Group2_2_ch11

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=Period of a Mass on a Spring Lab=

Lab: What is the relationship between the mass on a spring and its period of oscillation? toc Amanda, Michael, Ben, Kaila


 * 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 slope of a distance vs. force graph with varying masses will determine the spring force constant.
 * The period and mass should be directly related, therefore as one increases, so does the other with k remaining constant.
 * This spring force constant found using distance vs. force graph should be the same as the spring force constant found using oscillations with varying masses.


 * Materials and Procedures:**
 * First, we measured the spring force constant by measuring the length of the spring's elongation with a ruler when specific masses were added. We did this by placing a spring on a clamp/rod and placing a first mass on, once at equilibrium, we determined this position to be zero. We then added five separate masses, determined their respective positions with a ruler, and graphed them on a distance vs. force graph. The slope of the trend-line determined the spring force constant (k) for this spring. Next, we found the spring force constant by finding the period at different masses. We used the same spring and clamp/rod set up, though with the added masses, we pulled the mass downwards and allowed it to oscillate for ten cycles. After finding the time it took to complete ten oscillations with a stopwatch, we then recorded these times and divided them by 10 to get the period time. We then graphed the different data points on a period vs. mass graph, where the slope of the line represents the spring force constant.


 * Data and Graphs:**
 * Test 1:**


 * Test 2:**

Mass spring = 4.5 grams


 * Calculations**:

Method one: Hooke's Law: F=-kx When substituted for our graph: y = 12.601x Thus, k = 12.601 N/m
 * Discussion Questions:**
 * 1) Do the data for the displacement of the spring versus applied force indicate that the data for the spring constant is indeed constant for this range of forces?
 * There is a good linear fit and the value on the graph for the r 2 is .97769 which is large. Since the data points all follow the line, there is evidence that the spring constant is indeed constant.
 * 1) Why is the time for more than one period measured?
 * It is more accurate this way. It can be very difficult to get a correct and precise measurement when the time for the period is so small. Since we are using stopwatches, there is a certain degree of human error involved, so measuring more than one period and than dividing by the number of periods measured is a much more accurate method. We chose to take 10 periods and then divide that number by 10 to get the result for one period.
 * 1) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * In each of the cases, we got good results. There was evidence that the spring force constant was indeed constant for each method. The first graph, where we compared the force applied to the distance, is the more accurate method. We were using stopwatches to get the data for the second graphs, and because of human reaction time, it can be difficult to get perfect times.
 * 1) Generate the equations and the corresponding graphs for position with respect to time.
 * x = Acos(2(pi)f)t
 * 1) 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?
 * 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 the 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 m+1/3m (where m is the hanging mass and mx is the mass of the spring)? Redo graph #2 using sqroot m+1/3m, and explain these results.

By taking the mass of the spring into account, the results were much better. The new calculation of the spring force constant came out to be 13.005, which gave us a new percent error of 1.559%.

For this lab, we attempted to calculate k (the spring force constant) in two different ways. First, we used a spring upon which we added masses and then measured the distance the spring stretched with a ruler. By doing so, we could graph force vs. distance, from which we could determine k. Next, we pulled the spring down and allowed it to oscillate ten complete times. We took the time it took the spring to do this by using a stopwatch and then found the period of the spring's oscillation with different masses attached. We then graphed out period vs. mass and used known equations for period to find the spring constant again. We then compared the two values to prove whether or not they were identical.
 * Conclusion**:

The results of the experiment proved many of our hypotheses to be true. By finding the spring force constant by first finding the slope of the force vs. distance graph and using the equations of the period vs. mass graph and period itself, we found values with low percent difference. This demonstrates that relatively accurate calculations were obtained from our methods, proving that the spring force constant COULD be accurately determined from the way we performed the experiment. At the same time, though the two calculated values might not be exactly identical, the low percent difference for each demonstrates that in an ideal lab situation, the same value could be calculated twice.

To be more specific, we found percent difference between the spring force constants found through the position vs. force method/oscillation method to the average of the two. This was done instead of finding percent error because using percent difference determined whether or not our third hypothesis was true. We calculated a percent difference of 4.62% for the position vs. force method and a percent difference of 4.61% for the oscillation method. We also calculated percent error for the exponent we calculated from the period vs. mass graph (the theoretical value being .5) as a way of further demonstrating the validity of our results. The percent error for that was 4.76%.

There were various sources of error that could have increased the error in this experiment. First (in terms of the position vs. force method), we used a measuring stick that could not measure in hundredths of meters. Because of this, we were forced to estimate the lengths that the pendulum was stretched, which potentially could have given us measurements we might not have otherwise gotten. For the oscillation method, a potential source of error could have been the use of a stopwatch to measure the amount of time it took the spring to oscillate ten times. Because of the quick movement of the spring, it was extremely difficult to coordinate the spring exactly with the stopwatch; thus, the calculated periods could have been smaller or greater than the actual value. Therefore, our average period could be off, giving our results more error. To decrease the amount of error in this experiment, we could use a measuring tool that measures in hundredths of meters and use some sort of a device that could better find the exact time for ten oscillations than the stopwatch we used.

