Group4_2_ch11

=toc= =Lab: What is the relationship between the mass on a spring and its period of oscillation?= Ali, George, Jessica, Nicole The first procedure that we performed to determine the spring constant k was by measuring the elongation of a spring. We placed a spring on a rod connected to a clamp and initially added .02 kg to the spring. After the addition of this weight, we recorded the length, and called this zero. We then added five additional weights to the hanging mass and recorded the displacement. After, we graphed the displacement versus the force (weight of the masses). Hooke's Law states that F=-kx. From this graph, we were able to determine the spring force constant by looking at the slope of the line formed. The second procedure we performed to indirectly determine the spring constant k was by looking at variations in the period. We let the spring set at equilibrium, once again, and called this zero. We then pulled down the hanger, released it, and recorded the amount of time that it took for the hanger to oscillate 10 times. After, we divided by ten to get the time for one oscillation. For each trial, we would add more mass to the hanger. After, we graphed our data on a Period (T) versus mass graph. This slope of the line would represent the spring constant k.
 * Objective**
 * 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**
 * We will graph the displacement of the spring versus the force (weight) in order to find the spring force constant k
 * There should be a direct relationship. As more mass increases, time increases. As mass decreases, times decreases. K will always remain constant.
 * There should be little to no difference between the spring force constants.
 * Methods and Materials**
 * Data and Calculations**

//Force v. Displacement of Spring Graph//: Slope = 4.7877

This equals the spring force constant because the graphed equation y=4.7877x models Hooke's Law, F=-kx. If y is the force F and x is the displacement, then 4.79 N/m=k

R 2 = 0.98. The linear trendline accurately follows the equation F=-kx.

//Average Period v. Mass on Spring Graph// Slope = 2.9197 The slope in the graphed equation represents T/(m^0.5), which we can use to solve for k.

R 2 = 0.98. The power trendline follows the equation T=(2pi/k^0.5)*m^0.5.
 * Percent Difference/Error**


 * Analysis/ 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) The slope of the line on the displacement vs. force graph represents the spring constant k. Since it follows a linear form, we can form the conclusion that it does, in fact, hold true to its name: it is constant.
 * 3) Why is the time for more than one period measured?
 * 4) Time for more than one period is measured to erase the factor of human error. In order to ensure accuracy, we performed five trials for each mass. We would then record the time for ten periods, then divide this by ten to find the time for one period. The time for one period is two small to be measured manually by a handheld stopwatch.
 * 5) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * 6) Our first graph followed a linear fit, while the second graph followed a power fit. However, both graphs are similar in the fact that the spring force constant remains the same throughout. Based on the two procedures that were performed, we believed that the first method was more accurate. The second method involved more human error, with the reaction time of the "10th" oscillation and when to stop the watch, as well as the addition of massing the weights. In the first method, our tools were more accurate.
 * 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?
 * 12) [[image:Screen_shot_2012-05-02_at_9.07.00_AM.png]]
 * 13) 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?
 * 14) [[image:Screen_shot_2012-05-02_at_10.05.28_AM.png]]
 * 15) We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship [[image:equation1.png]] (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using [[image:eqiation2.png]], and explain these results.
 * 16) [[image:Screen_shot_2012-05-07_at_8.53.40_AM.png]]
 * 17) [[image:Screen_shot_2012-05-07_at_8.53.31_AM.png]]
 * 18) [[image:Screen_shot_2012-05-07_at_8.55.41_AM.png]]
 * 19) As you can see above, by using this new formula, we were able to get a more accurate k value.

The purpose of this lab was to find the spring force constant through two different methods and then compare the results of each. We thought that the two ks were going to be the same due to the fact every spring has only one spring force constant with a small range. The first method we used was the law of conservation and for our second method we used simple harmonic motion. The first method gave us a result that k= 4.79 N/m which was found through the slope of the graph's (force v. displacement) trendline. The second method gave us the result that k=4.63 N/m through calculations. The percent difference... There could have been many sources of error due to the difficulty of measuring a moving spring. Our readings may have been a little of due to the movement of the spring. During the oscillation, the cycles of the spring were very small which was definitely hard to count. We also had to use a stop watch and there is always human delay error because we react to things slower than when they happen unlike a motion detector. 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.
 * Conclusion**

=5/16 Lab: Speed of Sound (Resonance Tube) = Nicole, Ali, Ryan, and Andrew


 * 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.

As frequency increases, so will the resonance. The graph between frequency and resonance is a positively increasing linear graph.The closed tube should have a smaller slope than an open tube but they are both linear positively increasing graphs.  To find the effective lengths of closed and open tubes at which resonance occurs for a frequency, we will use a resonance tube with a length scale to find the frequencies and length at which resonance occurs. The frequency generator made these waves while the speaker produced sounds that would get louder at certain lengths when the white inner tube was being pulled out. We recorded the different lengths and which resonance occurred. We did the same method for open tubes and closed tubes except the closed tube had to be flipped because it stopped at a certain point and did not reach the full length of the tube.
 * Hypothesis: **
 * Methods and Materials: **

__Velocity (based on temperature in room: 24 degrees Celsius):__ __Wavelength:__ __Length of Tube:__ __Average:__ __Closed tube:__ __Open tube:__ __Closed tube:__ __Open tube:__
 * Data: **
 * Sample Calculations: **
 * Percent Difference/Error:**
 * Using slope of graphs to find speed of sound:**
 * Discussion Questions: **
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?

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.

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.


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

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.

Although the length of the tube was 1.32 meters, due to end shift, it ideally would have been 1.44 meters: 1.47 - 1.32 = 0.15 meters longer than physical tube. 1.47 - 1.44 = 0.03 meters longer than ideal tube. 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.
 * 1) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">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. [[image:Open_End_5_Nodes.png]]
 * 2) <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">What does this have to do with making music?


 * <span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">Conclusion: **

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 90%;">This was a particularly unique lab due to its nature. In order to preserve time, we collected data as a class and then proceeded to do calculations individually as part of a group. We hypothesized that as frequency increased, resonance would increase as well. As such, we would expect to observe a linear relationship between the two. We also hypothesized that the smaller tube would have a smaller slope than the open tube. Our hypothesis were proven true. After analyzing all of our data, a positive relationship was discovered between the slope, indicating a direct relationship between resonance and frequency. We calculated various percent errors for this lab. The percent error for the closed tube was 28.09%, which was rather high, and the percent error for the open tube was 5.26%, which was a modest amount. The percent differences and the percent error/difference for calculating the speed of sound are also above in the calculations section. There were several sources of error in this lab. For instance, since we worked on this lab as part of a larger group we had to use many different people to "hear" the tubes. Introducing this many stray variables into the equation inevitably leads to a larger percent error. For instance, certain people have better hearing than others and this could have impacted our data collection. Furthermore, the tube was moved by different people. Some people could have been gentler with it. This has to do with the issue of human interference and error. More people usually leads to less accurate results. The tube could've also been compromised, leading to bad data collection. This lab has many applications to the "real world". For instance, resonance and frequency are what make music possible. Woodwind instruments make use of these physics concepts to create different frequencies, allowing different types of music to be played. Guitarists also make use of these concepts by manipulating strings to produce different frequencies. It is clear that physics, above all scientific disciplines, has the most direct impact on daily life.