Group3_2_ch11

toc =Lab Mass on a Spring=

Part A: Danielle Part B: Sammy Part C: Jake Part D: Maddy


 * 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.


 * Hypotheses:**
 * The first spring force constant will be derived by measuring the elongation of the spring and using that value in the equation for Hooke's Law.
 * The second spring force constant, which will be equal to the first value, will be derived by calculating the slope of the trend-line of the relationship between period of each oscillation and mass.
 * The percent difference between the two derived spring force constant values will be very low.


 * Materials**
 * The materials we are going to use are springs, tape, clamps and rods, masses, balance, timers, meter stick.


 * Method**

media type="file" key="New Project 3 - Mobile.m4v" width="300" height="300"


 * Data**

Part 1 (Hooke's Law):





Part 2 (Periods of Oscillation):




 * Sample Calculations**
 * Analysis**


 * 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 suggest that the spring force constant is close to constant for the range of forces. Our graph shows a successful linear relationship with an R2 value of .997, which indicates that the trend line (whose slope is equal to the spring force constant) effectively describes 99.7% of our data. In the context of our lab, this means that the spring force constant of 3.8442 can be used to estimate 99.7% of the force values that we use. Also, we achieved very low percent difference values. Our average percent difference between the experimental spring force constants and the trend line slope was 7.027%, which is acceptable because it is significantly lower than 10%.
 * 3) Why is the time for more than one period measured?
 * 4) We measured the time for 10 periods, not just one period, because it would make our data more accurate. Each oscillation fluctuates from the previous one, with the spring’s oscillation getting lesser and lesser each time. Measuring the time for 10 periods allowed us to yield an average measurement for oscillation, so the final value is accurate with respect to the 10 individual measurements.
 * 5) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * 6) We calculated the percent difference between the average spring force constant derived from Method 1 and that derived from Method 2. The percent difference that we calculated was 8.002%, which is also significantly lower than 10%, so the k values were very similar. We can conclude that Methods 1 and 2 are effective in determining spring force constant, considering how close our spring force constant values were. That being said, the k values derived from the graph “Displacement v. Force” was more accurate. It had an R2 value of .99653, whereas the graph “Mass v. Period” had an R2 value of .99633, meaning that the results from Method 1 were .02% more accurate than the results from Method 2. Overall, both graphs achieved accurate results that were close to each other.
 * 7) Generate the equations and the corresponding graphs for
 * 8) position with respect to time.
 * 9) velocity with respect to time.
 * 10) acceleration with respect to time.
 * 11) 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?

Fspring = -kx Fspring = -(8.75)(-.15) Fspring = 1.31 N


 * 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 [[image:Untitled1.png]] (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using [[image:Untitled2.png]], and explain these results.
 * 2) Yes, our results are better when using the more accurate relationship. In the new graph, the value of the exponent is .4801, which is closer to the theoretical .5 than is the exponent from the original graph (.4683). The percent error between .4801 and .5 is 3.98%, compared to the 6.4% error between the original graph's .4683 and .5.


 * Conclusion**

The purpose of this lab was to find the spring force constant of a spring with two different methods and then compare them. The first method was to use conservation of energy to solve for the spring force constant and the second method was to use harmonic motion. We hypothesized that the two values would be equal because a spring only has one spring force constant. In order to complete the two different parts of the lab, we had to do two different types of tests. The first value of k, from the conservation of energy, was 21.853 N/m and the second value was 20.107 N/m. The percent difference between these values is 8.684, which is much larger than we would've liked. There are several things that can explain why we experienced this error. For the first part of the lab, it was hard to get exact measurements because the spring often oscillated a bit with the added weight. Due to human error, it was hard to get exact measurements. To get rid of this problem, we could have been more patient and more precise. The second part of the lab had more room for error. It was difficult to calculate the period because it was hard to tell exactly when the spring completed a ten cycles because it was such a tight spring that it didn’t make obvious movements. To get rid of this problem, we could have used a spring that stretched more. We also used a stopwatch rather than a motion detector, which wouldn’t result in that much error, but still could have contributed to it. This lab has many real life applications because there are spring force constants in anything with springs and we now how to calculate this k value if we ever need to. We also understand that a mass will affect the k value because we saw that as we added weight the spring oscillated more. This can help us understand how to stretch things more or less.

