Group1_4_ch11

Lab group 12 Period 4toc

Stephanie Wang, Jonathan Itskovitch, Hella Talas

=Lab: What is the relationship between a mass on a spring and its period of oscillation?= Task A: Stephanie Wang Task B: Stephanie Wang Task C: Jonathan Itskovitch Task C: Hella Talas


 * 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**: Since we will be using the same spring for both parts of the experiment, and the spring force constant doesn't change, the k value should be pretty similar for both parts of the experiments, regardless of the mass on the end of the spring. However, as the mass increases, the period should also increase. As the mass decreases, the period of oscillation should also decrease. However, the k value will still remain constant.

In the lab, we used two different methods to determine that the spring constant was constant, no matter how it is found. In the first procedure, we placed a spring on a spring stand. Ignoring the mass of the spring, we added masses to the spring, and calculated the displacement of the spring using a ruler. We used 5 masses. This allowed us to create a force v displacement graph on Excel and determine the average K value just by seeing the slope. In the second procedure, we allowed the spring to oscillate. Using a timer, we timed the period of 10 oscillations. This allowed us to find the period of 1 oscillation. We did 5 trials for each of the 5 masses. Using the formula for the period of a spring, we could calculate the spring constant. We then used the data to make a mass v period graph, and what resulted was a square root graph, because of the known square root relationship between period and mass in the equation.
 * Methods and Materials:**

Part I: Part II:
 * Pictures:**

Data Table: Hooke's Law Graph: Hooke's Law Data Table: Periods of Oscillation Graph: Periods of Oscillation
 * Data Tables & Graphs:**


 * Sample Calculations**:


 * Analysis**:
 * 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) Yes, since there is shown a linear relationship between the displacement and force applied. As force increases, displacement increases at the same rate. If this is put onto a graph, the slope is measured in N/m, which is the measure of the spring constant. This linear slope demonstrates that the spring constant is indeed constant.
 * 3) Why is the time for more than one period measured?
 * 4) We measured the time of 10 oscillations because by doing so we got a more accurate result. If we are to calculate just one oscillation, the reaction time to stop the timer has a larger impact on the result. By doing 10 times, we can find the average period and use a more accurate result in the data.
 * 5) Discuss the agreement between the k values derived from the two graphs. Which is more accurate?
 * 6) Derived from the graph itself, for the first method, we got a K of 4.00. For the second graph we got a K of 4.07. This indeed demonstrates that the 2 methods we did, applying masses and timing oscillations, prove that K is a constant ALWAYS. I think that the first method is more accurate because this result is closer to the actual data recorded. Also, keeping the spring still and accurate measuring is more accurate than an estimated time period (thrown off by timer issues).
 * 7) Generate the position with respect to time equation and the corresponding graph for x=0.32cos15.29t
 * 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 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 (where //m// is the hanging mass and //ms// is the mass of the spring)? Redo graph #2 using [[image:Screen_shot_2012-05-04_at_2.15.33_PM.png width="72" height="43"]], and explain these results.

The results were improved slightly. We achieved an exact 0.5 for the exponent. However, our k value was increased to about 6. We already had great results, so we didn’t need much improvement, but including this mass certainly helped making our results even better.

The purpose of the lab was to figure out the spring force constant using several methods. Our hypothesis states that no matter which way we calculate k, it should remain pretty similar. Fortunately, our lab results proved the hypothesis. We were able to achieve results that were very accurate and precise. The first method, using conservation of energy with the equation F=kx, we found a k value of 3.993. This may sound like a low value, but our spring was very flimsy. Through our 5 trials, we found that our biggest percent difference was a mere 0.586%. Our graph had an r^2 value of over 0.99. In the second method, our results still were excellent. Our average k value was 4.030, which is relatively close to 3.993. Our largest percent difference was 1.67%, proving that our results are close to the theoretical. Furthermore, the graph we obtained from the second graph was nearly a perfect square root, with an exponent of 0.4988. The percent error was only 0.24%. Despite our excellent results, sources of error may have prevented us from attaining perfect results. This was mostly due to the equipment we used, which do not provide exact results. For example, we had to use a stopwatch, which for everyone there is a difference reaction time to stop the stopwatch after the 10 oscillations. Using a motion detector from the Pasco and Data Studio would have greatly helped with our results. Error throughout this experiment could have derived from our lack of exact instruments. We had to eyeball some of our measurements due to the constant motion of the springs, which could have slightly skewed our results. We also were forced to use the stopwatch, which could have been slightly off considering that it didn’t have a motion detector attached that could send signals to stop it, and therefore make results more accurate. We did try to avoid error here though by making our period 5 oscillations, although some error should still be accounted for. If we used a motion detector, this would have greatly helped us in eliminating a great source of error throughout the experiment. Another source of error is derived from the spring itself. The spring may not give what it used to, and it may be damaged in some areas. This would throw off results greatly. Thankfully, our spring was not damaged. One real life application of the spring would be a shock-absorber in a car. When there is an accident, the car has a system whereby the springs give and there is a period where the spring compresses. This relieves the effect of the actual crash. Now that we know that mass and spring constant have a direct impact on the period of the spring, it is important to design the spring so there is a longer period. Longer periods help to stop the effect of the crash.
 * Conclusion:**

