Lab 1a: Experimental Uncertainties


 Harry Wade
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1 Physics 144 Chowdary How Things Work Spring 2006 Name: Partners Name(s): Lab 1a: Experimental Uncertainties Introduction Our first exploration today is intended to introduce you to the true nature of experimentation: no measurement is complete without some comment on the confidence with which it was measured. The key to doing experiments in any scientific field is making measurements. But when you make a measurement, there is always some uncertainty associated with it. Many times, these uncertainties are called experimental errors, but to me, that has always implied sloppiness. Of course, a good experimentalist will always seek to minimize the uncertainties in an experiment. But it is impossible ever to make a perfect measurement. In addition to minimizing errors, an experimentalist has to know how to determine the magnitude of the uncertainty. As we will see in this lab, knowing how large the uncertainty is in a measurement can be critical to making conclusions based on the data. In most of our everyday experience, the confidence that we have in our information is never explicitly stated. For instance, when asking someone what the temperature is outside, the response is always nearly a single number: It s 65 o today. This usually means that the true temperature outside is somewhere near 65 o, within a few degrees or so. The amount from which our measured value may be off from the actual value is called the experimental uncertainty or confidence limit of our measurement. When we hear that the temperature is 65o, we assume an uncertainty of a few degrees, and dress appropriately. Experience tells us that most times the temperature difference is rarely outside that range. You do often hear the confidence level given for polls. You probably expect to hear it. In fact, you should demand to hear it. And whenever you hear an experimental result obtained by measurement, you should demand to know the experimental uncertainty. The concepts and numerical tools illustrated in this lab have many applications in business (e.g., quality control and marketing surveys), politics (opinion polls), law (questions of evidence and proof), medicine (significance of changes in a patient's lab results), economics (measures of economic indicators), sociology (measurements of societal trends), etc. There are several goals of this lab: in particular, after finishing this lab, you should understand a) how measurement uncertainty can be reduced with the use of repeated measurements; b) how to characterize the magnitude of the uncertainty for a measured quantity; and c) how to make appropriate conclusions from your data, based on the uncertainty. Specifically, we ll develop the tools needed to answer the following question: How many brown M&M s would you expect to find in a typical bag? Later, you ll apply these tools to answer the following questions: Does the acceleration of an object sliding down a ramp depend on the mass of the object? Does the acceleration of an object sliding down a ramp depend on the angle of the ramp?
2 Instrumental Uncertainties vs. Statistical Uncertainties When you measure an object's length, you typically stick a ruler next to the object and read off the measurement from the ruler. There is some uncertainty in this measurement. First of all, the ruler only has markings down to a certain precision (say, a mark every millimeter), so your uncertainty is at least that amount. Second, there are frequently uncertainties associated with the fact that the scale may not be properly calibrated. For examples: if the ruler is bent, then the uncertainty might be even larger than the smallest increment on the ruler, or the end of the ruler might be worn down, so that zero isn t really zero. The same kinds of issues apply to weighing objects, timing intervals, etc. You can try to obtain a better ruler (or scale or watch), but you will still always have some minimum uncertainty associated with your measuring instrument. In many cases, uncertainties in measurements are due to random issues. For instance, if you are determining the duration of an event with a stopwatch, you might click the stopwatch on too early in some measurements and too late in other measurements. You might improve your timing method, say with practice or by using some form of automation via computer, but there will still be some uncertainty. The uncertainty due to this kind of random error cannot easily be estimated from a single measurement. If, on the other hand, you make several measurements, you can determine the uncertainty from the scatter in the individual measurements. Furthermore, if you average the individual measurements, you will get a value that is more accurate than any single measurement. That s why we try to make multiple measurements of some kinds of quantities, and apply the ideas of statistics to estimate the value and the uncertainty in the value. In this lab, we ll learn about statistical uncertainties. Procedure How many brown M&M s would you expect to find in a typical bag? Each person will be responsible for a particular bag of M&M s. Please don t eat your M&M s right away! (Don t worry, you ll get to eat them soon enough! And for those poor souls who don t like M&M s, your instructor will make the ultimate sacrifice and eat them for you ) 1) Open your bag and place the M&M s on a clean sheet of paper in front of you. Count the total number and the number of brown M&M s in your bag. Record your results below: Total Number of M&M s Number of Brown M&M s Question #1: Based on your sole measurement, what is your best estimate of the number of brown M&M s in a typical bag? Question #2: Based on your sole measurement, can you determine a confidence level for your answer to Question #1? Briefly explain. 2) Fill out a row on the table on the board for your results. When the table on the board is completed, copy the data to the following table:
