LAB1: A Scientific Model for Free Fall.

Transcription

1 LAB1: A Scientific Model for Free Fall. I. Overview. This lab explores the framework of the scientific method. The phenomenon studied is the free fall of an object released from rest at the surface of the Earth. The steps of the lab are as follows: 1- Gather experimental data for the falling object: Height of fall H and time T to fall from H, measured for various heights H. 2- Construct a model describing the phenomenon of free fall: From the data gathered in step 1, you will determine a general equation for free fall; this will be an empirical equation (i.e. an equation derived from observation, not from first pricinples). This equation will relate the two variables H and T. This equation will constitute your model. 3- You will then test your model by predicting the time of fall for a different H and check the prediction of your model against experimental evidence. In this lab you also learn about DATA ANALYSIS (error analysis (the following section), the least square fit, and how to quote results; as well as the use of Excel to analyze your data. II. of Errors. Background: Data Analysis and the Treatment II. A Intro. The measurement of physical concepts (like distance or velocity to cite only two) associated with any phenomenon is at the heart of the scientific method and therefore of physics. However no measurement can be exact: For instance it is in principle incorrect to write that the length of the table is L=1.65m because 1.65 is an exact number. As with any measurement, some error must associated with the measurement of the table. Thus we should write instead that, for instance L=1.65 +/- 0.03m where ΔL=0.03m is an estimate of the error we make when we say that the table is 1.65m long. ΔL is called the uncertainty in L. Thus to do experimental physics we must devise a scheme to estimate

2 2 Lab 1 Free fall Sc Models and Data Analysis.nb errors. Before we do that let us categorize measurement errors somewhat more precisely: II. B Sources of error. II.B.1 The fundamental source of error: There is a fundamental source of error that it is impossible to avoid: to associate a number with a concept we have we need a standard: like a bar with two scratches on it can represents what we mean by distance (the distance between the scratches). Thus for every concept, here distance, we define its meaning with a STANDARD, here the bar with scratches). That standard is thus a physical representations of our original concept and is unique since to compare measurements they should all refer to the same object (which justifies the choice of the name standard). So the measurement of every physical quantity refers to one or more standards: distance to the meter standard, time to the time standard (Vibration of atoms in a crystal), velocity to both distance and time standards etc... As such, standards and thus all measurements (since they are all derived from standards) have an inherent source of error. Let s see why: Take the meter bar (which is the former standard of length) as an example: The standard of length was defined as the distance between two scratches on the bar. Now, no matter how carefully these scratches were made, or how expensive and fancy were the tools that made them, these scratches MUST have a certain width. That width, no matter how small it is (say (10) -7 m), introduces a unavoidable source of error of the same order (thus about (10) -8 m) when any distance measurement is made. If you think carefully about it, you will find out that any "fix" you can think of in order to get around this problem, might increase the precision of the standard but CANNOT ENTIRELY REMOVE THIS SOURCE OF ERROR. Although this error is for most measurements much smaller than others sources of error, it is the ultimate limiting factor that allows us to say that no measurement can be exact. II.B.2 The two types of measurement errors: Errors fall into two broadly defined categories: systematic errors and random errors. Systematic errors are errors that will always offset a measurement by roughly the same amount. A common example is making distance measurements with a meter stick whose end is broken off, say at 3mm from the 0. Thus all distances you would measure with such a stick will be off by 3mm (measuring a 15.4cm pencil, you would find l=15.7cm). There isn't any data manipulation of your measurements that could estimate or even account for this source of error. However good care in your measurements will greatly help in eliminating systematic errors from your data. Random errors have many sources: limitations in the precision of the measuring instruments, fluctuations of some of the parameters of the experiment (for instance, when driving your car with a constant pressure on the gas pedal, gusts of wind can alter slightly your speed as they hit your car. As a result, a repeated measurement of your speed will exhibit random fluctuations), etc... When errors

