Group1_6_ch11

toc Andrea, Jake, Maddie, and Julie

= = =Resonance Lab= 5/15/12

==
 * Objectives:**
 * 1) Determine several effective length of 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:**
 * By using closed and open tubes we will be able to determine the effective lengths at which resonance occurs for a frequency. When the length increases, the resonance also increases.
 * Methods and Materials:**
 * We used a tube closed off with a small speaker. When finding the lengths for the closed tube, we closed off one end of the tube with the cap. To find the loudest points, we would pull the tube out to find the length at which there are maximums. The sound of the tube is oscillating to show the nodes and antinodes in the waves.


 * Graphs:**
 * Closed Tube**
 * Open Tube**






 * Sample Calculations:**
 * Discussion Questions:**
 * 1) What is the meaning of the slope of the graph for the open tube? For the closed tube?
 * 2) For the closed tube, the slope represents one fourth of the wavelength and for the open tube the slope represent one half of the wavelength.
 * 3) Why was the length of the tube always smaller than expected?
 * 4) The length is always smaller then expected because the theoretical values based of of the equation are can not be used in a real life situation. This inaccuracy is due to the difficulty of finding the exact amplitude of the wave because our perception generally estimates that it is lower then it actually is.
 * 5) Suppose that the temperature had been 10 degrees Celsius higher than the value measured for the room temperature. How much would that have changed the measure value of L?
 * 6) 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.
 * 7) Draw a figure showing 5th resonance in a tube closed at one end. Show also how length of the tube L, is related to wavelength.
 * 8) ( 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.
 * 9) Draw a figure showing 5th resonance in a tube open at one end. Show also how length of the tube L, is related to wavelength.
 * 10) )( 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.
 * 11) What does this have to do with making music?
 * 12) Guitars are one good example of how music is made using the concepts discussed in this lab. By changing the length of the string, Musicians are able to change the sound that it produces. By using our concepts on waves we would know that changing the wavelengths the frequencies would also change, which is why the sound that is produced also changes.

Conclusion:

After performing the lab, we found that our hypothesis was correct. We predicted that as length increased, the resonance would increase too. We are able to support our thesis because when we plotted our data, our graph was linear and positive.

Given our percent error values, we got very good results for our data. We found the percent error for the length of the open tube, which was only 5.6%, while the percent error for the length of the closed tube was only 4.48%. Given these values, it is clear that our results were accurate because they are below 10%.

There are many possible sources of error in this lab. Since we were basing our measurements on hearing, there weren't completely accurate values. Every person heard the maximums differently because they were either standing in different positions or didn't have the same sense of perception. Therefore, when trying to get the length of the tube at the loudest sound, our results could vary. The tube was also moved around while doing this lab, leading to miscalculated lengths. Both these errors lead to inaccurate lengths, which affect the overall lab. To fix these sources of error, we could have a sound sensor to measure when the volume is the highest. This way we could get a more precise length at the maximums. The tube also needs to be in a fixed position to the speaker and table so that the lengths would not change.

There are different real life applications for this lab. The most obvious would be through musical instruments such as woodwinds. As someone blows into it, a vibration follows and the sound is caused because of this. The sound is determined by whether the keys are blocked or open.

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

1) To directly determine the spring constant k of a spring by measuring the elongation of the spring for specific applied forces.
 * Objectives:**
 * We will be able to do this by graphing change in distance vs. force of added weights. The slope will represent the spring force constant.

2) 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.
 * As m increases, T increases
 * As m decreases, T decreases
 * They are directly related
 * K will remain constant

3) To compare the two values of spring constant k.
 * They should be the same.

First, we did a procedure to directly determine the spring constant k of a spring by measuring the elongation of the spring for specific applied forces. We did this by first hanging a spring on a clamp and rod. Next, we brought the spring to equilibrium by adding a mass. After adding the mass, we recorded the spring's position and called it zero. We then 5 additional masses, and measured the spring's change in distance from its original position. We graphed change in distance vs. force (weight of masses). The slope of this graph represents the spring force constant, because of Hooke's Law in which F=kx.
 * Materials and Procedure:**

Next we did a procedure 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.


 * Data Tables:**




 * Sample Calculations:**




 * 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?
 * The slope of the line on the displacement vs. force graph represents the spring constant. Because the line is linear (constant slope), we can conclude that the spring constant is indeed constant.
 * 1) Why is the time for more than one period measured?
 * We do this because it is difficult to measure the time for one period (very small time increment). We can more accurately measure the time for one period by taking the average of five periods.
 * 1) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * 2) The distance vs. force graph is more accurate because it lets us see the direct relationship between force and distance allowing us to find k. The mass vs. period graph is less accurate because we had to use a hand timer to measure the amount of time, which allows for a percent error and it doesn't directly give us the spring force constant.
 * 3) Generate the equations and the corresponding graphs for
 * 4) position with respect to time.
 * 5) velocity with respect to time.
 * 6) [[image:Photo_100.jpg]]
 * 7) acceleration with respect to time.
 * 8) [[image:Photo_1.jpg]]
 * 9) 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?
 * 10) [[image:Screen_shot_2012-05-01_at_6.31.58_PM.png width="162" height="95"]]
 * 11) 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?
 * 12) [[image:Screen_shot_2012-05-01_at_6.28.48_PM.png width="138" height="167"]]
 * 13) We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship [[image:http://honorsphysicsrocks.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]] (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using [[image:http://honorsphysicsrocks.wikispaces.com/site/embedthumbnail/placeholder?w=200&h=50 width="200" height="50"]], and explain these results.