There are various examples of masses on springs oscillating in the real world. For example, a bungee jumper oscillates after he makes his initial jump. The spring force constant for the bungee cord could be measured by using similar methods to the ones that were employed in this lab.

=Standing Waves Lab=
 * Objectives:** To determine the relationship between frequency and the tension of transverse waves traveling in a stretched string, the relationship between frequency and harmonic number, and the relationship between frequency and wavelength.


 * Hypotheses:** If the relationship between frequency and the tension is found, then it will be found to be a power relationship. If the relationship between frequency and harmonic number is found, then it will be found to be a linear relationship. If the relationship between frequency and the wavelength is found, then it will be found to be a power relationship. We believe these are the relationships based on equations that we have been given in class, which we have manipulated to show the relationship between certain variables.


 * Materials and Methods:** To determine these three relationships, we first set up a 25 yard string on an electronic oscillator that was positioned on a table by a pulley and a clamp. Then, we hung several weights from the end of the string that made its way over the pulley. From here, we ran several different tests during which we changed different variables. First, we kept the hanging mass and the linear density of the string constant and changed the harmonic number, frequency, wavelength, and wave speed to determine the relationship between frequency and harmonic number and the relationship between frequency and wavelength. Then, we ran a test in which we kept harmonic number and linear density constant and changed the hanging mass (which changed frequency and tension) to determine the relationship between frequency and tension. Once the data for these tests were collected, we set up data tables which were graphed on Microsoft Excel. We then found the equations of these graphs to determine the relationship between the various factors.


 * Data and Calculations:**

Analysis
 * Note: Percent "difference" should instead be percent error, that is a typo

Theoretical Fundamental Frequency: Percent Error (Fundamental Frequency):


 * The exponent should ideally be .5 because a power of 1/2 is equal to the square root of a value (in this case the force of tension).

Percent Error: Exponent: Wavelength (5 nodes):

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? Had the tension of the string increased, the velocity of the string would increase as well based on the equation v = square Root (T/(M/l)). Had the string stretched, it would have increased its length, which would make the string density smaller, and thus further raise the velocity. Because velocity equals frequency times wavelength, the frequency would have thus also increased.

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. Different strings have different linear densities, which has an effect on the hanging mass required to create a set number of nodes. Based on the equation v = square Root (T/(M/l)) as the density of the string increases, so must the tension, or the weight of the hanging mass in order to counter balance its effect in the denominator of the equation. The amount of nodes (represented by the harmonic number) is equal to 2L/wavelength. So if the string is stretched because of its makeup that would change its length, which would thus change the number of nodes. It is inherent that different strings will have different densities, masses, and amounts of stretch among other factors, which will all require different amounts of frequencies to achieve different node values. We used one type of string to take these instances of variability out of our lab.

4. What is the effect of changing frequency on the number of nodes? By increasing the frequency, we also increase the number of nodes, and thus the harmonic number. Ideally, they should increase in a linear fashion and their frequency should equal the frequency for the first harmonic times the number of harmonics.

5. What factors affect the number of nodes in a standing wave? Many factors affect the nodes on a standing wave. These factors include frequency, tension or hanging mass, the density and length as well as other factors of the string used will also have an effect on the number of nodes.

**Conclusion:** In this experiment, we attempted to determine the relationship between various qualities of a wave. More specifically, we attempted to find the relationship between frequency and tension force, frequency and harmonic number, and frequency and wavelength. To do this, we turned on an electric oscillator and changed several variables. The resulting experimental data was graphed on Excel, where we could then determine the type of relationship each was.

The data that we collected supported our hypotheses for each relationship. The relationship between frequency and tension is a power relationship, the one between frequency and harmonic number is linear, and the one between frequency and wavelength is power. We made sure this was true (by getting such equations from the data and then determining the r2 value). Our results were very consistent, as our r2 values were very high. The r2 values for each graph were above .99, showing that each set of values had a strong fit to the best fitting line or curve. To further gauge our accuracy, we found percent error for experimental values given to us from the data when compared to their theoretical values. These included the velocity from the frequency vs. wavelength graph, the fundamental frequency from the frequency vs. harmonic number graph, as well as the exponent and the wavelength for 5 nodes from the frequency vs. tension graph. These percent errors ranged from 0-11%. All the error of our results can be attributed to various possible sources of error.

There were a few sources of error in this lab. One was that our string was set up so that it was slightly tilted. While it was not much and most likely did not have a significant impact, the slant could have changed some data in very tiny amounts. The only way to try and fix this would be to redo the whole set up (we were unable to do this because the string was set up for us by another lab group and we had to leave this setup for another lab group who would be doing the experiment after us). Another source of error is that it is difficult to find the exact frequency for each trial. We are able to get to a point where it is very close, but it is hard to find the perfect value that it should be at. This is because the smallest increment that the frequency can be increased or decreased is by a tenth. To decrease this source of error, we could either use an electronic oscillator that can change frequency in smaller units (like hundredths) or perform multiple trials. This lab allowed us to visualize waves in a way that is difficult to do as well in an uncontrolled situation. It put frequency and how to calculate frequency into easy to see terms.

Standing waves can be observed in the real world and not just in an experimental situation. For example, they can be observed in musical instruments like guitars that make their music through the vibration of strings.