3 Lab 1 Free fall Sc Models and Data Analysis.nb 3 are random, mathematical analysis allows us to give estimates of their size. II. C Estimating random errors. Although the analysis and estimates of random errors is a complex mathematical subject if done carefully, we can easily justify the main results with common sense. In any case, whether derived from simple or sophisticated reasonning, the result used to estimate the magnitude of errors is an arbitrary choice. Errors in the measurement of a quantity Q will manifest themselves if we repeat the measurement of Q (say N times) in the same conditions: Because of random errors the outcome of the measurements won't be the same. Let's call these values: Q 1, Q 2, Q 3,...Q i,...q N. Since the errors are random, no measurement outcome is more likely, it is clear that the best estimate for the quantity Q itself is (and this is the first arbitrary choice that we justify by common sense) Q best =Q Ave 1 N i=1 N Q i ( indicates a definition ) Now let's define an estimate of the error we make in giving Q the value Q best by: ΔQ = σ N-1 ( Standard deviation of the Q i ) where the standard deviation is given by: N (Q σ N-1 i -Q Ave ) 2 i=1 N-1 Justifying the use of the standard deviation for the uncertainty is not hard but takes a little time. We do it in the next paragraph: Justification for the expression of the standard deviation as an valuable estimate of the error, or uncertainty in a series of measurements It is intuitively clear that the error is certainly related to how far the measurements stray from their average, hence the Q i - Q Ave in the formula. Since all the measurements are on the same footing (they must all be taken into account unless there is a KNOWN reason to throw one away) it makes N Q sense to take their average, hence arriving at: i -Q Ave i=1. But this sum has obviously a flaw: if some N Q's are larger than Q Ave and some smaller, a cancellation of the errors will result which is clearly unwelcome: you could have a zero uncertainty even though the individual values of Q could be all over the map! Not good! So, to take care of this problem we could just add the absolute values of the Q i - Q Ave 's, or equivalently, and this is what we do in the formula, sum the average of the square of the

4 4 Lab 1 Free fall Sc Models and Data Analysis.nb N (Q deviations and then take the square root of it. Hence: i -Q Ave ) 2 i=1 N. We're almost there: let's justify that N-1 that we are using instead of the N: Suppose we only made one measurement of Q. In that case we could not have any clue as to what the error on the measurement could be, thus σ should be undefined. Well, if we used the formula with N instead of N-1, we'd find σ, and thus our estimate of the uncertainty on Q, to be equal to zero since Q 1 - Q Ave =0 and N=1. It would obviously be stupid to estimate the uncertainty on a quantity to be zero if we made only one measurement of it, so the formula with N is not good, as least for small N! Now notice that if we use the N-1 formula instead for our one measurement case, the everything is fine because σ=0/0, which is undefined: Exactly what we wanted!! Hence the formula for σ. Try this out: Riding my bike to school on 4 consecutive days took me the following times: 24min, 20min, 19min, 19min. Compute, using your calculator and the formulae above the average time it takes me to ride to school and the uncertainty in the time it takes me to ride to school. Important footnote: Using Q=Qave +/-ΔQ gives us a bracket of values where we can say that Q lies, with a measure of confidence. However it does not say that Q is for sure in that bracket. Statistical analysis actually tells us that for random errors the chances of Q being within 1 standard deviation of the average are 67%. (and 90% within 2 standard deviations). lii. Square Fit. Background: Data Analysis and the Least In experimental science we very often need to find, or check, a relationship y(x) between two variables x and y that have been measured as a set of data pairs {(x1,y1), (x2, y2), etc... }. Let s assume that we are looking for the simplest possible relationship, a linear relationship where y(x) = ax + b (the case for higher order polynomial fit is similar). In general, our data points are not exactly aligned on a straight line because of experimental error, and thus we need find the best possible straight line through through these data points. Since a straight line is determined by its slope a and intercept b, the problem is thus to find the values for a and b that determine a line best fitting the data points. These best values can be determined from a number of schemes, the easiest of which is the least square fit. The least square fit that we'll use minimizes the sum of the "distances" squarred from each data point down (or up) to the fitting line.

5 Lab 1 Free fall Sc Models and Data Analysis.nb 5 Here s an example of how to proceed: First we have a table called datapoints of pairs {x,y} of experimental data: datapoints = List[{ , 0.250}, { , }, {-0.439, 0.698}, {-0.349, 0.932}, {-0.238, 1.099}] {{-0.705, 0.25}, {-0.63, }, {-0.439, 0.698}, {-0.349, 0.932}, {-0.238, 1.099}} We use Mathematica to plot y vs x for the 5 points: ListPlot[dataPoints] The problem now is to find a straight line that best fits this data set: So we are looking for y=ax+b through a data set {{x 1,y 1 },{x 2,y 2 },...,{x N,y N }}, the total distance from the data points to the best fit line is given by d total such that: d 2 total = n i=1 (y i - y (x i )) 2 where y i is the y value of the data point {x i,y i } and y (x i ) is the y value at x i as given by the line to be determined: y (x i ) = (a x i + b) We look at d total 2 as a function f (a, b) of 2 unknowns a and b. That is: f(a,b)=d 2 total = n i=1 (y i - (a x i + b)) 2 = (y 1 - a x 1 - b) 2 + (y 2 - a x 2 - b) 2 + (y 3 - a x 3 - b) We'll see (or already know) that to minimize a function we must set its derivative to zero so we obtain 2 equations to minimize the distance of the dat points to the line with respect to the slopa of the line and its intercept b: df da = 0 and df db = 0.

