Measurements & Uncertainties

Transcription

1 Measurements & Uncertainties Before you come to lab, make sure that you have read the Graphing Tutorial and the Uncertainty & Error Analysis Tutorial at the beginning of the lab manual. Introduction Experimental measurements and theoretical formalism provide the foundation of scientific disciplines. Experimental observations of the world around us motivate theorists to construct mathematical descriptions that explain the observations. In turn, the validity of a theoretical model is tested by the ability to confirm the mathematical predictions with experimental measurements. Experiment drives theory, and theory drives experiment. In order for this reciprocal relationship to function successfully, careful and reproducible measurements must be made. Because even the best measurements have some degree of uncertainty associated with them, experimental data are of little use unless they include a statement of the size of the uncertainties associated with the measurements and of how the uncertainties were determined. This tells others how much confidence they should have in the experimental measurements. This lab focuses on estimating and propagating uncertainties associated with experimental measurements in the context of the introductory physics laboratory. Equipment Background Meter stick Cylindrical objects Electronic balance Ruler The accuracy of a measurement depends on many factors including the precision of the measurement tool, the measurement method employed, and the person performing the measurement. Slight changes in how the measurement is performed and/or who performs it can yield different results for the same measurement. Even when the same person repeats a measurement multiple times using the same protocol, slight variations in the measurements are likely to occur. Because there is always some degree of uncertainty in measurements, no one measurement is precisely correct. Rather we report our best estimate for the measurement. In order for others to interpret the validity of our measurements, all measurements should include an estimate of uncertainty. Say we are measuring the value of some parameter X - we would report our measurement as X X best X where X best is our best estimate of the measured parameter, and X is the uncertainty we associate with the measurement of X. A statement of this nature indicates that we believe that any measurement of X is likely to fall somewhere in the range of (X best - X) and (X best + X). If we were to repeat this measurement multiple times, it is likely that our measurements would fall somewhere in this range. The best estimate of a measurement, X best, is typically obtained by taking the average value from a set of measurements that were repeated multiple times. For example, the data in Table 1 are from three independent measurements of the length of a rectangle. The best estimate of the length of the rectangle is the average value from the three measurements. When it is not possible to repeat 1

2 measurements multiple times, as will be the case in some of the labs for this course, the best estimate for the measurement is simply the most careful measurement you can perform. Trial Length (cm) l l 1 l l 3 3 cm Table 1: Determining the best estimate for the length of a rectangle: l =.15 cm Once the best estimate for a measurement is obtained, we need to estimate the uncertainty associated with the measurement. One form of uncertainty in experimental measurements has to do with the precision of the tool being used to perform the measurement. Consider, for example, measuring the height of a table using a meter stick. The smallest divisions on a meter stick are 1 mm increments, so we could make a precise measurement good to the nearest millimeter, 91.4 cm, for example. Typically, however, we record a measurement to the precision of the measurement tool and then estimate the next decimal place. Our measurement of the table height might then become cm where the hundredths place is an estimate. The meter stick permits us to measure the height of the table to the nearest millimeter (0.1 cm). The uncertainty associated with this measurement due to the precision of the meter stick is then less than this smallest division, on the order of 0.5 mm = 0.05 cm. We would report our measurement of the table height as cm. When a measurement is repeated multiple times and the errors are random, the standard deviation can be used as a measure of the uncertainty; this approach is illustrated in Table. In this example, the height of the table, h, is measured three different times. The best estimate of the table height is obtained from the average of the three measurements h h 1 h h The uncertainty in the measurement of the table height is obtained from the standard deviation h 1 h h h h 3 h 31 We then report the height of the table to be cm. Trial Height, h (cm) Average Standard Deviation 0.08 Table : Using repeated measurements to estimate the average uncertainty in the height of a table