3 Table 1: Class Results Name Total # # Brown It probably isn t surprising that both the Total # and # Brown might be different for each person. However, you should notice that the numbers are only slightly different. The real questions now are What is the best estimate for the number of Brown M&M s? How big is slightly? 3) There are lots of ways to present this information. One way is as in Table 1. Let s try a graphical way next. First, let s see how much variation there is in the individual measurements of the Brown M&M s: # Brown M&M's Student Number Graph 1: Variation in # Brown M&M s
4 4) Hopefully, from Graph 1 you notice that the # Brown seems to fluctuate about some value. It seems likely that that value would be a good estimate for the number of Brown M&M s in a typical bag. Question #3: What does that value seem to be, according to Graph 1? How could you calculate that value? What other common name would we give to that kind of calculation? Calculate this value using your calculator, and draw a dashed horizontal line representing this value on Graph 1. Please talk to your instructor. 5) So it seems to make sense that averaging together the number of Brown M&M s found in a number of different bags will give a good estimate for the number of Brown M&M s in a typical bag. However, to really know this number, we d need to measure the number of Brown M&M s in every bag ever created! Then, we could obtain the true mean. Of course, this is not practical (but it sure would be tasty), so we settle for looking at a subset of the overall number of bags. Obviously, the larger our subs et, the closer our experimental mean will be to the true mean. So we have our experimental average. However, we still haven t determined our confidence level: we don t know how much variation to expect in a single bag. Also, we don t know have a good estimate for how close our experimental mean is to the true mean. Luckily, these two estimates are closely linked. WARNING: Lots of people are confused by the distinction in the previous paragraph. We re actually trying to answer two questions, both related to the experimental mean: 1) In any given bag, by how many Brown M&M s do you expect to be off from the experimental mean? (i.e., for any single measurement, how far off do you expect that experimental result to be from the experimental mean). 2) How far away do you expect the experimental mean is from the true mean? Please ask your instructor if you don t understand these two questions. 6) From Graph 1, we get a sense of the variation in individual measurements from the experimental mean (which itself is our best estimate of the true mean). Now, we ll make another plot that shows this variation in a different way. We ll plot the number of times a particular number of Brown M&M s was found vs. that particular number. The number of times a particular number is found is called the frequency of that number. So for example, if there were 5 bags that had 16 Brown M&M s, the frequency of 16 would be 5. Here s an example: let s say we found 3 bags with 12 M&M s; 4 bags with 15 M&M s; 5 bags with 16 M&M s; and 3 bags with 17 M&M s. I d make a table with two columns: the first column would be a particular measurement (here the number of M&M s in a bag) and the second column would be the frequency of that measurement (here the number of bags with that number of M&M s). Then I d plot the first column on the horizontal axis and the second column on the vertical axis. I find it easy to use X s. This kind of a plot is called a frequency distribution plot or a histogram. You have almost certainly seen this when teachers report how students did on an exam.
5 Table 2: Example Frequency Distribution Table Graph 2: Example Histogram # of Brown M&M s Frequency X 13 0 X X 14 0 X X X X 15 4 X X X X 16 5 X X X X Please ask if you re unsure what s going on with a histogram. 7) Now, make a histogram of the class data as on the board or in your Table 1. I ve provided some axes below, but you ll need to fill in the numbers on the horizontal axis. Also, make sure you label the horizontal axis. Finally, draw a dashed vertical line for your experimental mean. Graph 3: Histogram of class data for # Brown M&M s 8) Both Graph 1 and Graph 3 tell us information about the spread in the experimental mean. In other words, they graphically represent the uncertainty in that measurement. To calculate a number that corresponds to that spread, we ll look at the spread of each individual result. We ll consider the spread of each individual result to be how far away each individual result is from the average. In Table 3, copy the appropriate data from Table 1, and also fill in the values for Column 3, labeled # Brown Average. 9) You ll notice that some of the values you just calculated in step 8 are positive, and some are negative. This makes sense, as the average is somewhere in the middle of the individual results. However, this means that we can t just average together the individual spreads. Instead, we ll get rid of the negative sign by squaring. So Column 4 is Column 3, squared. Then, we ll take the average. And then, we ll take the square root (to make up for squaring before). That might seem a little complicated, but just follow along with the table, and as the last step, take the square root of the shaded box. Please ask your instructor to check your work.
6 Column 1 Column 2 Column 3 Column 4 # Brown # Brown Average (# Brown Average) Average Table 3: Calculating the Uncertainty. 10) The square root of the number in the shaded box is known as the standard deviation. This is the best estimate of the spread in the class results, and tells you the confidence level that you have: for any single bag, there is a 68% chance that the number of Brown M&M s will fall within the range between the experimental mean the standard deviation and the experimental mean + the standard deviation. (Actually, this 68% is only true if the histogram is a bellshaped curve, known more formally as a Gaussian. Our calculation of the standard deviation is still correct, and it is still a good indicator of the confidence level; it s just that we might not be able to say 68%.) On Graph 3, draw a dashed vertical line representing the experimental mean the standard deviation and the experimental mean + the standard deviation. Note that some of our measurements fall outside this range. As mentioned above, this range tells us where to expect to find 68% of all M&M bags. If we wanted to know the range in which 95% of the measurements would fall, we d double the standard deviation. And if we wanted to be even more certain, a range of 3 standard deviations to either side of the experimental mean would include 99.7% of all possible results! 11) However, this only answers half our questions. We ve identified the experimental mean along with the standard deviation. This answers the first question: In any given bag, by how many Brown M&M s do you expect to be off from the experimental mean? (i.e., for any single measurement, how far off do you expect that experimental result to be from the experimental mean). What about the second question: How far away do you expect the experimental mean is from the true mean? Well, it turns out to be fairly straightforward, given that we ve already calculated the standard deviation. The quantity of interest that we want is called the standard deviation of the mean, and is given by the standard deviation divided by the square root of the number of measurements that were made. This makes some sense: the more measurements you make, the closer your experimental mean should be to the true mean. So the standard deviation of the mean = standard deviation number of measurements And you expect the true mean to be within one standard deviation of the mean of the experimental average.
7 12) Determine the standard deviation of the mean. Conclusions We ve learned that making multiple measurements of the same quantity gives a better estimate than a single measurement. Averaging together multiple measurements to get the experimental mean gives an estimate of the true mean. We ve seen several ways to represent the spread in measurements, via graphs and calculation. We re confident that any single measurement lies in the range: experiment al mean ± standard deviation. We re confident that the true mean lies in the range: experiment al mean ± standard deviation of the mean Questions a) Your instructor is about to open up a bag of M&M s. Write down your best estimate for the number of Brown M&M s your instructor will find. b) The other lab section does the same lab as your half, and determines that their true mean lies in the range 9 ± 3. Might it reasonable to conclude that their bags came from a different case than your lab section s bags?
8 Lab 1b: Graphing Motion Introduction Our second activity today will give you practice with the connection between position, velocity, and acceleration. You ll get exposure to graphing motion curves (position vs. time, velocity vs. time, and acceleration vs. time). You ll use a computer and an ultrasonic motion detector to generate motion curves. You ll use the techniques you develop here in the next section of today s lab activities. All the motion in this section is onedimensional. We ve seen onedimensional motion in the path of a ball as it travels up and down when thrown straight up. Here, we ll look at it for a cart on a low friction track as it moves back and forth. Position vs. Time At your lab station is a cart on a low friction track. You can see that you can roll the cart back and forth along the track. The track is marked with a ruler, so you can measure the position of the cart. Below is a proposed position vs. time graph for the cart moving back and forth. position time 1) With your partner(s), determine how you would have to move the cart along the track in order to get this position vs. time graph. When/where would you not be moving? When would you be moving fast or slow? When would you be moving to the left or to the right? Briefly describe how you would move the cart to get the position vs. time graph shown.
9 2) On the axes below, sketch the velocity vs. time curve that corresponds to the position vs. time curve above. Some things to consider: a) On the position vs. time graph, were there any times when the velocity was zero? How did you know? Draw those first. b) On the position vs. time graph, weere there any times when the velocity was constant? Here s how you could check: for constant velocity, the acceleration is zero. Zero acceleration is constant acceleration, so we 1 2 can use eq. (1.2.3): x = x0 + v0 t + 2 a t. What does this become for a = 0? That should look like the equation of a straight line. So a straight line on a position vs. time graph means constant velocity. The steeper the slope, the faster it moves. That makes sense, since a steep slope means a larger change in position. We can go one step further: if part of a position vs. time graph is curved, the velocity isn t constant; in other words there s an acceleration! c) How about positive velocity vs. negative velocity? Well, positive velocity means that the cart is moving in the positive direction, and negative velocity means that the cart is moving in the negative direction. d) Put that together and draw the sketch below. There aren t any numbers, but it should show zero, positive, negative; large vs. small, constant vs. non constant. velocity time 3) Are there any times when the acceleration is NOT zero? Mark those points on the curve above with arrows. If the acceleration is positive (here that means the cart is speeding up), mark it with a plus sign. If the acceleration is negative, mark it with a minus sign. Note that you can t really have abrupt changes like we re showing here. Some things to note: Velocity is the slope of a position vs. time graph Acceleration is the slope of a velocity vs. time graph
10 Graphing Motion with LabPro At your lab bench, you should see a green box labeled LabPro. This is an interface that uses various sensors to collect data and sends the data directly to the computer, where it can be analyzed in a variety of ways. You should see that the LabPro unit is connected to a rectangular blue box with a gold screen on it (the Motion Detector). The Motion Detector is mounted on a stand at one end of the ramp. The Motion Detector works on the same principle as the echolocation used by bats to hunt insects and avoid midair collisions, except the Motion Detector use sound waves, while bats likely use some complicated combination of senses. Basically, the Motion Detector sends out a sound wave, and the sound wave bounces off nearby objects, returning to the Motion Detector. The time between the sound wave being emitted and then being detected can be used to determine the position of the nearby object. In this way, the Motion Detector can create a position vs. time graph. Then, by using the slope argument we talked about earlier, the computer can create velocity vs. time and acceleration vs. time graphs as well. Procedure 1) Turn on the power to the LabPro unit by turning on the power to the power strip on the table. Log onto the computer. On the desktop is a folder called PHYS141. Open that folder, and then open the folder named Phys144. You should see an icon named Lab #1.MBL; doubleclick that icon to launch a data collection program called LoggerPro. You might get some initialization messages; just click Scan, then OK. 2) You should see a graph open up, with Distance on the vertical axis and Time on the horizontal axis, just like the position vs. time graphs we worked with earlier. You ll notice that the Distance axis goes from 1.5 to 1.5, but we can t actually get to less than zero since that would be the other side of the Motion Detector, which only works in one direction! Also, due to limitations on the machine, it doesn t work well for distances less than 50 cm from the detector. So make sure you limit the motion of the cart to between 50 cm from the detector and 1.5 m from the detector. 3) Position the cart approximately 70 cm from the detector. Note that the detector isn t set all the way at one end, so you need to be careful using the ruler on the track. Hold the cart fixed, and hit Enter on the keyboard. You should here the Motion Detector emitting a rapid clicking noise. The program collects data for 10 seconds; move the cart back and forth and you should see the position vs. time graph appear on the screen. 4) Now, as best you can, follow your description of how to move the cart to get the position vs. time graph you worked with earlier. It doesn t have to be perfect, but I would like to see a flat line, a steep positiveslope line, a shallow negativeslope line, and another flat line. Play around until you get something you like, and then call me over to look at it. 5) Looking at your position vs. time graph, what do you think the velocity vs. time graph will look like? You can make the computer plot velocity vs. time for you. Go to View, then Graph Options, then Axis Options, and on Vertical axis, click Velocity and hit OK. Does the velocity vs. time graph look like you d expect? Note that there may be some funny points. There are two reasons: the Motion Detector isn t perfect, and you might have noticed some funny bumps and wiggles in the position vs. time curve. So that will give some problems. Also, the slope calculation can t be perfect either. However, most of the results should seem quite reasonable to you. 6) Now, looking at your velocity vs. time graph, what do you think acceleration vs. time will look like? Again, you can make the computer plot this for you as well, following the same procedure as above. What does this graph look like? 7) Go back and remove the velocity and acceleration parts from the graph, leaving just distance vs. time.
11 Lab 1c: Motion on Ramps Introduction Our final activity today will have you collect data using the Motion Detector and analyze it using the techniques we developed for statistical uncertainties. We ll want to answer the following 2 questions to the best of our abilities, experimentally. Does the acceleration of an object sliding down a ramp depend on the mass of the object? Does the acceleration of an object sliding down a ramp depend on the angle of the ramp? Half the lab section will try to answer the first question, and half the lab section will try to answer the other; you ll share your results with another group in your lab section. Procedure (same for both questions) 1) Use the wooden block carefully to prop up the end of the ramp away from the motion detector. Maintain the general direction of the ramp. Prop it up using the short dimension of the block. 2) It turns out that you ll get better data if you start the cart low on the track, give it a push up and let it coast to its highest position and then coast back. Please don t let the cart bang into either end of the ramp; stop it with your hands if necessary. Try it a couple of times until you get good data; it should look something like this: Please show me your graph before proceeding. 3) Now, show velocity on your curve. Hopefully the velocity vs. time graph is linear in the region of interest (as we d expect for constant acceleration), and we can obtain the acceleration by getting the slope of the velocity vs. time graph. Ask your instructor how to get the slope, and write it down. 4) Repeat the experiment until you have at least 5 values for the acceleration of the cart. Write down your experimentally determined values for the acceleration of the cart.
12 5) At this point, your instructor should have told you which question you re going to answer. Perform your second experiment. Make sure you obtain and write down at least 5 values of acceleration. 6) Now, using the statistical tools we developed earlier, determine the experimental mean, the standard deviation, and the standard deviation of the mean for the acceleration of your two cases (your group and your partner group). Write down the experiment al mean ± standard deviation of the mean for each case. Do they overlap? If they do overlap, what can you conclude about the acceleration for your two cases? If they don t overlap, by how much are they apart? What can you conclude in this case? Please discuss your conclusions with your instructor, and turn in your lab packet.
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