Pre-Lab Required Homework Problem

Preparation for Experiment 1: Part 1

Study carefully Example 5.12 in Young and Freedman. Although the problem is simpleit is a classic one because it illustrates so many of the concepts and issues that you mustunderstand. It involves setting up and solving three equations in three unknowns. It involvesa constitutive relation: that is, some mechanical constraint between the bodies that couplestheir motions. In this case the fact that the string has a fixed length results in the equationthat the magnitude of the acceleration of m1 is equal to the magnitude of the accelerationof m2.

a) Explain in physical terms the form of the expression for the acceleration. It is adimensionless fraction times g. Why that particular fraction?

b) Is the tension in the string after m1 released from rest greater than, equal to, or lessthan its value before m1 is released? Explain physically why this must be so.

c) Make carefully labeled sketches of the position, speed and acceleration of m1 as afunction of t. Start the sketches at t = 0 and assume m1 is released from rest a shorttime afterward at t = t0.

d) Now imagine that m1 is acted upon not by a constant force, as in the example, but bya time varying force that has some positive value at t = t0 and subsequently decreasessmoothly to zero at a time tend. Again make carefully labeled sketches of the position,speed and acceleration of m1 as a function of t. For added clarity, indicate as a dashedcurve on your sketches the result you found in c) assuming the magnitude of the forceat t = t0 is the same as the value it had in c).

e) In the experiment, you will implement the situation in d) by replacing the discretemass m2 with a chain. The freely hanging portion of the chain decreases as m1 movesforward, thus giving rise to a smoothly decreasing force. What additional change in theequations of motion would have to be taken into account when modeling the motionof m1? [Hint: you may want to refer back to your answer to a).] Each of these changesalone is reasonably easy to handle using techniques we will learn later in the course.Together, they lead to a rather complicated analytic expression for x1(t) which we willnot deal with.

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Preparation for Experiment 1: Part 2

In 8.01 the apparatus you will use in the experiments is run by a computer application calledLogger Lite. Similar but much more powerful applications (LabVIEW is an example) are usedin many research laboratories to run instruments, acquire data, and perform sophisticatedanalyses of that data. Naturally, the applications have to be programmed to carry out eachof these tasks and that takes a fair amount of (usually graduate student) labor. Initially ourdata acquisition software was programmed to analyze as well as take the data. However, wefound that having the software do the analysis diminishes your understanding and enjoymentof the experiments. We now let you carry out most of the analysis of your data.

In class, you will use the spreadsheet application Excel to analyze your data. Excel is themost widely used application of this sort. Not all of you will have the opportunity to use aprogram like Logger Lite again, but it would be hard to imagine a student graduating fromMIT without having had to make extensive use of Excel or a similar spreadsheet application.

Most other spreadsheets, for example Apple’s Numbers and OpenOffice.org’s Chart, areremarkably similar to Excel. However, only Excel (as far as we know) is able to do apolynomial fit to data in a straightforward manner. That is why you will be using it in class.For this problem set we offer you two options. If you have Excel on your own computer, great.Go to the Excel option below. If you do not, you should use the OpenOffice spreadsheet. Itis on Athena. It can also be downloaded free of charge from http://www.openoffice.org/ .In this case, go to the OpenOffice option below. [Note: if you have a Mac and have Numberson it but not Excel, follow the OpenOffice instructions. They should apply almost exactly.]

Excel option

a) Downloading the data. We have taken two sample data sets, fallingMass.csv andfallingChain.csv. Download them from http://web.mit.edu/8.01t/www/Prelab%20Data/Open the Excel application. Choose Open from the File menu, navigate to the filefallingMass.csv, select it, and click on Open. In the dialogue box that appears choosein order Delimited → Tab → General (the default options) then click Finish. Fouradjacent columns appear in the speadsheet labeled Time, Position, Velocity and Ac-celeration.

b) Plotting the data. fallingMass.csv corresponds to dropping a fixed mass, thus to c) inPart 1. Use the cursor to highlight the rectangular block of data two columns wide,many rows deep, beginning with the title row – Time and Position– at the top. Eitherselect Chart from the Insert menu, or click on the Chart icon on the tool bar. On thefirst panel of the “Chart Wizard” dialog box which appears choose as Chart Type XY(Scatter)1. This will plot the data points without any distracting connecting lines.

1The most recent versions of Excel have replaced the “Chart Wizard” with a tab labeled Charts belowthe tool bar

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Move from one panel to the next (you do not have to enter anything along the way)until at the end you are given the option of creating the chart on the work sheet or ona separate sheet. Choose “As a new sheet”.2

Excel will choose by default to display the left most column on the x axis and the othercolumn will be displayed as the dependent variable plotted on the y axis. Note thatthe legend on the graph uses the titles at the heads of the selected block to label thepoints on the graph. Admire your results. Do they look like your first sketch in c) ofPart 1 (at least until the weight hit the floor)?

