Amanda,+Allison,+Erica,+Steven

toc =**Lab: Transverse Standing Waves on a String**=
 * Group Members:** Amanda Donaldson, Allison Irwin, Erica Levine
 * Class:** Period 2
 * Date Completed:** May 23, 2011
 * Date Due:** May 24, 2011

__**Objective:**__ 1. What is the relationship between frequency and the tension of transverse standing 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?

__**Hypothesis:**__ 1, As tension increases, frequency increases as shown in the equation below. 2. As frequency increases, the harmonic number will increase. 3, As frequency increases, wavelength should decrease as demonstrated by the equation below.

For this lab, we used an electric oscillator (string vibrator and sine wave generator), pulley and table clamp, weights, string, and a meter stick.
 * __Materials:__**


 * __Procedure:__**
 * 1) Gather the above listed materials. Connect the string vibrator to the sine wave generator. Hang the string that is tied to the string vibrator over the pulley on the other end of the set up. Hang a large mass (1000 g) to the end of the pulley.
 * 2) Turn on sine wave generator and turn the frequency dials. Set the amplitude at max.
 * 3) Begin with one harmonic and record the measured frequency. Increase the harmonic number and find the frequency for each. At frequency, the string should be quiet. This will test the frequency and harmonic number.
 * 4) Simultaneously, once the frequency and harmonic number is determined, measure the wavelength in the string. At frequency, the string should be quiet. This will test the frequency and wavelength.
 * 5) To test the frequency and tension, change the tension hanging from the end of the string over the pulley. Adjust the frequency so that the string is quiet. Record this value.

__**Data:**__




 * __Graphs:__**



__**Discussion Questions:**__ 1. Calculate the tension T that would be required to produce the n=1 standing wave for the red braided string. The above calculations show how we calculated T if n = 1 in a standing wave.

2. What would be the effect if the string stretched significantly as the tension increased? How would that have affected the data? As shown by the equation v=√(elastic/inelastic), we can conclude that as the string stretches and tension increases, velocity also increases. An increase in velocity would affect the harmonic number at a particular frequency. This means that our results would be drastically different if the string was stretchy.

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. Each string has its own varying properties such as its elasticity and its natural frequency. Therefore, by changing the type of string, these variables change. As the hanging mass increases, so does the tension, and therefore the velocity is greater thus resulting in more nodes. Strings with a higher elastic property can handle more mass while still producing the same number of nodes.

4. What is the effect of changing frequency on the number of nodes? As the frequency increases, so does the number of nodes. There are more periods in the standing wave at a greater frequency displaying more nodes and antinodes.

5. What factors affect the number of nodes in a standing wave? Frequency affects the harmonic number. Therefore, anything such as length, elasticity, or tension, that changes this variable will also change the number of nodes.

__**Conclusion:**__ Based on the data we collected, we were able to confirm that our hypotheses were correct. First, we hypothesized that as frequency increased, tension would also increase based on a square root trend. The graph that our data produced, with an R2 value of .9987, and an exponential value of .44 strongly validates our hypothesis. This conclusion can be justified by the equation. The nature of this equation states that as tension increases, so will the velocity. An increase in velocity causes a direct increase in frequency. Second, we hypothesized that as frequency increased, the harmonic number would also increase in a linear fashion. Our graph produced from collected data has an R2 value of .9986, which validates our hypothesis. This makes sense based on the equation. As frequency increases, the wavelength will decrease based on the fact that velocity must stay constant. As a result of the wavelength decreasing with a greater frequency, there is room for more nodes, therefore increasing the harmonic number. This relationship is based off of the equation y=mx. The value of the slope in this equation represents the natural frequency of the string. Our value for natural frequency at the 1st harmonic was 27.7 Hz. Lastly, we hypothesized that as frequency increased, wavelength would decrease. Our inverse graph with an R2 value of .9976, and an exponential value of -1.01 validates our hypothesis. The slope represents the speed of the wave in the string. **verify velocity with V=root T/m/L equation.** f=v/lamda

=Lab: Period of Mass on a Spring =

 **Group Members:** Amanda Donaldson, Allison Irwin, Erica Levine, Steven Thorwarth **Class:** Period 2 **Date Completed:** May 16, 2011 **Date Due:** May 17, 2011

__**Objective:**__ What is the relationship between the mass on a spring and its period of oscillation? In this lab, we will determine k, the spring constant, of a spring two different ways. First, we will, determine this value by measuring the the distance the spring stretches. Secondly, we will measure the period of oscillation based on different masses.

__**Hypothesis:**__ The greater the mass, the longer the time of the period. Additionally, there is a direct relationship between the mass hanging on the spring and the period of oscillation. On the graph of force times distance, the slope is the spring constant, k.

__**Materials:**__ For this lab, the necessary materials include the spring, a spring stand (with a meter stick), masses, a balance, and a timer. 

__**Procedure:**__
 * 1) Gather the materials listed above.
 * 2) Set up with spring on the stand and attach a mass.
 * 3) <span style="font-family: Tahoma,Geneva,sans-serif;">Have one person time a certain number of oscillations while another member measures the distance the spring stretches from its' point of equilibrium.
 * 4) <span style="font-family: Tahoma,Geneva,sans-serif;">Record these values in an excel spreadsheet and calculate the value of the spring constant, k.
 * 5) <span style="font-family: Tahoma,Geneva,sans-serif;">Repeat this process with varying masses.

<span style="font-family: Tahoma,Geneva,sans-serif;">__**Data:**__ <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">__**Sample Calculations:**__

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<span style="font-family: Tahoma,Geneva,sans-serif;">__**Conclusion:**__ <span style="font-family: Tahoma,Geneva,sans-serif;">After completing this lab, we concluded that our hypothesis was correct. First, we proved through our experiments that as we increased the mass hanging from the spring, the length of the period increased. For example, a hanging mass of 0.02 kg, had a period of 0.414 s, while a mass of 0.04 kg hanging on the spring had a period of 0.655 s. As seen by these numbers, the greater the mass, the more time in takes for the spring to stretch and compress. When doing the experiments using Hooke's law this same theory was demonstrated, because as the Force (mass times gravity) increased, so did the distance. Two such values that demonstrate this increase are a force of 0.2352 N with a distance of 0.097 m; while a force of 0.343 N had a distance of 0.147 m. Additionally from our data, the graphs we produced show a direct relationship between the mass and period, and the force and distance. As shown in the above Force vs. Distance graph and the data table, depending on which way we found the spring constant, the value for k was slightly different. When we measured the period of the spring, we found k to be 2.953 and when we used the Hooke's law, we found the spring constant to be 2.377. Although these values are relatively close, there is a 21.62% difference. Within parts of this lab, there is room for error. One form of error occurred in both trials - that is, the reading/measuring of the distance in which the spring was stretched. Additionally, when measuring the period, the timer may have started or stopped the timer to early or to late thus resulting in a varying time, and in the end, changing our results.

<span style="font-family: Tahoma,Geneva,sans-serif;">__**Discussion Questions:**__ <span style="font-family: Tahoma,Geneva,sans-serif;">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? <span style="font-family: Tahoma,Geneva,sans-serif;">The data for the displacement of the spring versus the applied force is a clear indicator of how the spring constant is a constant because there is a linear relationship between the displacement and mass in each trial. When the data is plotted on a graph of displacement versus applied force, the slope of this line is the value of the spring force, and it is one constant value. The spring force constant can be considered constant because it does not change in the spring, even when the applied force does.

<span style="font-family: Tahoma,Geneva,sans-serif;">2. Why is the time for more than one period measured? <span style="font-family: Tahoma,Geneva,sans-serif;">The time for more than one period is measured due to accuracy. It is easier, and more accurate to measure a greater number of oscillations than to only time one cycle.

<span style="font-family: Tahoma,Geneva,sans-serif;">3. Discuss the agreement between the k values derived from the two graphs. Which is more accurate? <span style="font-family: Tahoma,Geneva,sans-serif;">The trial when we calculated the period, gave us a k value of 2.953 N/m and when we used Hooke's Law, our k value was 2.377 N/m. The percent difference between the two values is 21.62%. The experiment using Hooke's law was probably more accurate because we were more likely to record accurate distance rather than time. When recording the period, we did not factor in our reaction time.

<span style="font-family: Tahoma,Geneva,sans-serif;">4. Generate the equations and the corresponding graphs for: (a) position with respect to time, (b) velocity with respect to time, and (c) acceleration with respect to time. <span style="font-family: Tahoma,Geneva,sans-serif;">a. x(t) = 0.085cos(15.18t) <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">b.V(t)=-1.29sin(15.2t) <span style="font-family: Tahoma,Geneva,sans-serif;"> <span style="font-family: Tahoma,Geneva,sans-serif;">c. a(t) = -19.58cos(15.18t) <span style="font-family: Tahoma,Geneva,sans-serif;">

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

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

<span style="font-family: Tahoma,Geneva,sans-serif;">7. We neglected to take into account the mass of the spring itself. Are your results any better when using the more accurate relationship, m+(1/3)ms, (where m is the hanging mass and ms is the mass of the spring)? Redo graph #2 using √(m+(1/3)ms), and explain these results. <span style="font-family: Tahoma,Geneva,sans-serif;">Even after taking into account the mass of the spring itself, our results remained the same. Our slope remained at 13.355 and our k valued stayed at 2.953 N/m. By consistently neglecting to include the mass of the spring in every trial, out slope would remain the same because the masses in all of our trials increased by a constant rate (1/3 of the mass of the spring) and the period squared remained the same as before. This resulted in the slope of our graph remaining the same. <span style="font-family: Tahoma,Geneva,sans-serif;">