Group5_2_ch11

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 * Dani Rubenstein, Jenna Malley, Ryan Hall, Andrew Chung**

=Lab: Spring Force Constant 5/2/12= What is the relationship between the mass on a spring and its period of oscillation?

- 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 the spring constant k
 * Objectives:**


 * Hypothesis:**
 * We hypothesize that spring force constant will be equal through both methods of determining it.
 * To find the value of K, we will both graph the force (gravity) versus the displacement, and period versus mass.
 * As mass increases, so should the period (and vice versa), but K should remain constant.

For this lab, we completed two procedures in order to compare two different results for the spring force constant. In the first method, we used a spring, a a meterstick, a spring stand, and different masses. First, we took the spring and attacked it to the stand. Then, we measured the distance that the spring was at during equilibrium in order to get our stretched distances later. We added on five different masses and each time recorded the displacement that it moved from its equilibrium position. Next, we recorded this data in a table on Microsoft Excel and create a Force vs. Displacement graph to get our spring force constant. We then moved on to method 2. In this method, we used a spring-stand, a spring, masses, and a stop watch. We attached the spring to the stand and added masses to the bottom. Then, we pulled down the spring in order to start its movement. We counted ten oscillations and and then divided by 10 to get the period. We took these results and created another table on Excel. We made a graph of Mass vs. Period and used this to find our spring force constant for the second method.
 * Methods and Materials:**

media type="file" key="method 1.mov" width="300" height="300"
 * Video of Method 1:**

media type="file" key="method 2.mov" width="300" height="300"
 * Video of Method 2:**


 * Data Tables:**
 * METHOD ONE**


 * METHOD TWO**
 * Graphs:**
 * METHOD ONE**
 * METHOD TWO**

Method One Method Two Percent Difference
 * Sample Calculations:**


 * Discussion Questions:**


 * 1.) Does 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?**

Yes, because not only are the two results of spring forces constants are close to equal, but the slope of the line of best fit, representing the spring force constant, for our displacement versus force graph is linear, meaning that the value of K is in fact constant.


 * 2.) Why is the time for more than one period measured?**

By measuring the time it took the spring to make 10 full periods and then dividing it by 10 to find a single oscillation, our results were more accurate and lessened the effects of human error. Moreover, we did 5 trials for each mass, to further eliminate possible human error.


 * 3.) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?**

For Method 1, our K value was 12.554, and for method 2, it was 11.603. We believe that the K value based on method 1 (using displacement) is more accurate, because for the second method, we got a wide range of periods for the same mass, which could have thrown our results for the spring force constant off.


 * 4.) Generate the equations and the corresponding graphs for**
 * 1) **position with respect to time.**
 * 2) X = 0.32cos15.29t
 * 3) **velocity with respect to time.**
 * 4) **acceleration with respect to time.**


 * 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?**
 * 6.) 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?**

The power that the x is raised to is better than in the previous graph, considering that it's closer to the desired value of .5. However, this gives us an 11.25 N/m for the spring force constant, which is lower than both of our previously calculated spring force constants but still in the right range.
 * 7.) The following is the graph and trend line for when the mass of the spring was accounted for:**


 * Conclusion:**

This lab was one of the more straightforward ones that we have. In it, we attempted to find the spring force constant using two various methods. This was done in an effort to see which one yielded better results and whether the two methods produced the same value. We hypothesized that our values would indeed be equal for each method and that the spring force constant should remain true to its name (constant). In the beginning, we started with various springs that we measured using rulers. We had to add masses to the springs using the materials listed in the aptly named section above and recorded all the oscillation data. After collecting all the data for this, which you can see above, we graphed various values. All of this is explained in the methods section above. Eventually, it was revealed that our methods did not yield exactly similar results, though they should have. We experienced a 7.6% error, getting a K of 12.554 N/M for method one and a K of 11.60 N/M for method two. We were thus able to conclude that the spring force constant remains the same for a range of forces. We also discovered that method one was a more accurate way to calculate the spring force constant. There were several sources of error for this experiment. For instance, while stretching the coils they could have been permanently damaged, leading to incorrect values for the spring force constant. Also, it was difficult to get excellent measurements because of the nature of the lab. With coils moving up and down it is hard to find actual values, which probably contributed to our percent difference. In order to change this in the future, we would use materials that perhaps allowed for more precise measurements to be taken. An application of this could be spring shoes. Basically, you add coils/springs to the bottom of shoes to jump up and down. Because we now know that weight affects oscillation, we will be able to reason why some people can jump higher than others with the springs.

=Lab: Transverse Standing Waves on a String=

- What is the relationship between frequency and the tension of transverse waves traveling in a stretched spring? - What is the relationship between frequency and harmonic number? - What is the relationship between frequency and wavelength?
 * Objective:**


 * Hypothesis:**

1. As the tension increases, the increase in frequency will increase by a power of 1/2. 2. As the harmonic number increases, the frequency will increase linearly. 3. As the wavelength increases, the frequency will decrease by a power of -1.