3 Uncertainties estimated from repeated measurements are often better indicators of the actual uncertainty associated with a measurement because they take into account more than just the precision of the measurement tool. Most experimental parameters of interest are derived from one or more measured values, each of which have an uncertainty associated with them. For example, the density of an object, (the Greek letter rho ), is related to the mass, M, and volume, V, by M V Measuring the mass and volume of the object will permit us to calculate the density. Both M and V have uncertainties associated with them, and these uncertainties will propagate through the calculation of the density and influence its uncertainty. In order to determine the uncertainty of the calculated value for the density, we need to address how uncertainties are propagated when measured parameters are combined arithmetically or evaluated in functions. The Uncertainty & Error Analysis Tutorial at the beginning of the lab manual discusses this topic in detail. Read this tutorial carefully before proceeding. Appendix 1 (p.10) summarizes some important rules for propagating uncertainties that were developed in that tutorial. For a detailed discussion of propagation of uncertainties, see An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements by J.R. Taylor. Procedure In this lab we will investigate three different approaches for handling uncertainty in experimental measurements. In the first two sets of measurements, we will consider propagation of uncertainties when the measurement tools are the limiting factors in the precision of our measurements. In the third exercise, we will acquire a data set that has some inherent variability due to the person performing the measurement and the actual measurement technique. In this case, we will appeal to repeated measurements to estimate the uncertainty in our measurements. In the final example, we will investigate how graphical analysis of a data set can eliminate systematic errors in measurements. Archimedes Principle The density of an object,, can be determined from the relationship M V where M is the object s mass and V is the object s volume. When both M and V are easily measured quantities, the density of an object is simple to determine with a scale and a ruler. However, when an object has a complicated shape, it may not be possible to measure its volume or density. This was the problem faced by Archimedes of Syracuse, a Greek mathematician, physicist, and inventor who lived c. 00 BCE. Legend has it that the king asked Archimedes to determine the amount of gold in the king s irregularly shaped crown without damaging it. As Archimedes was contemplating this problem in the bath, he noticed that the water level rose as more of his body was submerged. He realized that this principle could be used to measure the volume of the crown. Archimedes published these findings in his treatise On Floating Bodies and the relationship between buoyancy and displacement became known as Archimedes Principle. It states that when an object is immersed in a liquid of density ρ it experiences an upward buoyant force F b given by F b = Vρg 3

4 Where V is the volume of the liquid displaced by the object, and g is the acceleration of gravity near the Earth s surface. (Vρ) is the mass of the displaced liquid. The vector diagram in Figure 1 shows the forces acting on the sample when it is submerged. F T is the force exerted by the string on the mass (tension). F T F b mg Figure 1 Since the object is not accelerating, Newton s Second Law leads us to conclude that the net force on the object is zero. In the diagram, the magnitude of F T must equal the weight of the object (Mg) minus the buoyant force F b (weight of liquid displaced): ΣF=0 F T + F b Mg = 0 F T = Mg - F b Rewriting the last equation in terms of densities gives: F T = Vρ g Vρg Where ρ is the density of the unknown metal and ρ is the density of the water. Dividing by the weight of the metal, Mg, (which equals Vρ g) yields the ratio of the weight in the water to the weight in the air: 4

5 FT Mg 1 Attach the sample to a hand scale with a string, and measure its mass in air and in water (Note: The scales are calibrated in mass units of grams). The scale reading in water is F T, the force exerted by the string (tension) on the mass. The density of water is 1000 kg/m 3 at standard temerature and pressure (STP). Compute the value of the density of the metal based on your measurments. Determine an algebraic expression for the fractional uncertainty of the density of the metal. Calculate a numerical value for the absolute uncertainty of the density of the metal and then report the value for the density of the metal in the form of X best ΔX. Calculate the Percent Difference between X best and the actual value of the density of the metal, listed in Table 3. Metal Density, ρ (g/cm 3 ) Brass (B or BB) 8.69 Aluminum (A or AA).70 Titanium (C or CC) 4.67 Table 3: Density of metals used for Archimedes Principle Measuring Human Reaction Time We will now consider an experiment to estimate human reaction time. These data will exhibit more variability from measurement to measurement than those associated with measuring the density of the metal cylinder because many factors associated with how the experiment is performed may change slightly from trial to trial. Because of this variability in the measurements, we will use repeated measurements to obtain average values for the reaction time and the associated uncertainty. In order to estimate human reaction time, we will consider how long it takes one person to catch a meter stick that is dropped through their hands by a second person. One person should hold the meter stick vertically at the 100 cm end. A second person, whose reaction time is to be determined, should place their hand around, but not quite touching, the meter stick at the 40 cm mark. The person catching the meter stick should watch the meter stick near their hand (and not the person holding the stick). As soon as the person catching the stick sees the meter stick start to move, he/she should grab the meter stick. We are interested in measuring the distance that the meter stick drops relative to the 40 cm starting point d y final y initial d y final 40cm where y final is the final position of the person s hand and y initial = 40 cm is the initial position of the person s hand. For each person in your group, perform 10 trials and record the final position in a data table in your notebook. Table 4 illustrates a representative data table for the reaction time experiment. 5