6 6 Lab 1 Free fall Sc Models and Data Analysis.nb These 2 equations determine the 2 unkowns a and b (so the problem is well posed: 2 unknowns; 2 equations). In case the curve is a higher degreee polynomial, we have more coefficients but also correspondingly more equations. To see an example of using Mathematica to fit a data set see the code at the end of this handout in Appendix 1. IV. Background: Quoting Measurements. IV. A Significant Figures : Recall that the number of significant figures (sig figs) is equal to the number of figures in a number, counting from the rightmost digit in the number towards the left until you reach the last non-zero digit. eg has 3 sig figs: from right to left we get 2, then 0, then 1 and 200 also has 3 sig figs: 0, 0 and 2. The zeros to the right of a number are the reason why you must use scientific notation 2*10 2 has 1 sig fig but 200 has three, so does 2.00*10 2. In 200 the zeros are there to tell us that it is not 199 nor 201, but 200; wheras 2*10 2 is consistent with any number between 150 and 249. And 2.0*10 2 is consistent with any number between 195 and 204. Standard deviations can be quoted with 1 or 2 sig figs, more and it becomes ridiculous! IV. B Quoting Measurements: Experimental results are quoted as Q=Q - ±ΔQ with ΔQ given by the standard deviation quoted with 1 or 2 sig figs. Since ΔQ gives the uncertainty in Q, it is clear that the number of sig figs in Q - should be consistent with ΔQ. NEVER EVER quote a result without writing down units, even for intermediate results; unless of course it is a dimensionless quantity. Angles that are dimensionless also require units, however: 90degrees is very different from 90 radians! For instance if ΔQ= s (s stands for seconds) and we wish to use 1 sig fig in the standard deviation, then we first round off ΔQ to ΔQ=0.03s. Then, assuming that we found Q - = s, ΔQ=0.03s tells us that the second digit after the decimal point is uncertain in Q - and thus the 4 is uncertain. It is clear that all the digits to the right of the 4 in Q - are therefore meaningless. So we must now round off Q - to its uncertain digit: we get 1.25s and thus quote the result as Q=1.25s±0.03s. In case you want to quote the standard deviation with 2 sig figs, say ΔQ=0.029s then we keep the

7 Lab 1 Free fall Sc Models and Data Analysis.nb 7 corresponding digits in Q -. We thus have the result Q=1.248s±0.029s. It is important to respect these rules. There are few things that look more idiotic than a result with 15% uncertainty (comon in many of our labs) quoted with 9 significant figures because that is what the calculator display shows at the end of the calculation. IV. C Propagation of Uncertainties There are rules concerning how uncertainties are propagated. For instance if Q=1.26±0.02 and R=0.05±0.01 how should we quote the product P=Q*R? The rules are simple and are derived quickly with some calculus - and we shall do it later. However the widespread use of software tools such as Microsoft's Excel has made it easy to perform large number of computations and avoid dealing with this technicality. In our example we would just compute Q i R i for each of the N values and compute the average and standard deviation of these N values of Q*R rather than use Q - R - and use the rule for getting Δ(QR) from ΔQ and ΔR. V. Free fall: the experiment. The Lab: Take a meter stick and a stop watch as well as an object of your choice (make sure it is dense and compact; e.g. take a 100g weight). The purpose of the lab is the following: you will use your data to propose a power law between time of drop and height of drop. Then, following the framework of the scientific method, you will make a prediction using the proposed law, for the time of drop from about 5m. You will then measure the time for that drop 6 times to test the predictio of your model and cooclude on its validity V.A Experimental study of the phenomenon. Drop the object from 5 different heights from about 0.8m to about 3m. Drop the object 5 times from each height and measure the time it takes to fall to the ground. It is strongly suggested that the same person drop the object and time the drop in order to minimize delays and thus systematic errors. V.B Analysis of the data and determination of the model Since in this analysis you will propose a power law for the relationship between H and T, you will look for a law of the form

