Group4_6_ch11

= = toc

**What is the relationship between the mass on a spring and its period of oscillation? **
By: Molly, Joey, Magna, and Sarah

**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:**
 * By finding the value for k using a force distance graph, we will see that it is the almost exactly the same as we measuered using the oscillations and graphing them.

**Equipment List:** Springs, tape, clamps and rods, masses, balance, timers, meter stick. First we set up a rod with a spring on it, and the called the point where it hung equilibium. Then we added masses to the spring to find the displacement that each mass caused. Using those displacements we found the force and then graphed the data points. Using Excel we found the spring constant value, which was the slope of the line. To do the second part of the lab we used the same set up, except this time we pulled the mass downwards and recorded the amount of time it took for it to oscillate ten times. Once we recorded the oscillation times we divided it by ten to get the period. By making another graph which is period by mass we found the k value a different way.
 * Methods and Materials:**

**Calculations**  

Yes it does. Our spring force constant for the displacement was about 3.5 while the constant for the applied force was 3.8. Our low percent error of 10% indicates that the k or constant is indeed constant. We also did not take into account errors in timing which created this error.
 * 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? **

We recorded the time for 10 oscillations in order to limit the possibility for error. We took the total time and divided by 10 to get a more accurate period.
 * 2. Why is the time for more than one period measured? **

Both graphs are very similar in that they both show a spring force constant that is very similar. However we think that the force vs. distance graph is more accurate because it doesn't involve calculating the period. Calculating for the period leaves more room for human error, than just simply reading a ruler.
 * 3. Discuss the agreement between the k values derived from the two graphs. Which is more accurate? **


 * 4. Generate the equations and the corresponding graphs for**
 * position with respect to time.

x=Acos(wt) x=Acos(2πft) A is the amplitued, 2πt takes over for w, since f=(1/2π)√(k/m). (2πf)^2 = k/m, which then gets plugged in to make the graph.

This problem can be solved using Hooke’s Law, which is the equation we used to find "k" directly, as long as we assume the spring is at equilibrium: F = - k * x F = - (8.75 N/m) * (- .150 m ) F = 1.3125 N
 * 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? **


 * <span style="font-family: Arial,Helvetica,sans-serif;">5. 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? **
 * <span style="font-family: Arial,Helvetica,sans-serif;">6. We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship [[image:honorsphysicsrocks/Screen_shot_2012-05-01_at_12.56.27_PM.png width="66" height="27"]] (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using [[image:honorsphysicsrocks/Screen_shot_2012-05-01_at_12.56.32_PM.png width="74" height="30"]]and explain these results. **



The slope of this graph and the other period vs mass graph are used to find the k value. From this graph we can conclude that our k value is 3.67 and from the original graph we can conclude our k is 3.83. When we compare the results from this graph to our theoretical k value of 3.4771, we get a lower percent error of 5.55%. Therefore we can conclude the results are better when using the more accurate relationship.

__**Conclusion:**__ For this lab our job was to determine the spring force constant or (k) for a specific applied force. We also had to 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. We then compared the results. We believed that by finding the value for k using a force distance graph, we will see that it is the almost exactly the same as we measuered using the oscillations and graphing them. We found that our hypothesis was true. Using the two methods we were able to compare the two values getting pretty close values thus proving our hypothesis.

After we collected the k value both ways we compared the two values using percent error. For the first test we used a distance vs. force graph and the k value was the slope of the line (3.4771). For the second test we used a Period vs. Mass graph, which produced a slope of 3.2093. We then plugged this into the equation. We used the values to solve for a k of (3.833). We compared the two values and got a reasonable percent error at 10.2%. This in my opinion is a respectable percent error. There were a few places in the experiment that we can list as the main sources of error. On the first test we added masses to a holder and measured the amount of stretch created. Getting the exact location is difficult and it is likely that the results we recorded were slightly off. For the second test we used a stop watch to record 10 oscillations of the spring and then we determined the period. This improves the chance of getting a good period, however it is not completely error free as each persons reaction time still plays a significant role in the trial. These are the two most prominent sources of error, which if fixed would significantly reduce the resulting percent error. We could try to fix them by having multiple people record the times and distances and then take the average of those values to be used. We could also have ran more than 1 trial at each mass to lessen the chance for error.

