet:**__ __**Sample Calculations:**__

__**Discussion Questions:**__ __**Conclusion:**__ Our results were very good. Part 1 yielded a percent error of of 4.6, and Part 2 4.05. The percent difference was 1.56 for wave speed and 3.5 for fundamental frequency. Our hypotheses were also correct. Frequency and tension are indirectly related, so as frequency increases tension decreases, yielding a power fit graph. The frequency of a wave and its harmonic number, or the number of antinodes, are directly proportional, so as the frequency increases, the harmonic increases, yielding a linear graph. As the frequency increases, the wavelength decreases, yielding a power fit graph.
 * 1) Calculate the tension that would be required to produce the n=1 standing wave for the red braided string.
 * 2) [[image:Screen_shot_2012-05-11_at_12.55.07_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) If the string stretched significantly as the tension increased, the velocity would increase, too. This would have changed our results by changing the harmonic number at each frequency. Also, an increase in tension would lead to an increased velocity, given the direct relationship that they have.
 * 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) The type of string impacts the amount of mass required to make a certain amount of nodes. Each type of string can have its own wave length, frequency, and other similar properties. Therefore, the values from different strings could all be different, which would make a group lab difficult (in terms of comparison). By using the same type of string, each lab group can get similar results to help support the overall class data.
 * 7) What is the effect of changing frequency on the number of nodes?
 * 8) Changing the frequency would cause an increases in the number of nodes. This is due to the direct relationship that they share. The harmonic number (amount of nodes), is equal to the number of noes times the frequency.
 * 9) What factors affect the number of nodes in a standing wave?
 * 10) There are numerous factors that affect the number of nodes in a standing eave. For example, as the frequency increases, so does the number of nodes (as mentioned above). Other factors that affect the number of nodes in such a wave include the string length, the string type, and the density of the string.

One source of error was the difficulty in assessing what frequency gave us the clearest waves. For example, in Part 1 where we added mass and found the new frequency that gave us the same harmonic number as with the previous mass, a frequency of 20 Hz may have appeared to give us the desired number of antinodes while 20.5 Hz would have given us the best and clearest wave. This error can be minimized by either ensuring to test frequencies within 5 or so Hz of the one we think is the best to make sure it really is the best frequency or by having some type of instrument that indicates when the wave is the smoothest and clearest. Another source of error came from the instability of the hanging mass. It would sometimes swing, altering the tension in the string and slightly skewing our results. To fix this, we could put the mass hanger within a barrier that keeps it from swinging without actually holding it so that the weight force remains constant.

**LAB: Resonance Tubes (Speed of Sound)**
Task A: Hella Talas Task B: Hella Talas Task C: Lauren Barinksy and Stephanie Wang Task D: Maddie Margulies


 * 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:**
 * 1) Resonance should occur at even intervals along the tube for the closed tube.
 * 2) Resonance should also occur at even intervals for an open tube.
 * 3) The speed of sound that we measure should be the same as the speed of sound in air


 * Methods and Materials:**

In this lab, we used a resonance tube with a sliding inner tube to find the length at which resonance occurs for a certain frequency. At first we used a closed tube by covering it with a cap, so the theoretical length would only be the measure of how much we pull the tube out. However, for the open tube was with a cap, so we had to account the length of the actual tube in addition to how much we increased the length of the tube. This technique was used to find all the resonance points which is where the pitch is highest.



__**Video:**__ media type="file" key="Movie on 2012-05-16 at 11.25.mov" width="300" height="300"
 * Data:**






 * Calculations:**






 * Graph:**



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 half of the wavelength. The equation that relates L, n, and wavelength is L=.5*n*wavelength. Since n (harmonic number) is the x variable and L is the y variable on the graph, the slope is .5*wavelength. For the closed tube, the slope is a fourth of the wavelength because the equation is L=.25*n*wavelength.
 * 3) Why was the length of the tube always smaller than expected?
 * 4) The diameter of the tube has an effect. As the diameter increases, the length requirement decreases. To find the corrected values of L, we use L = 0.5*n*wavelength - 0.8d for the open tube and L = 0.25*n*wavelength - 0.4d for the closed tube.
 * 5) 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?
 * 6) The velocity would increase if the temperature is increased. Since velocity directly affects the wavelength, the wavelength would increase proportionally. In our case, the wavelength would have increased by a factor of 6/700.
 * 7) 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.
 * 8) <><>< The left is the closed end and the right is the open end. From the equation L=1.2*n*wavelength, we can see that each arrow is a fourth of a wave, and because it is the fifth harmonic number, we draw 5/4 of a wavelength.
 * 9) 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.
 * 10) ><><><><>< Both ends are open. From the equation L=1/4*n*wavelength, we see that each arrow is 1/5 of a wavelength, and we need 5/2 wavelengths for the fifth harmonic number.
 * 11) What does this have to do with making music?
 * 12) All of these concepts are used in instruments. For example, a guitarist is able to change the length of the string that the vibrations are moving through to change the sound. This causes the frequency to change by changing the wavelength, resulting in a different sound. Woodwind instruments are able to produce various sounds because because the length of the tube can be changed as well as the end being changed from open to closed or vice versa. This allows for different wavelengths, causing different frequencies.

Our results for the lab were very good. The highest percent error was 6% which is very low. However, this error could be due to a lack of hearing the difference between the lower tones and higher tones. Also, the measuring could have been slightly off. This could be fixed by using data studio and looking at the amplitude to find the highest one. The highest amplitude will show you the loudest tone which represents the antinode. The measurement errors could be fixed by having a second person evaluating the length of the tube. A second set of eyes will be more accurate.
 * Conclusion:**