Standing Waves. Miscellaneous Cables and Adapters. Capstone Software Clamp and Pulley White Flexible String

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1 Partner 1: Partner 2: Section: Partner 3 (if applicable): Purpose: Continuous waves traveling along a string are reflected when they arrive at the (in this case fixed) end of a string. The reflected wave interferes with the incoming wave, resulting in most cases in an erratic and irregular movement of the string. Under particular conditions (coordinated length of the string, wave frequency, linear mass density of the string, and tension of the string) a standing wave pattern emerges and can be observed very clearly. Under these conditions the wavelength of the standing wave (and thus that of the incoming wave) can be measured. The theoretical relationship between the aforementioned parameters is examined and tested for its consistency in a real experimental setting. Apparatus List: PASCO Mechanical Wave driver Miscellaneous Cables and Adapters Clamp and rod Capstone Software Clamp and Pulley White Flexible String Hooked weight set Two C-Clamps Digital Scale (on shared table) Pasco 750 Interface Pasco Power Amplifier II (CI-6552A) Theoretical Background What Are Waves And How Can We Classify Them? In very general terms, a wave is a disturbance that travels from one place to another (for example, a stone thrown into a calm pond creates a disturbance that travels along the surface of the pond, away from the site of impact). In addition, if the disturbance originates from a continuous source, then a continuous wave is created (for example, a wave created by a running motor of a boat). Furthermore, some waves are created by a continuous source that moves sinusoidally. The wave can then be classified as a harmonic wave. In this experiment we generate sinusoidal waves with the help of a computer, Capstone software, the 750 Pasco interface, and a Pasco amplifier. We will use sinusoidal waves in this experiment 5

2 because a sinusoidal (harmonic) wave has a distinct (single) frequency associated with it. In contrast, rectangular or saw-tooth waves are composed by superposition of many sinusoidal waves of different frequency and amplitude. Use of the harmonic waves therefore reduces greatly the number of parameters in our experiment. In other words, we can clearly identify the frequency of the wave by using harmonic waves. Waves can be classified also by the direction of the disturbance. If the disturbance is perpendicular to the direction in which the wave travels, the wave is called transverse (an example would be a wave traveling along a string). If the disturbance is in the same direction as the direction in which the wave travels, the wave is called longitudinal (example: Sound). Reflection of Waves, When a wave encounters a discontinuity of the medium in which it travels (sound hitting a wall, the end of a string, the water waves hitting the beach) a certain portion of the wave is reflected backwards. The pattern of the reflected wave resembles closely that of the incoming wave. In cases where the discontinuity forces the waves amplitude to be zero at the location of the discontinuity (e.g., when the end of a string is tied to a fixed place so its end cannot move) the reflected wave is inverted. In the opposite case (e.g., a string with a loose end that can move) the reflected wave is not inverted. In either case, if the wave is continuous, there will be basically two waves traveling simultaneously along the medium: The incoming wave and the reflected wave. The two waves will interfere, which means that their respective time and space dependent displacements will add up. In general, adding the incoming and reflected wave results in an irregular wave pattern. However, if the conditions are right, the interference results in a regular wave pattern along the medium. Such a regular pattern is called a standing wave. The Speed Of A Wave Traveling On A String The speed (v) of a wave on a String is determined by two properties: 1) The tension (T) in the string. 2) The linear mass density of the string. The linear mass density can be calculated from the mass (m) of the string and it s length (l):. 1. The speed can be calculated with this equation:. 2. In addition, the relationship between the frequency f, wavelength, and speed v of the wave is given by the equation. 3. 6

3 N A N Standing Wave Patterns On A String With Fixed Ends A standing wave can be characterized by the presence of areas where the string does not move at all (called nodes ) and areas where the string moves with maximum amplitude ( anti-nodes ). The most simple standing wave thus consists of two nodes and one anti-node. The nodes are at the two sides that are fixed and the anti-node is halfway between the nodes Note: The distance between two nodes corresponds to half a wavelength Suppose, starting from this standing wave pattern, you increase the frequency of the wave slowly, you will notice that first the standing wave pattern will disappear and then, when the frequency is right, another standing wave pattern will emerge. This pattern has two Anti-nodes and three nodes. Again, the distance between two nodes corresponds to half a wavelength. Clearly, the wavelength is reduced for this second standing wave pattern by a factor of 2. Problem 1: [1 pt] Assuming that the speed of the wave has not changed when going from the standing wave pattern with one Anti-node to the one with two Anti-nodes, how would the frequency have to change? Problem 2: [ 1 pt] Make a drawing of the next expected possible standing wave pattern that would occur at the next higher frequency. 7

4 Setup: The mechanical wave driver is positioned underneath a string. The string is attached to a rod on one end of the table and redirected over a pulley at the other end of the table. At the end of the string (on the pulley side) a weight hanger (and weight) is attached that creates the required tension in the string. Make sure that the small plug with the U-shaped groove is attached to the moving rod of the mechanical wave driver. The string should run through the groove of that plug. Thus, the mechanical wave driver only lifts and lowers the string periodically. Never attach the string directly to the wave driver s arm. This may damage the vibrating membrane within the wave driver by creating a sideways tension on the rod. The wave driver itself should be clamped to the table with two C-clamps. N A N A N 8

5 Electrical Connections: Computer/Capstone Software Pasco 750 Interface Channel A Pasco Power Amplifier II Mechanical Wave Driver Procedure: Activity I - Creation of In this activity you will familiarize yourself with the apparatus and experience a few of the basic properties of standing waves. You should have a white elastic string that is long enough to reach from the rod at one end of the table over the mechanical wave driver and over the pulley at the other end of the table and several inches further down. A special banana plug with a center groove is provided as a guide for the string. That plug is to be positioned in the center rod of the wave driver. 9

6 CAUTION! Make sure you carefully put the locking mechanism of the wave driver into the locked position (you may need to pull up or push down the shaft slightly to be able to properly lock it) during plugging and unplugging of the banana plug. Otherwise the membrane of the wave driver may be damaged. Don t forget to unlock that mechanism again afterwards (before starting vibrations). Route the string over the pulley and attach the mass hanger set to it. Position the pulley on the table such that the distance from where the string is attached on the rod next to the wave driver to the top of the pulley is exactly 2.00m. The mass hangers plus attached masses provide the tension in the string. Make sure that the mass does not touch the floor when you hang a mass of 250g on the string (note: make sure you take the mass of the hanger itself into account as well, it is usually 50g). If it does touch the floor, you probably need to attach the mass a bit further up on the string. Problem 3: [ 1 pt] Write down the equation with which you can calculate the tension in the string from the attached mass (M). (Hint: Don t use Eq. 2 for this! Instead consider the free body diagram of the mass. It is very simple!). T = The mechanical wave driver is driven by the power amplifier, which in turn is controlled by the computer through the Pasco 750 interface. To start the oscillations of the mechanical wave driver, you need to open the file.cap. It is available on the elementary lab web page. Please follow the choice of browser indicated on the website. Click on Capstone Files and then click on.cap. A file dialog opens asking you whether you want to open, save or cancel. Click Open. It may take 20 seconds or so for the file to load and start up in the Capstone software. Follow the instructions - given in the Capstone file that you have opened - to control the frequency at which the wave driver operates. Do not exceed an amplitude of 5Volts this limit protects the mechanical wave driver from damage. If the mechanical wave driver rattles, you may be using too high of an amplitude and it may bad for the wave driver. These wave drivers are basically speakers like on a stereo and they can be damaged if the volume (amplitude) is cranked up too much. Keep that in mind and reduce the amplitude if necessary even below 5 Volts. With a total of 250 g mass attached to the string start running the wave driver at a low frequency (a few Hz). Slowly increase the frequency until you see a standing wave pattern that has one antinode in the middle. That means that half a wavelength of the standing wave occupies the length of the string (2.00m). Use the frequency reading of that standing wave together with the information from Q1 to estimate the frequency of this standing wave. Then slowly increase the frequency and search for the standing wave that has a node in the middle of the string so that the wavelength equals 2.00m. Try to find standing waves with multiple nodes at even higher frequencies. 10