8 8 Lab 1 Free fall Sc Models and Data Analysis.nb H = αt n where both n and α are to be determined from the data. To look for a power relationship, the easiest is to take the log of the equation yielding: Log[H] = n Log[T] + Log[α] where Log is the natural log. Thus n is determined from the slope of the graph of Log[H] vs. Log[T] and α from the intercept of the graph. You will be doing the analysis using Microsoft Excel. Open an Excel spreadsheet, and enter your raw data in table form (see last page for an example). Using Excel's Logarithm built-in function make the corresponding table of log T's. DO NOT compute the log(t)'s individually!! Enter an equal sign in the spreadsheet cell where the table of log's is starting and enter LN(address of cell of 1st time). Then enter your formula and drag that cell across the table and in a couple seconds you have a table of the logs. Then compute the average of these values for each height and the standard deviation of each data set. Compute the corresponding AVERAGE() and STDEV() of the log of the data set for each height (See last page Appendix 2 for an example of a layout). Now you can propose an empirical model for the power that governs free fall from rest, by graphing Log(H) vs Log(T). From the Excel graph, right click on the data, add a Trendline, and format the trendline to make sure the following is taken in account by Excel: Trendline is linear (we need a straight line), equation is displayed. Get get slope "n" and intercept "Log[α]" and give your expression for that power law: H = αt n V.C Model prediction and testing Now use your model to predict the time of fall from 5m: t predicted. Carry the experiment and measure the corresponding time of fall 6 times. Compute average and standard dev. to get t experimental ±Δt. If the 2 values of t don t agree compute the percent difference between them. V.D Conclusion Make sure to address the following questions: 1-Restate your model (write down the equation relating H and t using the parameters you found) and how well it followed your data (did the graph go through all the data points. This is the model you choose to represent free fall.

9 Lab 1 Free fall Sc Models and Data Analysis.nb 9 3-Test your model: Make a time of fall prediction from you chosen model for free fall and do this for a height significantly larger than the heights here and measure it experimentally (say 5m). Do you have agreement? If you don't it invalidates your theory; and if you do have agreement it doesn't prove your theory but increases confidence in it. Remenber: Scientific models, laws or theories cannot be proven correct, but only disproven by experiment. However as a model passes more and more experimental tests our confidence in its validity certainly grows.

10 10 Lab 1 Free fall Sc Models and Data Analysis.nb APPENDIX 1 : Code to compute a linear least square fit : Assume the folllowing experimental data set : datapoints = List[{ , 0.250}, { , }, {-0.439, 0.698}, {-0.349, 0.932}, {-0.238, 1.099}] {{-0.705, 0.25}, {-0.63, }, {-0.439, 0.698}, {-0.349, 0.932}, {-0.238, 1.099}} Let s plot this data set for y vs x: According to Mathematica we can access these values of the data points the following way: For y1 and x5 we would call: datapoints[[1, 2]] datapoints[[5, 1]] Let s plot these points to get an idea of the (x,y) relationship: dataplot = ListPlot[dataPoints] For these 5 data points, the sum of the distance squared of each to the line ax+b is (note that distance is computed along the y axis) dist2[a_, b_] = datapoints[[1, 2]] - a datapoints[[1, 1]] - b ^2 + datapoints[[2, 2]] - a datapoints[[2, 1]] - b ^2 + datapoints[[3, 2]] - a datapoints[[3, 1]] - b ^2 + datapoints[[4, 2]] - a datapoints[[4, 1]] - b ^2 + datapoints[[5, 2]] - a datapoints[[5, 1]] - b ^2 ( a - b) 2 + ( a - b) 2 + ( a - b) 2 + ( a - b) 2 + ( a - b) 2 Now we let Mathematica compute the derivatives of this function with respect to a then with respect to b: fa[a_, b_] = D[dist2[a, b], a]; fb[a_, b_] = D[dist2[a, b], b]; And now we set fa and fb equal to 0 simultaneously to find the line that minimizes the distance betwen

11 Lab 1 Free fall Sc Models and Data Analysis.nb 11 itself and the data points. Again, we let Mathematica do the hard work and find: NSolve[{fa[a, b] 0, fb[a, b] 0}, {a, b}] {{a , b }} To finish we now plot the data set as well as the best fit line through the set: Show[dataPlot, Plot[1.72 x , {x, -0.8, -0.2}]]

12 12 Lab 1 Free fall Sc Models and Data Analysis.nb Appendix 2: Example of Microsoft Excel layout for the lab data: