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Recall from previous labs, that resonant frequencies are proportional to the velocity (v), length of the resonant object (L), and harmonic number (n) n v fn (4) 2 L V= From this definition of the wavelength (2), and the previous equation for resonant frequencies of a string, the relationship for the speed of the wave propagating down the string can be found, = af (5) This experiment will verify the relationship in equation (1) by examining resonant standing waves on a string. The string used will be a stretchy elastic cord connected to a variable frequency String Vibrator. If we combine equations (1) and (5) then we find that the propagating wave is described by, Ft v = 2 f = (6) р According to equation (5), the wavelength (2) and frequency (M) are both proportional to the speed at which the wave propagates down the string, and this in turn is proportional to the tension (Ft) and mass per length (u) of the string. By setting these two equations equal to each other, we can verify that there is indeed a relationship between frequency (f) and wavelength (2), and that the tension (Ft) and mass per length (u) can then be determined from this relationship.
Procedure Part A - Equipment Setup 1. Set up your apparatus as shown in Figure 1 below. The force transducer should be at one end of the stretchy string and the string vibrator at the other. Attach the stretchy string in the following way. Tie a knot in the string and pull the string through the hole in the vibrator blade. Make sure the knot cannot slide through the hole in the vibrator blade. Then tie a loop in the other end of the string so it can fit over the hook on the force transducer. You will want to tie the loop so that there is a small amount of tension in the string when it is hooked to the force transducer. Note that the string vibrator can be moved back and forth to vary the tension in the string. string vibrator force transducer Figure 1 2. Connect the Force and Voltage Sensors to the 850 Universal Interface, and run leads from the String Vibrator to Output 1. Make sure the interface is on and connected to one of the laptop USB ports. 3. Start up Capstone and open the Hardware Setup tab to make sure all the devices are added. Choose the One Small, One Large Display template from the QuickStart window, and select a digits display to show force on the left side and a graph display for the right side. On the graph display, create a second y-axis to show both force and voltage vs time. Make sure that the sampling rate for data collection on both sensors is high enough to detect the quick pulses we will be performing (1000Hz). 4. Remember that before recording data, the Force Sensor needs to be reset by pressing the ZERO button on the sensor when there is no string attached.
Part B - Find wave velocity from mass per length tension Note: As you start entering data into your Maple worksheet, please use the variable names in parentheses. 1. Find a nearby scale and a length of stretchy string. Measure the total mass (Mtotal) and length of an identical unstretched piece of stretchy string (Ltotal) in order to later be able to calculate the mass per unit length (jaunstretched) for the string. 2. Make sure that Force Sensor has been zeroed by using the ZERO button on the sensor with the cord unhooked. Then collect force data on a graph for a few seconds. To determine the average force (String Tension FT) select the force data on the Force vs. Time graph and find the average agnit or use the digits display created above to show the force value. You do not need to screenshot this graph. Note: You should not move either the force sensor or the string vibrator after this measurement is made so as not to change the string tension. If you later change the length of the string you will need to re- measure the force or tension and recalculate the mass per unit length! 3. You also need to be able to determine the mass per unit length of the stretched string ((ustretched). To make this calculation possible, first measure the stretched length (Lstretched) (that will be vibrating) and then unhook the string at the Force Sensor end and measure the unstretched length (Lunstretched) of the portion of the string that is actually going to be vibrating. Remember that this will have to be recalculated if you change the length of the vibrating string.
Part C - Speed of a single pulse The most basic way to find the speed of a wave is to send a pulse down a string and measure the time it takes to travel a known distance. That is what will be done in this part of the procedure. You will use the force sensor and the voltage sensor, to time a pulse traveling down the string. 1. Make sure the voltage and force sensors are connected to the 850 Universal Interface and linked to the laptop via one of the USB ports. From your graph that shows force and voltage vs time, check that your data sampling rate is high enough to detect the changes from the string pulse. 2. To create a pulse, pull the string vertically downward with your finger as close as possible to the force sensor. Then release the string suddenly. See Figure 2 below. Because you are pulling down on the string the Force Transducer will measures a larger than normal force until you release the string. Notice also that when the pulse reaches the String Vibrator, it makes the blade move up and down; this motion moves a magnet inside a coil, which generates a voltage spike that the Voltage Sensor will measure. Figure 2 3. Start recording data just before you pluck the string, then immediately stop recording after you see the peaks register on the graph.
