1. "If reproducibility may be a problem, conduct the test only once."
2. "If a straight line fit is required, obtain only two data points."
"Vesilind's Laws of Experimentation," cited by Arthur Bloch, Murphy's Law and
Other Reasons Why Things Go Wrong!, Price/Stern/Sloan (1977), p. 75.
Preparation
Due 12:00 PM before start of lab
Prelab assignment 4 (*.html)
Equipment
laboratory laptop, PASCO Capstone
table clamp, rightangle clamp, rods (for pulley)
collision track, cart, "cargo blocks," pulley, string
10 g1000 g mass set
PASCO Motion Sensor II
"750 Interface" box, power and USB cables
Microsoft Excel
Big Ideas
A system of objects subject to an external net force along a fixed direction will accelerate along that direction (Newton's second law). Ultrasound sensors can be used to track the motion of objects. Data subject to random experimental errors can be characterized statistically (average, standard deviation) and represented graphically using error bars.
Goals
Students work in (selfassigned?) groups to understand how to apply Newton's second law to the behavior of a twobody system (modified Atwood's machine).
Students learn how to use ultrasound sensors to track the motion of objects.
Students learn how to handle data subject to random experimental errors by using spreadsheets to calculate the average and standard deviation of repeated measurements, and to display these using error bars on graphs.
Students build upon previous knowledge of graphing and curvefitting software, and best practices for data collection and graphical analysis.
Tasks
(Record your lab partners' names on your worksheet for tasks 12, to be turned in at the end of today's lab for randomly selected grading for your group.)
1. Modified Atwood's Machine Setup Turn a carts over, and test the wheels by spinning them (if they don't freely spin, or noticeably slow down right away, the axles will need to be replaced). Set up a modified Atwood's machine, with a 10 g mass hanging from a string that goes over a pulley, with the cart attached to the other side of the string. The cart should be free to move horizontally on the track as the hanging mass falls downwards. (For each run, prepare to catch the cart just before it hits the pulley, and also check that the string still runs over the pulley wheel.) Place the motion sensor at the other end of the track (for later use).
 Release the cart from rest at a point 15 cm from the motion sensor, and observe the behavior of the hanging mass and cart. Take the direction of the hanging mass and the cart's motion to be the positive xdirection (so the horizontal direction "turns the corner" at the pulley and continues downwards). Assuming that the string does not stretch, the hanging mass and the cart will always have the same velocity and the same acceleration, and this twobody system can be considered a single mass subject to the "horizontal" force of the hanging mass weight. After the cart is released, is the sign of the velocity v_{x} of the cart and hanging mass system positive or negative (or zero)?
Sign of system v_{x}: [ + − 0 ]
Brief explanation:
 Is the sign of the acceleration a_{x} of the cart and hanging mass system positive or negative (or zero)?
Sign of system a_{x}: [ + − 0 ]
Brief explanation:
 Now attach a 50 g hanging mass from the string, and again release the cart from rest. Is the acceleration of this twobody system (hanging mass and cart) less than, the same as, or greater than the acceleration with the 10 g hanging mass? Use Newton's second law to explain why this is true, by identifying which quantities (total system mass, external net force value) change or remain constant.
Acceleration with 50 g hanging mass: [ < = > ] with 10 g hanging mass
Brief explanation:
 Now with a 50 g hanging mass attached to the string, load the cart with two "cargo blocks," and again release the cart from rest. Is the acceleration of this twobody system (50 g hanging mass and loaded cart) less than, the same as, or greater than the acceleration with the 50 g hanging mass and unloaded cart? Use Newton's second law to explain why this is true, by identifying which quantities (total system mass, external net force value) change or remain constant.