Through this experiment, we were able to identify the relationship between the period of oscillation and mass. We observed that as you increase the mass on a spring, you also increase the period. This is not a linear relationship, however, as we had to use a power-fit line on our graph. With regards to our line, we could find the spring force contant (k) from the slope. We could derive a lot of information about the spring itself and the oscillation of the mass through this simple experiment. Our data itself was precise, with an R2 value of .988. However, our calculations did not factor in the mass of the spring. This could have contributed to the error in our experiment. Also, using a stopwatch is not the best method for keeping track of time, since human error will always be a factor. If we were to do this experiment again, we could use a motion sensor underneath the spring, which would directly measure distance over time, and would improve the accuracy of our results. Overall, our experiment yielded good results, but there was room for improvement. This sort of testing occurs regularly at amusement parks. For example, the Oakwood Park has a ride called The Bounce. (http://farm4.staticflickr.com/3009/2771613826_1a4b1687e2_z.jpg). This ride simulates bouncing up and down on a spring. From our results, we know that increasing the mass, or adding more riders, will alter the period of oscillation. Therefore, the computers that control this ride must take into account all of the factors that we measured in our experiment, in order to properly simulate oscillating motion. More or less force must be exerted by the brakes depending on the mass of the riders, and knowing when to engage the brakes would depend on the period of oscillation.
 * Conclusion**

= 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: We hypothesize that frequency and tension are inversely proportional, so as tension increases, frequency will decrease. We also predict that as frequency increases the harmonic number will increase as well. As the frequency increases, the wavelength will get smaller because interference should create more nodes.

Method and Materials: We attached a string to a generator on one side and a pulley with a mass on the other side. We used the generator to test different frequencies that altered the harmonic numbers and the amplitude of the waves.In order to increase the amplitude to its maximum, we used the tuning dials.

Data Tables: Velocity m/L Frequency and Percent Error Percent Error of Exponents
 * Analysis and Sample Calcs-**
 * Tension**


 * 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? 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? 5. What factors affect the number of nodes in a standing wave?
 * [[image:Screen_shot_2012-05-12_at_3.34.57_PM.png width="236" height="254"]]
 * As the tension increased, stretching the string, the velocity of the wave would increase. If we look at the equation freq.=wavelength*wave speed, we can see that as the velocity increased so would the harmonic number. THis would have changed the data because in order to get the same harmonic number of waves that we used in the lab, we would've had to use lower frequencies.
 * No two strings are the same, so each new string will have a different widths and other differences. Just the different widths of the strings will change our results because it will alter the frequencies that they resonate at. Because the mass increases the tension, it will also alter the frequency that the string resonates at.
 * Frequency and the number of nodes are directly related. As frequency increases, the number of nodes increases.
 * Frequency, the fundamental frequency, and the number of anti-nodes.

Conclusion:

Unfortunately, the majority of our hypothesis was proved false by this experiment. First, we hypothesized that as tension increases, frequency decreases. However, our graphs clearly show that increasing tension led to an increase in frequency. As we added mass to the hanger, we also had to dial up the frequency on our sine wave generator in order to maintain the same number of anti nodes. We were correct with our second hypothesis, which was that as harmonic number increases, so does frequency. This was evident because when we would dial up the sine wave generator's frequency, we would form more and more anti nodes.

Our third prediction was backwards; rather than the biggest wavelength occurring at the lowest frequency, the biggest wavelength was at the highest frequency. With regard to error, all of our % errors were low, all below 10%. The sine wave generator did not operate perfectly, and may have been the source of our error. Rather than solely moving the string up and down as it should have, it also move side to side. Our string was creating a 3 dimensional tube-like shape, rather than a 2 dimensional wave form. Also, measuring wavelength required us to physically measure, which introduced human error. To fix these sources of error, we could have used a better sine wave generator, and fixed the ruler down to the table.

This lab has real-world applications as well. When movie theaters are built, they need to test their speaker systems in order to place them all in the correct place. Failing to do so would create unwanted interference, with spots of constructive interference where the movie would be much louder, and other of destructive interference where sounds would not be heard. Using their own more advanced instruments, they measure the waves generated by the speakers, similar to the waves generated in our lab. This allows them to find the perfect positions for the speakers in the room to minimize interference.