Situations like this occur in various points throughout our lives. One example would be a pull down attic roof hatch. The spring holds the ladder portion tight against the door when at rest. When the ladder is straightened out the spring hold the ladder to the door allowing you to climb up. If the ladder company doesn't test the spring and determine its maximum support the ladder would collapse if a heavy person were to walk on it. Concepts like this are incorporated all around us.

= Lab: Transverse Standing Waves on a String = By: Molly, Joe, and Sarah

**Objective**: 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?

1. We believe that as Tension increases so does frequency, they form a direct relationship. 2. We believe that as Frequency increase so will harmonic number, they form a direct relationship 3. We believe that as frequency increases the wavelength will decrease, they form an inverse relationship.
 * Hypothesis: **



After collecting the materials, we connected the two metal poles to the clamps and secured them to the table. We then attached the plastic pulley wheel to one side securing it firmly. On the other side we attached the electronic oscillator and an electric monitor. We connected the wires from the monitor screen to oscillator and then to the power supply. We then retrieved several masses and/or hangers. We ran the string from the oscillator to the pulley where the masses were then attached. Then we turned on the oscillator and began the laboratory.
 * Materials and Methods: **



%Error of power on frequency vs tension graph: Fundamental frequency: Percent error of fundamental frequency on frequency vs harmonic number graph: Velocity: Percent Error of velocity on frequency vs wavelength graph: Percent error of power on frequency vs wavelength graph:
 * Frequency vs. Tension Calculations**

1. Calculate the tension T that would be required to produce the n = 1 standing wave for the red braided string.
 * Discussion Questions: **



2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? As the string stretches the tension of the string increases which would also mean an increased velocity. The increased velocity would influence the harmonic number at a given frequency. If the string had been stretchy the results for this lab would have been rather different.

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. The type of string certainly plays a role in the lab. We all selected the same string to minimize this. We did this so we can compare results; because if we had all selected different strings the properties of each string would have effected the results and made it harder to compare results. As we increase the amount of mass that is hanging from the string the tension increases, this creates a greater velocity which results in more nodes.

4. What is the effect of changing frequency on the number of nodes? As the frequency increases the string vibrates more which increases the number of nodes.

5. What factors affect the number of nodes in a standing wave? Frequency is the main factor that affects the number of nodes, it also affects the harmonic number. Thus factors such as string length, tension, or even the string elasticity will change the number of nodes.

**Conclusion:**

In this lab we are trying to determine the relationship between Frequency and tension, harmonic number and wavelength for transverse waves traveling in a stretched string. For the first relationship, frequency vs tension, we hypothesized that this would be a direct relationship because as tension increases so does frequency. Although it is true that as one increases so does the other, it is not in a direct relationship. Instead, our graph of f vs T was that of a power fit. As you look at the equation to find the frequency, the tension is to the ½ power (square root), so because of this the exponent on the graph should be .5. However, it is actually .5043, so by finding the percent error we can see that there is only a .86% error.

The second relationship is frequency vs harmonic number. We hypothesized that this would have a direct relationship and we were right. This is because the equation for frequency using harmonic number is f=f 1 (n). In this case our coefficient is 9.9089 which is the fundamental frequency, or the frequency of one antinode. However our calculated value of the fundamental frequency is 9.95. By taking the percent error we can find that there is only a .413 percent error. Our R squared value is also .99973, which considering how close that is to one, shows that these results are statistically significant.

Our last relationship was between frequency and wavelength. We hypothesized that this would be an inverse relationship and we were correct. For the equation of our graph, the coefficient shows the velocity/ wave speed, which was 60.94. However, our calculated value for velocity is 61.3. This results in a 1.6% error. Also the power on this graph should theoretically be -1 because it is an inverse relationship. However, we got an exponent of -.941. This results in a percent error of 5.9%

There was a very small amount of error in the first (frequency vs tension) and second graph (frequency vs harmonic number). This percent error could have been from the machine because we noticed that the string looked as if it could have also been rotating instead of just going up and down like it should have. Also, we may thought we found the right frequency but were off by a couple decimal places. However, due to the high R squared values and the very low percent errors, it shows that the results that we got for this lab are statistically significant and did not just happened due to chance. The last graph had a slightly higher percent error. This is most likely because this involved measuring the wavelength. Because the string was constantly moving, it was difficult to measure the wavelength and because of this there was a higher chance of human error than in the other two parts of the lab.