Lab: Transverse Standing Waves on a String
4/9/12 __Task A__: Jake __Task B__: Danielle __Task C__: Danielle/George __Task D__: George

__Objectives__: __Hypothesis__:
 * What is the relationship between Frequency and the tension of transverse waves traveling in a streched string?
 * What is the relationship between frequency and harmonic number?
 * What is the relationship between frequency and wavelength?
 * As the tension increases, so does the frequency. Therefore, it should create a power fit on the graph.
 * They are directly related, and should have a linear relationship because as frequency increases so does the harmonic number.
 * They should show an inverse power fit because as the frequency increases, the wavelength decreases.

__Materials__:
 * We used a 25 yard string and first set it up on an electronic oscillator that was attached to the table by a pulley and a clamp. After adding the weights to the bottom of the string, we ran several different tests in order to figure out the objectives. To figure out 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 changed all of these in order to determine the relationship between frequency and harmonic number as well as the relationship between frequency and wavelength. In order 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.

__Video__: media type="file" key="Movie on 2012-05-09 at 08.26.mov" width="300" height="300" Picture:

__Data__:

Part 1: Frequency & Tension

The trendline for this graph is a power fit because it models y=Ax^b, where y is Frequency, x is Tension, b is 0.5, and b is 1/(lamda* sqrt(m/L)).

Percent Error:

Part 2 (Frequency & Harmonic Number) & Part 3 (Frequency & Wavelength)



Frequency v. Harmonic number has a linear fit because it follows the equation f n = f 1 *n

Percent Error:



__Percent Error:__



__Discussion Questions__: 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 the tension increased then the string would stretch significantly meaning that the velocity would also increase. If the string was stretched then this lab would have completely different results because the increased string would change the harmonic number 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. 4) What is the effect of changing frequency on the number of nodes?
 * If we all selected a different type of string then the results would be greatly effected because of the certain properties of each string. This would make it very hard to compare results with other groups. There would be an increase number of nodes if we increased the amount of mass that is hanging from the string. This would then lead to more tension, which creates a greater velocity, resulting in more nodes.
 * The number of nodes increases as the frequency increases because the string vibrates more.

5) What factors affect the number of nodes in a standing wave?
 * Some factors that may affect the number of nodes in a standing wave are tension, length of string, and also frequency. Frequency is the main factor though because not only does it affect the number of nodes, it also affects the harmonic number.

__Conclusion:__

The results for this experiment reflected the theory behind standing waves very accurately. Just as we hypothesized, frequency and tension had an exponential relationship (power fit). This we proved by graphing several points and algebraically showing that frequency and tension are related in an equation resembling y=Ax^0.5. We were also correct in hypothesizing that frequency and harmonic number were directly proportional (linear fit). This we proved by graphing and showing that the two values in an equation resemble y=Ax. Lastly, we were also correct in hypothesizing that frequency and wavelength would be inversely related in a similar way (y=Ax^-1). These results make sense logically because if the string is tightened (increase tension), there is more inertia to overcome and the particles are more spread out and therefore takes a higher frequency to produce the same wave. Similarly, a higher frequency produces a higher harmonic number because higher frequencies means shorter wavelengths. Shorter wavelengths means more nodes appear on the standing wave; thus, a higher harmonic number. This holds true for the last relationship as well. As the frequency increases (more cycles per second), and the velocity remains constant, the wavelength must get shorter.

We had very little error in this lab. For the first equation we examined the percent error between coefficients and exponents. The coefficient, 1/(lamda* sqrt(m/L)), had only a 5.26% error between theoretical and experimental values. The exponent had a 0.26% error. In the second relationship, there was a 3.68% error in the coefficients, which was supposed to be fundamental frequency. In the last relationship, the exponents had a 2.6% error. The error occurred mainly due to friction. Also, it was difficult to discern which exact frequency produced the highest amplitude. The most error occurred in the first relationship because there were many variables that were hard to calculate.

To address these errors, perhaps it would be wise to experiment with different strings to see which produces the least friction. Also, to make the standing wave easier to see, place a black sheet behind it. The nodes will be very easy to spot that way. It is important to know these concepts because this is how all stringed instruments work. shortening the string shortens the length and increases the frequency, producing a higher note.

=Speed of Sound - Resonance Tube=


 * Objectives**
 * 1) Determine several effective lengths of the closed tube at which resonance occurs for a frequency.
 * 2) Determince several effective lengths onf the closed tube at which resonance occurs for a frequency.
 * 3) Determine the speed of sound from the measured wavelengths and known frequency of the sound.

Frequency and resonance have a positive linear relationship. As one increases, so will the other. When it is a closed tube there should be a smaller slope in the "node vs. length" graph than when it is an open tube.
 * Hypotheses:**

For this lab we are using resonance tubes with length scale marked on the tube, frequency generator, speaker, class thermometer.
 * Materials:**

We used a resonance tube with a sliding inner circle 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. We pulled the tube to be longer and longer finding all the resonance points (where the pitch is at its highest). Then we performed the same procedure with an open tube (no cap on the end).
 * Methods:**



Analysis: Discussion Questions:
 * Data**

The slope of the open tube is equal to ½ wavelength, because the graph fits the equation L = n/2 x wavelength, where ½ times wavelength is the slope. Similarly the closed tube’s slope equals ¼ wavelength, as it fits the equation L = n/4 x wavelength.
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?

2. Why was the length of the tube always smaller than expected? The length of the tube was often smaller because we used the equation wavelength = 4l/n to solve for our experimental wavelength. This did not take into account the length ofour tube, which has an effect on our data. For an open tube, the shape will shift the antinode into the end of the tube, so the tube is shorter than it needs to be. The equation that would take this into account is L=n/2 x wavelength - .8d. For a closed tube, the antinode is again shifted in, so we again only need a shorter length to hear the antinode. This is modeled by L = ¼ x wavelength - .4d. This, as the diameter of the tube increases, the length requirement decreases.

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? Theoretically, had the temperature been 10 degrees higher, the velocity would have increased by .607(10). This increased value would have then been used in the equation velocity = frequency x wavelength, and increased the wavelength by .09. Wen length is then solved for, the Length requirement increases by .09 as well for the second harmonic, as an example.

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. Figure: ( l L=n/4 x wavelength  L(5) = 5/4wavelength

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. Figure: )( L=n/2x wavelength L(5) = 5/2wavelength

6. What does this have to do with making music? In music, instruments, specifically woodwinds, often allow one to change the length of the tube, which would thus change the sound that we hear amplified by the tube. The musician can change the closed or open quality of the tube, and thus the wave frequency and what we hear.

Conclusion: Our hypothesis for this lab was that frequency & resonance have a positive linear relationship. Meaning that as one increases, so will the other. We said that when it is a closed tube there should be a smaller slope in the "node vs. length" graph than when it is an open tube. After completing the lab, we found this hypothesis to be true. This is because the slope of our graph was positive & linear, so the two formed a direct relationship, which was exactly what we hypothesized. The percent error that we calculated was 11.22% for the open tube. *The percent error that we calculated for the closed tube was___*. Although this is not the lowest value, we still tried to do many things to minimize the error. For example, we took a large amount of trials & then instead of using just one, we averaged them together in order to get the most accurate results. There are many things that we could have done differently to lower those numbers. Some sources of error could be the measurement of the tube because it was not an exact value each time, and it was hard to see exactly what point it was on. We also could have not stopped at the high point, and can blame our hearing for that. Something we could have done to fix these errors could have been maybe more than one person measuring the location on the tube. We also could have had other people working on different tubes because only a small amount of people was needed to gather the data. An example of a major real life relations with this lab could be musical instruments like woodwinds. This is because when someone blows into an instrument there is a vibration and that is where the sound comes from, and it is determined by the keys that are either open or blocked.