There are many changes one can make to the way the data is plotted, but at this stageone is more important than the others: the size of the symbols used to mark eachpoint. The symbols are probably too large. Correct this as follows. Double click onone of the symbols on the chart used to plot the data. This opens a Format Data Seriesdialog box. Choose the Patterns panel and reduce the Marker Size to an appropriatevalue, probably the smallest available. Click OK to see the results.

2The most recent version of Excel first places the chart in the sheet you are working on. To move it to aseparate sheet, go to Chart in the Menu bar and click on Move Chart.

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Repeat this procedure to get plots of Velocity and Acceleration. Here you must choosenon-adjacent columns for plotting. First select the portion of the Time column youwant to plot. On a Mac you must Command click on the corresponding portion ofthe non-adjacent column to add it to your first selection. Otherwise, everything inbetween is selected as well. On other operating systems the method may be different.You should compare the Velocity and Acceleration plots with your sketches from c) inPart 1.

It is a fact of life that determining the derivative of a function from a discrete represen-tation of that function (that is, a set of data points) is a subtle business at best, andparticularly difficult when there is noise added to the signal. We see this here. Theposition represents the actual data taken by the apparatus. The velocity and accelera-tion were computed from this data by computing first one – then a second – derivative.Logger Lite uses a special smoothing procedure to do this, but the representation ofhigher order derivatives gets successively noisier, and thus less useful. The questionarises, is it better to determine the (constant) acceleration by averaging the values onthe computed Acceleration curve or to determine it from the fit of analytic functionsto the Velocity or Position curves? You will study this question below.

c) Preparing to fit the data. The results you sketched in c) of Part 1 are “piece-wiseanalytic”: that is, they are represented by continuous analytic functions in two ormore separate regions. But the functions change from region to region. The fittingroutine in Excel is not sophisticated enough to handle this. Thus, we must deal onlywith the second of the regions, that for t ≥ t0. Moreover, we must adjust the time tobe zero at the beginning of the fitted region and to end before m2 hits the floor.

There are many was of doing this. Here is a simple one that preserves the originaldata. Navigate back to the sheet with the data by clicking on the appropriate tabin the lower left hand corner of the Excel window. Click on the letter at the verytop of the Position column to select the entire column. Then choose Column from theInsert menu. A blank column will be inserted between the Time and Position columns.Determine from the data – perhaps from the velocity graph you just made – the timewhen motion began, that is, when m1 was released, and the time when the falling mass

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hit the floor, the “start” and “stop” times. [Note that as you move the cursor over adata point on a graph in Excel, a small dialogue box apears giving the coordinates ofthat point.] Make a note of those two times. Next on the data sheet select the emptycell just to the right of the start time. Imagine for example that the start time is in cell

A23. You will then just have selected cell B23. In that cell type = A23 - $A$23 andhit enter. The equal sign signifies that this is a computed mathematical expression.The A23 is the relative location of a cell. The $A$23 is the absolute location of thatcell. The cell should now show the result 0 . Click on that box – the one showing0 – and activate Copy. Now highlight all the subsequent cells in that column down toand including the one opposite your stop time. Activate Paste. You now have a timeseries, starting at t = 0, corresponding to the portion of the data you want to fit.

Highlight a rectangular block of data two columns wide and many rows deep wherethe left hand column is the time series you have just created.

Plot this data as you did before and put it on a new chart sheet. Using the same setof times, make separate plots of Velocity and Acceleration. You now have data setsrepresenting results that should be describable by simple analytical functions startingat t = 0. The only things of value you have lost are the names in the graph legend,but you can add those back using the Chart Wizard dialog box if you want.

d) Fitting the data. Go to your new chart sheet for the Velocity. Click on one of thepoints. A few representative points from that series will become highlighted. Go tothe Chart menu and click on Add Trendline. A Trendline dialog box will appear. Onthe Type panel choose Linear.

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Move to the Options panel and check the box for Display equation on chart. Click OK.[R-squared is an analytic measure of the “goodness” of the fit to the data. It requiressome interpretation and would just be a distraction here. We will not use it.]

The best fit liner representation of the Velocity data will appear as a solid line throughthe data and the resulting analytic expression will be displayed. Copy down the co-efficient of the t1 power (displayed as x1 on the chart). That is the fitted value forthe magnitude of the acceleration based on the Velocity data. Now click on the linethat was just generated. That takes you back to the Trendline dialog box. Go to theType panel and choose Polynomial and set the Order to 2. Remember to check DisplayEquation. Click on OK. You will now see the best fit using terms through t2. The fitshould not be much better. You can change the position of the displayed formula inthe graph by clicking on the formula, then moving the bounding box that appears.