Acceleration with loaded cart: [ < = > ] with unloaded cart
Brief explanation:
 Select the narrow beam setting (the cart symbol on the "person/cart" switch") on top of the motion sensor. Plug the yellow and black motion sensor plugs into the "1" and "2" ports on the front of the interface box, and connect the USB cable from the back of the interface box to the laptop. Plug in the AC power into the interface box as well, and flip the switch in the back to "on." Make sure that the motion sensor is angled to look horizontally down the track, by setting the direction knob to "0".
 Run the PASCO Capstone software package. From "Tools" on the lefthand side of the screen, click on "Hardware Setup," and click on the Bluetooth button to stop it from looking for wireless devices. Click on the "1" port on the interface box, in the dropdown window, select "Motion Sensor II." The interface box window should now depict both "1" and "2" ports connected to a motion sensor icon. Click on "Hardware Setup" again to hide this display. (At the bottom border of the screen, you may need to increase the sampling frequency from "20.00 Hz" to "40.00 Hz" to get better results.)
 From "Displays" on righthand side of screen, drag a "Graph" window onto the blank page. Click on "< Select Measurement >" on the horizontal axis of the graph, and in the popup window, under "Time" select "Time (s)." Click on "< Select Measurement >" on the vertical axis of the graph, and in the popup window, under "Motion Sensor II" select "Position (m)."
(At this point, and at other critical points, be sure save this workspace as a file to the desktop. PASCO Capstone may freeze or crash, but you can then exit the program completely, and click on the saved file on the desktop to start where you left off.)
 Now use a 20 g hanging mass with the cart still loaded. Release the cart from rest at a point 15 cm from the motion sensor (this is the minimum detectable distance for the motion sensor), and at the same time the cart is released, press the "Record" button at the bottom of the screen. (It's okay to press the "Record" button just a little after the cart is released, but not before the cart is actually released; this may take some practice.) Make sure to catch the cart just before it hits the pulley, and then press the "Stop" button to end the data run.
 Click on "Highlight range of points in active data," and resize the area to enclose only of the data points where the cart is accelerating down the track (leave out the flat portions). Click on "Apply selected curve fits to active data  Select curve fits to be displayed" and choose "Quadratic: At^{2} + Bt + C."
 This software may report uncertainties with two significant figures. Truncate/round all results for the horizontal motion parameters with just one uncertain significant figure (e.g., rewrite "−4.85 ± 0.18" to "−4.9 ± 0.2," or "0.0275 ± 0.0076" to "0.028 ± 0.008").
C = x_{0} = __________ ± __________ m.
B = v_{0x} = __________ ± __________ m/s.
A = (1/2)⋅a_{x} = __________ ± __________ m/s^{2}.
Note that this is just matching the variables from the quadratic curve fit equation:
x(t) = A⋅t^{2} + B⋅t + C,
with the horizontal motion equation for constant acceleration motion:
x(t) = (1/2)⋅a_{x}⋅t^{2} + v_{0x}⋅t + y_{0}.
 What is the numerical value of the horizontal acceleration a_{x} of the cart? (Make sure you have the correct positive or negative sign, use the proper number of significant figures/decimal places.) Briefly explain how this sign is consistent with the placement of the motion sensor and the initial motion of the cart (and also your answer for 1(c) above).
a_{x} = __________ m/s^{2}.
Brief explanation:
 Repeat 1(i)(l) again, and report the remeasured value of the horizontal acceleration a_{x} of the cart for four more trials (for a total of five measurements). You should typically get slightly different results; this variation should be due to random experimental error, which can be characterized statistically. You don't need to record the uncertainties for each measurement here.
Horizontal cart acceleration with 20 g hanging mass
Trial → 
1 
2 
3 
4 
5 
a_{x} (m/s^{2}): 