7 Try the following: Change the frequency such that you have a nice standing wave with 4 nodes and 3 anti-nodes. Take a ruler or a pencil and hold it a small distance directly underneath the anti-node that is closest to the pulley. Now slowly move the ruler/pencil upwards. Problem 4: [ 1 pt] What happens to the standing wave pattern? Repeat this procedure but now position the ruler/pencil under the node that is half a wavelength away from the pulley. Problem 5: [ 1 pt] What happens to the standing wave pattern? Problem 6: [ 1 pt] Try to explain the different results you got (if you did). 11

8 Activity II - Experimental Determination of the Speed of a Wave Equation (3) describes how the frequency and wavelength are related to the speed of the wave. If you solve Equation (3) for f, you can see clearly that a plot of frequency versus 1/wavelength should be a straight line with a slope that equals the speed of the wave. The idea behind this activity is to measure the wavelength of the standing wave (which equals that of the traveling wave) for various frequencies, then plot f (on vertical axis) versus 1/ (on horizontal axis), and finally determine v from the slope of that plot. Of course it would be possible to calculate v from a single frequency/wavelength pair. However the approach of using several frequencies will reduce the uncertainty in the calculated speed. In addition, we can then verify that indeed the speed of the wave is reasonably independent of the frequency of the wave (this is not necessarily the case for all kinds of waves in all media; for example the speed of light is frequency dependent, unless in a vacuum). Start out with the lowest fundamental standing wave. Measure frequency and wavelength and estimate the uncertainty in both quantities. Continue to increase the frequency and try to measure frequency and wavelength for higher fundamentals. You can go at least to standing waves with 4 or 5 nodes, probably higher. However the amplitude of the standing wave will decrease for higher number of nodes to the point where you won t be able to see it at some point. The graphing for the results of the 250g mass needs to be done by using an Excel spreadsheet. The uncertainty needs to be determined in the same way (high slope and low slope) as you did in the homework assignment. Results: (6 pts, including all calculations and Excel Spreadsheet) M = 250 g T = L= L= f f (estimate) 1/ (1/ ) v = ± (Include the uncertainty, see Appendix I for hints). 12

9 After you have determined the speed of your wave experimentally, you should now determine the theoretical speed of your wave based on Equation (2):. 2. As you can see, beside the tension T that you should have already determined, you need to determine the linear mass density stretched of the STRETCHED string. To get that, you should first determine the mass and length of the UNSTRETCHED string (use a ruler and use the digital scale to get the length and mass of the unstretched string). Results: m unstretched = l unstretched = Now you need to determine how short the string-length becomes that originally stretched from the metal rod on one end to the top of the wheel on the right side. So, the STRETCHED length was 2.0 meters. To get the UN-STRETCHED length of that segment of elastic string, mark the location of the string that is on the top of the wheel by grabbing it with your fingers (keep holding on to it) and then moving that point of the string towards the mechanical wave driver until the string is no longer stretched. Now measure the distance between that point of the string and where the string is attached at the rod at the other end. That new un-stretched length is obviously less than 2.0 meters. Record that new length which we will call L unstretched. L unstretched = To determine stretched you need to realize that, where m stretched is the mass of the string between the two end-nodes and L stretched is the length of the string between the two end-nodes. L stretched = 2.0m if you have set up everything correctly before. However, 13

10 Combining these two equations it follows that. That means that the linear mass density of the stretched string is a bit less than the linear mass density of the un-stretched string, which hopefully makes sense to you. Finally you can see that the theoretical value of the speed of the wave is Calculate the theoretical velocity for hanger mass M = 250g: Calculate the % difference between your measured velocity and the theoretical velocity: % 100 Print out your f versus 1 spreadsheet calculations and graph for the 250g hanger mass (make sure it all fits on one page and shows high slope, low slope, average slope, standard deviation of slope, standard deviation of the mean of the slope etc. similar to the homework) and include it with this lab report. 14