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Fit the Position data using the Polynomial representation and Order = 2. Two timesthe coefficient of the t2 term is the fitted value for the acceleration based on the Positiondata.

Finding the acceleration from the Acceleration data requires some common sense.Although we expect a roughly constant set of values, you may see the points rise,plateau for most of the time span, then begin to fall before the end time. This may bedue to transient effects in the motion or it may be an artifact of the algorithm usedto compute the derivatives. In any case it is easer to determine the acceleration byeye from the data in the plateau region than it would be to try to fit the data recordto some analytic expression. Compare the value you find here with the values youdetermined from the position and velocity data.

e) The falling chain. fallingChain.csv corresponds to a time varying force. It came froma hanging chain experiment similar to the one you will perform in the classroom. Thegraphs of Position, Velocity and Acceleration should look like those you sketched ind) of Part 1. Proceeding as you did in b) though d) of the current part, isolate andgraph the Position, Velocity and Acceleration for the times between the start time andthe end time. The graph of the Velocity should show an obvious curvature downward,indicating a decrease in the force moving the cart. Fit the Velocity to an Order 3polynomial. The Acceleration data is noisy but its decreasing trend should be clear.Fit the Acceleration with an Order 2 polynomial. To judge the consistency of youranalytic fits, compute derivative of the expression you get for the Velocity and compareit with the expression you get for the Acceleration.

f) Print out the graphs with the fitted curves and resulting expressions you obtained in d)

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and e). Bring them to class and hand it in along with your in class experimental writeup. Add a statement to the graphs you hand in saying that you did the work on yourown, then sign it.No credit will be given for the experiment without these graphs.

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OpenOffice option

a) Downloading the data. We have taken two sample data sets, fallingMass.csv andfallingChain.csv. Download them from http://web.mit.edu/8.01t/www/Prelab%20Data/Open the OpenOffice.org Spreadsheet application. In that spreadsheet, choose Openfrom the File menu, navigate to the file fallingMass.csv, select it, and click on Open.In the dialogue box that appears the specification Comma Separated should alreadybe checked. Click OK. Four adjacent columns appear in the speadsheet labeled Time,Position, Velocity and Acceleration. You should select these columns and choose Cellsfrom the Format menu. Use the dialogue box that appears to make sure you have atleast 4 places after the decimal point for all your numbers.

b) Plotting the data. fallingMass.csv corresponds to dropping a fixed mass, thus to c) inPart 1. Use the cursor to highlight the rectangular block of data two columns wide,many rows deep, beginning with the title row – Time and Position– at the top. Eitherselect Chart from the Insert menu, or click on the Chart icon on the tool bar. On thefirst panel of the “Chart Wizard” dialog box which appears choose as Chart Type XY(Scatter). This will plot the data points without any distracting connecting lines.

Move from one panel to the next (you do not have to enter anything along the way)until you click on “finish”. The resulting chart appears as a figure on the sheet youare working on. It is convenient to have each chart you create on a separate sheet. InOpenOffice, one does not have the option of saving the chart to a separate sheet fromwithin Chart Wizard as one has in Excel. Rather, one must do it by hand as follows.Go to the bottom left hand corner of the spreadsheet window and click in the openspace just to the right of the tab labeled Sheet 1. A dialog box opens allowing you tocreate a second sheet. Go back to the first sheet. Click on the chart to select it. Copyit. Go back to sheet 2 and Paste the chart there. You may wish to expand the size ofthe chart by using the tiny squares that appear on the corners and edges of the chart.You should then delete that chart from the original sheet.

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OpenOffice will choose by default to display the left most column on the x axis andthe other column will be displayed as the dependent variable plotted on the y axis.Note that the legend on the graph uses the titles at the heads of the selected block tolabel the points on the graph. Admire your results. Do they look like your first sketchin c) of Part 1 (at least until the weight hit the floor)?

There are many changes one can make to the way the data is plotted, but at this stageone is more important than the others: the size of the symbols used to mark eachpoint. The symbols are probably too large. Correct this as follows. Double click onone of the symbols on the chart used to plot the data. This opens a Format Data Seriesdialog box. Choose the Patterns panel and reduce the Marker Size to an appropriatevalue, probably the smallest available. Click OK to see the results.

Repeat this procedure to get plots of Velocity and Acceleration. Here you must choosenon-adjacent columns for plotting. First select the portion of the Time column youwant to plot. On a Mac you must Command click on the corresponding portion ofthe non-adjacent column to add it to your first selection. Otherwise, everything inbetween is selected as well. On other operating systems the method may be different.You should compare the Velocity and Acceleration plots with your sketches from c) inPart 1.