4. When you have the force and voltage data on a graph you will need to adjust the graph quite a bit to spread it out enough so you can see the time between pulses clearly. Use the coordinates tool on each graph to carefully find the elapsed time, At, between the sudden change in tension and the initial change in voltage. Try this several times and make sure that you have a reliable and repeatable value for At. It might be good to record several values and get an average value. When you get a good graph put your names on the graph and take a screenshot. Later in the Analysis section you will calculate the pulse speed. Part D - Wave speed by measuring wavelength and frequency In this part you will vary the frequency carefully to find the various resonant modes of the string. You will know that you have a resonant mode when you get a large oscillation of the string that looks like figure 3 below. Figure 3 1. Open the Signal Generator tab and choose the 850 Output 1 to vary the frequency applied to the vibrating string. Set the initial voltage to about 4 volts and adjust as necessary. To manually input a frequency, type the value desired and then hit Enter. Otherwise the arrows to the right of the frequency window will allow you to adjust up or down. All the frequencies used in today's experiment will range from 0 Hz to 100 Hz. 2. In this part you want to adjust the frequency carefully to find resonant frequencies and therefore good standing waves. As you do this you have to be quite careful. Initially, start somewhere near 10 Hz and adjust the frequency slowly upward or downward until you find a resonant frequency. Note that you have to carefully adjust the Good Node frequency so that the node at the vibrator end of the string is right at the end of the vibrator arm. See Figure 4. It is tempting to look only at the amplitude of the wave and concentrate on making it as large as possible; but it is also important to check that the nodes are "clean” and well defined, especially the node at the vibrating blade. Check the end of the vibrating blade. There should be a node at the point where the cord attaches, as shown in the first picture below. An example of a bad node is shown in the second picture. The blade rattling against Bad Node the plastic case indicates a bad node or too much amplitude (voltage). Figure 4 3. Now that you know how to carefully find standing waves and resonant frequencies, it is time to find 5 or 6 resonant frequencies. Start at about 5 Hz and gradually increase the frequency. Find the first resonance (1 antinode) and then work your way up trying to find each successive resonant frequency. Record data in two data arrays, with the NA values (number of antinodes) in one and the corresponding resonant frequency () values in the other.
Analysis Part B - Find mass per length and wave velocity from a pulse 1. Calculate the unstretched mass per length (Luunstretched) by applying equation (2). Note that you will use Mtotal and Ltotal to make these calculations. 2. Now calculate the stretched mass per length (jstretched). You will to do this by using jaunstretched as well as Lstretched and Lunstretched. (Hint: how do you determine the mass of the string segment used?) 3. Use the tension you measured (FT) and the mass per unit length (ustretched) to find the wave speed (vuF). Part C - Speed of a single pulse 1. Use the stretched length (Lstretched) and the time of travel (At) that you measured in Part A to calculate the pulse wave speed (vpulse). 2. Compare this wave speed (vpulse) with the one obtained using the mass per unit length and tension (VF), by finding a percent error. Part D - Wave speed by measuring wavelength and frequency 1. Use equation (3) along with your array of NA data and the stretched length of the string (Lstretched) to calculate an array of the wavelengths for the vibrating string. 2. Use equation (5), the array of wavelengths and your frequency array to make a graph to find the wave speed (vgraph). Find the wave speed from the slope of the graph, and compare this wave speed to the two determined earlier by calculating percent errors with each. 3. Use the graph created in the previous step, and equation (6) to determine the mass per length (ugraph). Remember that since wavelength and frequency are proportional to the speed of the wave propagating down the string, we can use the slope of the graph to solve for the mass per length (ugraph). 4. Compare the %Eofd for the two values you found for the mass per unit length of the string.
Questions 1. Between the three ways you found wave speed in this experiment, which value do you think is the most reliable? Why? 2. Look at the pulse graph in Part A, and look for the reflections as the pulse bounces back and forth. How could you use these to make your calculation for vpulse more accurate? 3. Between the two ways you found mass per length, one you got by measuring string lengths and string mass and the other you found by using a graph, which do you think is the most accurate and why?