=Transverse Standing Waves on a String =

Task A: Matt Ordover Task B: Hella Talas Task C: Maxx Grunfeld Task D: Stephanie Wang

Objectives:

1) What is the relationship between Frequency and the tension of transverse waves traveling in a stretched string? 2) What is the relationship between frequency and harmonic number? 3) What is the relationship between frequency and wavelength?

Hypotheses:


 * Since frequency and tension are indirectly related, as frequency increases, the tension should decrease. Also, as the tension increases, the frequency should decrease.
 * Since frequency of a wave and its harmonic number are directly proportional, as one increases, so should the other. As one decreases, the other will also decrease.
 * The relationship between frequency and wavelength can be described with a power fit graph. As the frequency increases, the wavelength will decrease.

Procedure: A string was attached to a pulley and mass on one end and an electronic oscillator on the other end. We then did two different experiments with this set up. Part 1: We changed the mass for each trial and found the frequency for max amplitude of a certain harmonic number. We chose the harmonic number two, so we changed the masses and found which frequency gave the biggest amplitude of a standing wave with a harmonic number of two. This gave us the first relationship that we were looking for.

Part 2: We kept the mass constant this time and found which frequencies gave the max amplitude for various harmonic numbers. This gave us the second and third relationships that we were looking for.



Graphs:





 Data Tables

Part 1:



Part 2:

Sample Calculations:

<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;"> <span style="font-family: Verdana,Geneva,sans-serif;"> <span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;"> <span style="font-family: Verdana,Geneva,sans-serif;">

<span style="font-family: Verdana,Geneva,sans-serif;">Analysis:
 * 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?** If the string stretched significantly as the tension increased, it would have affected the data in several ways. An increase in tension would also result in an increase in wave speed. This can be seen in the equation for wave speed. Additionally, the harmonic number would change. This is because harmonic number is related to frequency, and frequency is related to wave speed. Therefore, if the string stretched as the tension in creased, many other variables would change.

<span style="font-family: Arial,Helvetica,sans-serif;">The type of string could really change in this lab. If a string is stretchy and not rough, it would affect the results of the lab because the tension would be different for other masses, resulting in different velocities. In addition, the velocities were changed, the number of nodes at frequencies would be changed also.
 * <span style="font-family: Arial,Helvetica,sans-serif;">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. **

<span style="font-family: Arial,Helvetica,sans-serif;">When the frequency increases, the string is vibrates more which increases the number of nodes because there are more waves being sent through the string.
 * <span style="font-family: Arial,Helvetica,sans-serif;">4. What is the effect of changing frequency on the number of nodes? **

<span style="font-family: Arial,Helvetica,sans-serif;">The frequency, string length, tension, elasticity, and mass per unit length all affect the number of nodes.
 * <span style="font-family: Arial,Helvetica,sans-serif;">5. What factors affect the number of nodes in a standing wave? **

<span style="font-family: Verdana,Geneva,sans-serif;">Conclusion: Our hypotheses were correct. We found out that frequency and tension are indirectly related. As we increased the tension, the frequency decreased. This relationship was represented with a power fit graph. We also found that as the frequency increased, the harmonic number of the wave also increased. This resulted in a linear graph. Also, we used a power fit graph to find the relationship between wavelength and frequency. As the frequency increased, the wavelength decreased.

Our results were excellent. For part 1, we got a percent error of 9.02%. We also got a percent error of 0.9% for part 2. As for percent difference, we got 6.95% for the wave speed, 6.96% for the wavelength, and 1.53% for the fundamental frequency. This is very good, but it is obviously not perfect. This means that there were some sources of error. First, it was difficult to see what frequency gave the largest amplitude since we could not measure with an instrument and could only guess for the most part. To remedy this, we should do this experiment against a large sheet of graph paper. Then, we could tell for sure what frequency gave the largest amplitude. Also, the hanging mass was not very stable. It would constantly swing around, which would alter the results slightly. To fix this, we should put two pieces of cardboard next to the hanging mass so it would keep the mass in place without affecting its weight or tension. There are many real life applications of this lab. We see standing waves everyday. An example would be the vibration of a violin string. Also, other waves are present in our daily lives too. If waves didn't exist, we wouldn't even be able to hear sound!