11 Perform similar measurements for several different hanger masses M. Due to time restrictions of this lab we will no longer determine errors in frequency and wavelength in the following tasks but instead use a simpler approach: In Capstone, there are tabs called 50 g Data, 100 g Data etc.. Under these tabs you will find empty data tables and associated graphs that we have preformatted for you. Simply enter your frequency and 1/ data and the corresponding graph will be automatically created for you. Then you can simply use the fit-feature of Capstone to do a linear data fit and determine the slope of your data. This linear regression, as it is often called, also generates a value for the uncertainty in the slope m. You may ask yourself: How can it be that Capstone calculates an uncertainty when we never even enter any uncertainty values for the frequency and for 1/? The answer is: In linear regressions, the uncertainty is based simply on how far the data points are away from your fit-line. (Note: If you have estimated reasonable error bars in an experiment, then the scatter of data away from the fit-line should be very similar in size compared to the estimated error bars.) 15

12 Measurements for M = kg: (2 pts, including the calculations) Results: M = 50 g T = L= L= f 1/ v = ± (determine from linear fit slope in Capstone) L unstretched = (re-measure, this will be different now!) v theoretical = (re-calculate) % difference = Perform your calculations of v theoretical and % difference in the shaded area: Measurements for M = kg: 16

13 Measurements for M = kg: (2 pts, including the calculations) Results: M = 100 g T = L= L= f 1/ v = ± (determine from linear fit slope in Capstone) L unstretched = (re-measure, this will be different now!) v theoretical = (re-calculate) % difference = Perform your calculations of v theoretical and % difference in the shaded area: 17

14 Measurements for M = kg: (2 pts, including the calculations) Results: M = 150 g T = L= L= f 1/ v = ± (determine from linear fit slope in Capstone) L unstretched = (re-measure, this will be different now!) v theoretical = (re-calculate) % difference = Perform your calculations of v theoretical and % difference in the shaded area: 18

15 Measurements for M = kg: (2 pts, including the calculations) Results: M = 200 g T = L= L= f 1/ v = ± (determine from linear fit slope in Capstone) L unstretched = (re-measure, this will be different now!) v theoretical = (re-calculate) % difference = Perform your calculations of v theoretical and % difference in the shaded area: 19

16 Appendix I: Hints regarding the estimations/calculations of the uncertainties: 1) The uncertainty in the frequency ( f) can be estimated based on the precision with which you can pinpoint the frequency at which the best standing wave occurs. For example, you may find that you get a very good standing wave at 6.2 Hz, but you can t really tell whether 6.3 Hz or 6.1 Hz are any different. However, you may find that at 6.0 Hz the standing wave is clearly lower in amplitude. So, in this case you would estimate that the uncertainty in the frequency is 0.1 Hz or 0.2 Hz. 2) The uncertainty in the inverse of the wavelength,, is harder to obtain. It basically originates from the precision with which you can measure the distance between the two nodes that are furthest apart in other words, the distance between the place where the string is attached to the rod next to the mechanical wave driver and the location of the node that is on top of the wheel at the other end. You have to consider the precision of your meter stick, but you also may find that it is somewhat unclear where the end-nodes are exactly located and that may create the larger uncertainty. Suppose that length between the end-nodes is called L and the uncertainty in that length is L. You first need to estimate L. Then you can calculate the uncertainty in (1/ ), which we call (1/ ), from L, L, and the wavelength inverse 1/ as follows (we provide this formula without proof here because it requires calculus): δ 1 λ 1 λ δl L 3) Estimating the uncertainty in the velocity: The uncertainty in the velocity is equal to the uncertainty with which the slope of the f versus 1/ can be determined. To estimate this uncertainty, you need to enter error bars in both frequency-direction ( y-axis ) and 1/ -axis ( x-axis ) to your plot. Then you need to find the slopes for the steepest and for the shallowest lines that reasonably fit the data (reasonable means that these lines should each miss no more than 33% of the data points including their error bars. You will get a maximum slope and a minimum slope using this method. Let s call these slopes s max and s min. The mean slope (which equals the speed of the wave) is s s s 2. The standard deviation of the slope is: δs s s 2. 20