 Calculate the average of these five measurements (keep the raw result for now, to be truncated later).
Average a_{x} = __________ m/s^{2}.
 Calculate the standard deviation σ of these five measurements, which is a measure of the "spread" of these data points (assuming that the error is random in nature). Use WolframAlpha (*.html) to calculate the standard deviation, by replacing the example numbers in the text box with your data points from above (keep the raw result for now, to be truncated later).
Standard deviation σ_{ax} = __________ m/s^{2}.
 The standard deviation of the random experimental errors can be considered the uncertainty of the average. Truncate/round the average and standard deviation of a_{x} to one uncertain significant figure.
Data set a_{x} = __________ m/s^{2} ± __________ m/s^{2}.
2. Plotting a Data Set with Random Experimental Errors
 The data set below lists the results of dropping a mass from different heights onto rock salt (independent variable) and the resulting impact crater depths (dependent variable). Because slightly different crater depths would result from random experimental error, the mass was dropped several times for each height.
Crater depth vs. impactor drop height
 A  B  C  D  E  F 
1 
Drop height (cm) 
Crater width 1 (cm) 
Crater width 2 (cm) 
Crater width 3 (cm) 
Average width (cm) 
Std dev (cm) 
2 
10.0 
1.5 
1.0 
1.5 


3 
20.0 
1.5 
1.0 
2.0 


4 
30.0 
2.0 
2.5 
1.5 


5 
40.0 
2.5 
3.0 
1.8 


6 
50.0 
3.5 
4.0 
4.5 


7 
60.0 
3.5 
4.0 
4.5 


8 
70.0 
5.0 
4.5 
5.5 


9 
80.0 
4.5 
5.0 
5.0 


10 
90.0 
5.0 
4.5 
5.0 


11 
100.0 
6.0 
5.5 
5.0 


(Data from B. Nelson, H. Moreno, A. Stitzer, "How Deep of a Crater Does an Impactor Make in Rock Salt When Dropped from Different Heights?" (April 2017).)
Use a software package (such as Excel, instructions below, which may be slightly different depending on the version) to graph this data set, and to apply a linear trendline with vertical error bars.
 Manually enter (or cutandpaste linebyline) the data from cells A1:D11 (the ":" denotes the range of data starting from cell A1 in the upper left corner to D11 in the lower right corner).
 For cell E2, type in "=AVERAGE(B2:D2)" and hit return. This will calculate the average of the B1, C1, and D1 cells. Copy and paste this cell onto cells E3 through E11.
 For Cell F2, type in "=STDEV(B2:D2)" and hit return. This will calculate the standard deviation (a measure of the "spread" of the random experimental errors). Copy and paste this cell onto cells F3 through F11.
 Highlight all of the cells, and then rightclick, and then in the popup window, select "Format Cells...," and under "Category" select "Number," and set "Decimal places" to 1. This will properly display all the cells with one decimal place.
 Select an unused blank cell anywhere else in the spreadsheet. Under "Insert" on the top bar menu, click on "Scatter," and in the popup window select "Scatter with only Markers" (plotting data points only, no connecting lines or curves). A (blank) graph should appear.
 Click on the graph. Under "Design" on the top bar menu, click on the "Select Data" button. In the popup window, click on the "Chart data range" box, and on the spreadsheet, clickdragrelease to select the cells A2 through A11; and then while pressing the "Ctrl" key, clickdragrelease to select the cells E2 through E11. (Check to see if "=Sheet1!$A$2:$A$11,Sheet1!$E$2:$E$11" is in the box; this is how to select independent and dependent parameters that are not in adjacent columns.) Click on "OK" when done. You should now see the data points plotted on the graph.
 Rightclick on any of the data points in the graph, then select "Add Trendline...," and in the popup window, select "Linear." Click on the "Options" tab, check both "Display Equation on chart" and "Display Rsquared value on chart" options. Then click "OK" when done.
 Click on the graph. Under "Layout" on the top bar menu, click on "Error Bars," and in the popup window click on "More Error Bars Options..." In the popup window, select "Vertical Error Bars," set "Display Direction" to "Both," set "End Style" to "Cap," and under "Error Amount" select "Custom" and click on "Specify Value." In the popup window, under "Positive Error Value" clickdragrelease to select the cells F2 through F11, and under "Negative Error Value" clickdragrelease to also select the cells F2 through F11, and then click "OK" when done. The vertical error bars should now correctly display the standard deviation for each data point. (As for horizontal error bars, typically independent variables will not be subject to random experimental errors that vary from measurement to measurement, so we can neglect them.)
 Print out one copy of this graph (to be attached to the one report from your group to be selected later for grading today).
 Which data points are outside of one standard deviation from the linear trendline? (These points would be candidates for taking more measurements, if time allowed, and this would ideally beneficially result in smaller error bars and a better linear trendline.)
Brief discussion:
3. Modified Atwood's Machine with Different Hanging Masses
(Done on whiteboard only, to be worked on and presented as a group.) Develop an experimental trendline equation (not necessarily linear?) of how the acceleration of a cart loaded with two "cargo blocks" (dependent variable) depends on the amount of hanging mass (independent variable), starting from 20 g. Refer to the example data table below to create your spreadsheet.
Acceleration vs. hanging mass
 A  B  C  D  E  F 
1 
Hanging mass m (kg) 
Cart a_{x} trial 1 (m/s^{2}) 
Cart a_{x} trial 2 (m/s^{2}) 
Cart a_{x} trial 3 (m/s^{2}) 
Average a_{x} (m/s^{2}) 
Std dev (m/s^{2}) 
2 






3 






4 






5 






6 






7 






8 






9 






10 






11 






(Due to the softwareintensive nature of today's lab, it is not necessary to write a procedure for your group whiteboard project other than to cite the motion sensor type and software package used.)
 Print out one copy of your data table, and print out one copy of your graph (with trendline equation and error bars); you do not need to record the data table or draw the graph on a whiteboard. Note that you should follow the suggested bestpractice guidelines for data collection and graphical analysis:
 Minimummaximum data range (spanning a factor of at least 5×, 10× is better).
 Has minimum number of data points (10).
 Concentrated data in rapidly changing portions of graph.
 Variable data points should be an average of repeated measurements (in this case, three), with a standard deviation reported in the data table, and represented with vertical error bars on the graph.
 Outlying data points should be replaced/removed.
 Proper choice of trendline fit type.
 Write out a concluding statement on the whiteboard regarding the validity of your mathematical modeltesting it by measuring the acceleration using a hanging mass greater than your maximum experimental hanging mass, and comparing the percent discrepancies between the measured acceleration, and predicted acceleration from your trendline equation. Include the specific relevant numbers in this statement, such that it can be read (and cited) on its own without referring to the above calculations and numbers.
 Bring up your whiteboards to the front of the class, to be presented to the instructor, which should include:
 A descriptive abstract: a brief summary of the purpose and methods used, but not of the data nor the conclusions. (Use the sample abstract below for today's lab.)
"We used an ultrasound sensor to track the motion of a modified Atwood's machine, where a mass hangs from a string and pulley that pulls on a cart allowed to move horizontally. We then constructed a mathematical model of how the horizontal mass accelerated due to a range of different hanging masses, and tested the validity of this model by predicting and measuring the cart acceleration for a hanging mass value outside the range of the original data set." Stepbystep procedure.
 Data table, calculations and/or results.
 Evidencebased conclusion statement.
 Documentation Rubric (tasks 12)
(Graded from randomly selected group member)
Score  Description 
3  Explanations complete and calculations correct, or very nearly so. 
2  Essentially complete; few explanations/calculations missing or incorrect. 
1  Substandard effort; substantive amount of explanations/calculations missing or incorrect. 
0  Unacceptable or no significant effort. 
 Whiteboard Rubric (task 3)
(Graded as a group, evaluated by instructor during debrief session)
Score  Description 
3  Complete, thorough, understandable, with little or no clarification needed during verbal instructor critique (can be resubmitted and presented again with requested corrections/revisions made). 
2  Minor problems; some corrections/revisions requested by instructor still needed, but not completed. 
1  Minimally acceptable effort, essential/critical revisions still needed. 
0  Unacceptable or no significant effort beyond experimental work. 
Followup
Complete this week's lab report and postlab assignment, next week's prelab assignment, and review lab instructions.
Due 12:00 PM before start of next lab
Postlab assignment 4 (*.html)