It is a fact of life that determining the derivative of a function from a discrete represen-tation of that function (that is, a set of data points) is a subtle business at best, andparticularly difficult when there is noise added to the signal. We see this here. Theposition represents the actual data taken by the apparatus. The velocity and accelera-tion were computed from this data by computing first one – then a second – derivative.Logger Lite uses a special smoothing procedure to do this, but the representation ofhigher order derivatives gets successively noisier, and thus less useful. The questionarises, is it better to determine the (constant) acceleration by averaging the values onthe computed Acceleration curve or to determine it from the fit of analytic functionsto the Velocity or Position curves? You will study this question below.

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c) Preparing to fit the data. The results you sketched in c) of Part 1 are “piece-wiseanalytic”: that is, they are represented by continuous analytic functions in two ormore separate regions. But the functions change from region to region. The fittingroutine in OpenOffice is not sophisticated enough to handle this. Thus, we must dealonly with the second of the regions, that for t ≥ t0. Moreover, we must adjust the timeto be zero at the beginning of the fitted region and to end before m2 hits the floor.

There are many was of doing this. Here is a simple one that preserves the originaldata. Navigate back to the sheet with the data by clicking on the appropriate tab inthe lower left hand corner of the OpenOffice window. Click on the letter at the verytop of the Position column to select the entire column. Then choose Column from theInsert menu. A blank column will be inserted between the Time and Position columns.Determine from the data – perhaps from the velocity graph you just made – the timewhen motion began, that is, when m1 was released, and the time when the falling masshit the floor, the “start” and “stop” times. [Note that as you move the cursor over adata point on a graph in Excel, a small dialogue box apears giving the coordinates ofthat point.] Make a note of those two times. Next on the data sheet select the emptycell just to the right of the start time. Imagine for example that the start time is in cell

A23. You will then just have selected cell B23. In that cell type = A23 - $A$23 andhit enter. The equal sign signifies that this is a computed mathematical expression.The A23 is the relative location of a cell. The $A$23 is the absolute location of thatcell. The cell should now show the result 0 . Click on that box – the one showing0 – and activate Copy. Now highlight all the subsequent cells in that column down toand including the one opposite your stop time. Activate Paste. You now have a timeseries, starting at t = 0, corresponding to the portion of the data you want to fit.

Finally, highlight a rectangular block of data two columns wide and many rows deepwhere the left hand column is the time series you have just created.

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Plot this data as you did before and put it on a new chart sheet. Using the same setof times, make separate plots of Velocity and Acceleration. You now have data setsrepresenting results that should be describable by simple analytical functions startingat t = 0. The only things of value you have lost are the names in the graph legend,but you can add those back using the Chart Wizard dialog box if you want.

d) Fitting the data. Go to your new chart sheet for the Velocity. Double click on thechart and the cursor turns into an arrow. Point the arrow at one of the the pointsrepresenting Velocity and a message identifying that data series will appear. Rightclick and a dialog box will appear. In that box, click on Insert Trendline. A Trendlinedialog box will appear. On the Type panel choose Linear and check the box for Displayequation on chart. Click OK. [R-squared is an analytic measure of the “goodness” ofthe fit to the data. It requires some interpretation and would just be a distractionhere. We will not use it.]

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The best fit liner representation of the Velocity data will appear as a solid line throughthe data and the resulting analytic expression will be displayed. Copy down the coef-ficient of the t1 power (displayed as x1 on the chart). That is the fitted value for themagnitude of the acceleration based on the Velocity data. You can change the positionof the displayed formula in the graph by clicking on the formula, then moving thebounding box that appears. By double clicking on the equation you open a dialoguebox that allows you to change the number of decimal places in the coefficients of theterms. Four should be sufficient.

Finding the acceleration from the Acceleration data requires some common sense.Although we expect a roughly constant set of values, you may see the points rise,plateau for most of the time span, then begin to fall before the end time. This may bedue to transient effects in the motion or it may be an artifact of the algorithm usedto compute the derivatives. In any case it is easer to determine the acceleration byeye from the data in the plateau region than it would be to try to fit the data recordto some analytic expression. Compare the value you find here with the values youdetermined from the velocity data.

e) Chain.txt corresponds to a time varying force. It came from a hanging chain experimentsimilar to the one you will perform in the classroom. The graphs of Position, Velocityand Acceleration should look like those you sketched in part d) of Part 1. Proceed asyou did in b) though d) of the current problem: fit the Velocity to a linear function.The linear fit to the velocity should not represent the data very well. You can seethis by eye, taking into account the deviation of the line from the general trend of thedata, relative to the “noise” in the data. The acceleration data, absent any analyticfit, should still be clear enough to convince you that it is not a constant. Rather, it isdecreasing in magnitude as the amount of freely hanging chain decreases.

f) Print out the graph with the fitted curves and resulting expressions you obtained in d)and e). Bring this to class and hand it in along with your in class experimental writeup. Add a statement to the graphs you hand in saying that you did the work on yourown, then sign it. No credit will be given for the experiment without this graph.

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