17 The standard deviation of the mean of the slope (and therefore the uncertainty in the speed) is: δs δs N, where N is the number of data points in your f versus 1/ plot. 4) You may ask yourself: How can I estimate s max and s min needed in the previous step from my data? First, you need to plot your data in Excel and include error bars in both frequency and in 1/ in your plot. To add error bars to your Excel graph, you can create another column of data for the uncertainties in the measured frequencies and another column of data for the uncertainties in 1/. Error bars can be added in Excel 2007 as follows: Click on the graph so that it is selected. Now, you should see Chart Tools at the top of your Excel sheet. Underneath Chart Tools there is a tab called Layout. Once you click Layout you should see Error bars. Click on Error bars and select More Error Bars Options. A window will pop up and on the upper left corner it will display either Vertical Error Bars or Horizontal Error Bars. On the left side of your main Excel sheet you should see a menu just above Format Selection. From that menu you can choose whether you want to define the X error bars or the Y error bars. The popup window changes accordingly. In the popup window choose Custom and then click on Specify Value. Another small window pops up called Custom error bars. To choose the positive error bars, click on the box located to the right of ={1} underneath Positive Error Value. Then select the column that contains your uncertainties and click on the little box again. Do the same with the Negative Error Values since they should be the same in this case. Then click OK to close the Custom Error Bars Window. Now do the same for the other direction. Once you have added error bars in both directions, You need to create a column that calculates a fit-line with a slope that you can choose. First, choose a field in your Excel spreadsheet in which you can put your slope-value. Let s say the field is K2. Populate the field with a value, for example 1.3. Next, calculate the frequency values of your fit as a function of your 1/ values: In a new column to the right of your data calculate your frequency values using $K$2 as a reference of your slope. For example, if your 1/ values start in the field B5 and you want to calculate your frequency of your fit in column N, then you would enter in the field N5 the equation =$K$2*B5 and hit enter. You can calculate the fields below in a similar manner by clicking on field N5 and then dragging the small black square in the lower right boundary of field N5 downwards as far as you need and then releasing it. If you click on N6, it should now have the equation =$K$2*B6 in it, etc.. 21

18 To add your fit-column to your graph, right-click inside the graph area and select Select Data. Add a series to your data by clicking on Add. Call the series name Myfit or whatever you like and then select for the series x-values the column with the 1/ values and for the series y-values your fit column. Once you have added your fit-data to your graph, you may want to right-click on one of the fit data points in the graph and select Format Data Series to add a line between the fit points if you like. By changing the value in the slope-field, you can now change your fit-line. To find s max find the maximum slope that is still in agreement with your data including error bars (Note, approximately 33% of your data points will be missed by the fit-line. The fitline should NOT go through ALL data points with error bars.). Similarly, find the lowest slope to get s min. Appendix II: Hints Regarding Trendline Analysis with Excel (2007): 1) Click on the graph. 2) Under Chart Tools click Layout. 3) Under Trendline select Linear Trendline. 4) Once the trendline appears, right-click on the trendline and select Format Trendline. 5) From the popup window select Set Intercept = 0.0 and select Display Equation on Chart. 6) Close the popup window. 7) The trendline equation should appear in the graph giving you the slope of the trendline. 8) You can use the linest function of Excel to calculate statistics on trendlines, which includes an uncertainty in the trendline slope and y-intercept. How to do that is not obvious, so exact instructions are given in the Excel tutorial at the beginning of this lab manual on how to do that. Note that these uncertainties calculated by Excel are NOT based on your error bars. They are based only on the scatter of the data around the trendline. Ideally, if your uncertainties are estimated reasonably, the trendline uncertainties of Excel and those obtained by your graphical method above should actually be comparable in size. 22