 * Methods and Materials:**

Materials: Electrically driven oscillator, pulley & table clamp assembly, weight holder & selection of masses, string, electronic balance.

The nice thing about this lab is that all the materials involved are relatively straight forward, at least compared to the plethora of labs that we have already completed. First, we set up a generator at one end of two lab tables set together. The generator was attached to a string that ran the length of the two tables put together, finally ending at a pulley on the opposite end of the table. There were several weights attached to the end of the string beneath the pulley. The oscillator is electric and thus allows us to change various variables on the system. Using the electric oscillator, we varied the frequency to get data about the wavelength, wave speed, mass and length of the mass plus the string (some of these values do not need to be calculated with the oscillator, we simply mean that this all played together to create a final data sheet).


 * Video:**

media type="file" key="Movie on 2012-05-09 at 08.36.mov" width="300" height="300"

Trial One:
 * Data:**

Trial Two:

Based on Trial One:
 * Graphs:**

Based on Trial Two:

__Wavelength (for 2 nodes with a Tension of 4.90 N):__ __Wave Speed (for 2 nodes, with a Tension of 4.90 N):__ __Fundamental Frequency (for 2 nodes, with a Tension of 4.90 N):__ __Between velocities:__ __Between wavelengths:__ __Between fundamental frequencies:__ __Between theoretical and experimental values for exponent in frequency v. Tension graph:__ __Between theoretical and experimental values for exponent in frequency v. wavelength graph:__
 * Sample Calculations:**
 * Percent Difference:**
 * 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 this have affected the data?
 * Several things would change if the string were stretched significantly as the tension increased. For one, an increase in tension results in an increase of velocity. We see this through the equation velocity is equal to the square root of tension divided by the (mass / length). Intuitively, we see that as tension increases velocity increases as well. Furthermore, the harmonic number would change as a result. This is because harmonic number is equal to the amount of nodes multiplied by the frequency. Frequency is related to velocity in that it is velocity divided by lambda, which is the wavelength. As such, the amount of harmonics would change.

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 plays a large effect on the amount of hanging mass needed to create a set number of nodes. Different strings could have different wave lengths, different frequencies, and different physical properties. As such, the values gained from a set of different strings would all be very different, making a group laboratory rather useless. It is for this reason that we all used the same string in our labs, to try and replicate similar results to other people in our class to effectively prove our hypothesis.

4. What is the effect of changing frequency on the number of nodes?
 * The harmonic number is equal to the amount of nodes multiplied by the frequency. Thus, we see a direct relationship between the changing frequency and the number of nodes. This means that as one increases, the other also increases and vice versa. This, of course, is if the length of the string is specified and doesn't change throughout the process, though even if the string length does change the relationship should be the same.

5. What factors affect the number of nodes in a standing wave?
 * A variety of factors affect the number of nodes in a standing wave. As we saw in the last discussion question, as the frequency increases the number of nodes also increases because they share a direct relationship. The harmonic number is also related to the number of nodes multiplied by the frequency, indicating that the harmonic number plays a part in the number of nodes in a standing wave. Velocity plays a part into the former equation, as it is equal to the square root of the tension force divided by the (mass/length). Since this plays into the equation velocity is equal to the frequency times the wavelength, it easily becomes clear that all of these factors affect the number of nodes in a standing wave.


 * Conclusion:**

Through this lab, we were able to answer three objectives that were given to us at the beginning. The first objective was to find the relationship between tension and frequency of a transverse wave. We hypothesized that as tension increased, the frequency would increase by a power of 1/2. The second objective was to find the relationship between frequency and harmonic number. We hypothesized that as the harmonic number increased, the frequency would increase linearly. Finally, the last objective was to was to find the relationship between frequency and wave length. We hypothesized that as wavelength increases, frequency will decrease by a power of -1. To find out whether our predictions were correct, we attached a string to an electrically driven oscillator and used this to gather the needed information. After completing the lab, we were able to see that our hypotheses were, in fact, correct. Additionally, the data we gathered was extremely precise, which can be seen between our percent error and percent differences. These calculations showed that our error ranged between 0 and 8.8%, thus showing that each result fell within a 10% error range. Although we received excellent results, there were some sources of error that contributed to our results. For example, human error definitely played a part in this lab. While twisting the nob to create different frequencies, it is often difficult for one to get the measurement that is exactly correct. Although it might look stable, the frequency could have been a few decimal places off from what it actually should have been. To fix this error, we could have solved for frequency before and simply set the machine to that number to get the exact result. In this lab, however, it was difficult to have many sources of error. The materials were simple, the task was easily completed, and there weren't many measurements that we had to take. Therefore, it was easy to get precise results which our group definitely did.