If I were to change this lab to account for the errors, I would tape a ruler down underneath the string. I would also use some type of marking device (even something like a paper clip) to use as an indicator for where the nodes are so we could more easily measure the wavelength. Even though this would probably interrupt the movement of the string, it would make it much easier to measure. However, I would not change much else because we really didn't have very high percent errors. This concept of waves is obviously very important in our lives because many substances move as waves. One of the most important of these would probably be light. We can use labs like this that use a string so we can visualize what is happening to waves that we cannot see.

=Lab: Speed of Sound-Resonance Tube= 5/15/12 By: Sarah, Magna, Joey, Molly Period 6


 * 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:** We will be able to find the lengths at which resonance occurs at a certain frequency. We believe that as length increases, the resonance will too.

For this lab we are using a tube and a speaker. To find the lengths at which the sound is the loudest we are increasing the length of the tube by pulling it outwards. As we pull the tube the sound oscillates, which represents the nodes and anitnodes of the waves. We are using both a closed and open tube. In order to create the closed tube we are using a cap.
 * Methods and Materials:**
 * Data:**
 * Graph:**


 * Sample Calculations:**
 * Analysis:**


 * Open Tube**


 * Closed Tube**

>> For this lab we created one hypothesis. We believed that as length increases, the resonance will too. We found this hypothesis to be true. We believe that it is true because the slope of our graph was positive and linear thus we concluded the two formed a direct relationship.
 * Discussion Questions:**
 * 1) **What is the meaning of the slope of the graph for the open tube? For the closed tube?**
 * <span style="font-family: Arial,Helvetica,sans-serif; font-size: 90%;">For the open tube, the slope is equal to one half of the wavelength. The equation for the length of an open tube is L=(1/2λ)n, and the equation of the line is y=mx. Because L is being graphed on the y-axis on n on the x-axis, m, or the slope, is equal to 1/2λ, 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. Because the equation for the length of an open tube is L=(1/4λ)n, and the equation of the line is y=mx. Because L is being graphed on the y-axis on n on the x-axis, m, or the slope, is equal to 1/4λ, 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?**
 * Because when we calculated the theoretical values it was based off of an equation that is not valid in a real life situation. So we had to adjust our answer by taking into account the end shift.
 * 1) **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?**
 * The length of the open tube when the room was 27 ˚C was 0.3477 meters and the length of the closed tube was 0.17385 meters. By increasing the temperature by ten degrees the speed of sound would increase to 353.7 m/s. This would then cause the wavelength to increase to 0.7074. Plugging this into the equation for the length of an open tube the length would be 0.3537, an increase of 0.006. For the closed tube the new length would be 0.17685, an increase of 0.003.
 * 1) **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.**
 * |( the line at the left end is the closing, the parentheses are quarters of a wave. The length of the tube is related to the wavelength because at the 5th harmonic, there are 5/4 of a wave fitting in the tube so the length of the tube is 5/4lambda.
 * 1) **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.**
 * )( This shows 5 half waves in the open tube, therefore there are 5/2 wavelengths in the tube, so the length of the tube is 5/2lambda
 * 1) **What does this have to do with making music?**
 * The workings of woodwind instruments have a lot to do with these concepts. Musicians are able to play with the length of the tube or instrument as well as changing it from an open to a closed end. This allows for different wavelengths and results in differing frequencies. The variation creates the different notes, which in turn creates music.
 * Conclusion:**

Our error results are pretty good for this lab. For the open tube we had a good percent error of 6.4%. For the closed tube we also had a low percent error of 4.5%. I was not surprised by the low percent errors because we took several steps to minimize error as much possible. We took several trials and then averaged them to get the most accurate results possible. The steps we took we successful because we were able to conduct the experiment well retaining accuracy throughout.

There were several possible sources of error. One of the most significant sources of error was the measurement of the tube. It was easy to have some variation in measurement, which would have caused some difference in our results. Another source error is that we misheard the high point and then incorrectly measured the location based on that false position. These are two sources of error that definitely affected our results. We could have improved the results for this lab by having multiple people measuring the location. We could also have had multiple tubes and tried the lab in different rooms and then averaged those results.

One of the major real life relations with this lab would be musical instruments especially woodwinds. When the musician blows into the instrument a vibration is created. The sound travels through the instrument. The sound of the note is the determined by the keys that are blocked/open. This is true for most musical instruments from guitars to woodwinds.