6 Trial y final (cm) d (cm) 1 10 Average Standard Deviation Table 4: Sample data table for human reaction time measurements For an object falling freely from rest under the influence of gravity, the distance, d, that it travels in some time interval, t, is described by d gt Rearranging the previous equation, we can solve for the reaction time in terms of our experimentally measured distance t d g g d 0.5 As you will see when you perform multiple measurements, your reaction time will vary slightly from trial to trial. Compute the average value of d and its standard deviation ( d ) for your set of ten measurements. Compute the best estimate of your reaction time. Determine an algebraic expression for the fractional uncertainty of your reaction time in terms of the fractional uncertainty of d. Report your reaction time in the form t = t best t. Eliminating Systematic Error through Graphical Analysis In the two previous examples, we addressed how to propagate uncertainties for single and repeated measurements. If a systematic error in our measurements existed (for example, a balance was consistently reading 0.5 g less than the actual masses of objects), both of these earlier approaches would be influenced by this systematic error. Even if we had taken great care in performing the measurements, and had correspondingly small uncertainties in our final results, our answers could be offset from the actual value of the parameter, and we would have no way of knowing it. Before proceeding, carefully read the Graphing Tutorial. One way of avoiding systematic errors is to appeal to graphical analysis. Let us consider the following example to illustrate how graphing a data set can eliminate systematic errors. Hooke s law states that the extension of a spring is proportional to the force causing the extension. F k x where F is the magnitude of the force applied to obtain the extension of the spring, x, and k is the spring constant that describes the stiffness of the spring. 6

7 If we hang a spring vertically and attach masses to the end of it, the restoring force of the spring balances the weight of the masses, so Mg k x In this experiment our independent variable is the mass (M) that we hang on the spring. The resulting extension of the spring ( x ) depends on the mass and is, thus, our dependent variable. Solving for the dependent variable, we find that x g M k If we were to plot x vs. M we would obtain a straight line with a slope equal to the ratio of the acceleration of gravity to the spring constant. From a linear fit to this data set, we could determine the spring constant from the slope of our line, slope g g k k slope If the hanging masses were not carefully calibrated and had a systematic error such that what we perceived to be a mass M was really (M + M error ), then the previous equation would become g x ( M M k slope intercept error g g x m M k k error Inspection of the above equation shows that we could still estimate the spring constant from the slope of a line fit to our data without the systematic error of the balance affecting our results. What we would observe in our graph is that our line would have the same slope as before, but the systematic error introduces a vertical offset into the data. This would show up in the y-intercept of the graph. The data in Table 5 are measurements of the period of a pendulum, T, when the pendulum length, L, is varied. Mathematically, we expect these two parameters to be related by ) T L g Manipulate the equation for the period of a simple pendulum so that it becomes linear in the independent variable L. Create a plot 1 of the pendulum data in Table 5 that is linear in L. Add a best-fit line to the plot, and determine the acceleration of gravity (g) from the fit to the data. 1 Refer to the Graphing Tutorial for tips on graphing. 7

8 Length, L (m) Period, T (s) Table 5: Period measurements for a pendulum with varying lengths Concluding Questions When responding to the questions/exercises below, your responses need to be complete and coherent. Full credit will only be awarded for correct answers that are accompanied by an explanation and/or justification. Include enough of the question/exercise in your response that it is clear to your teaching assistant to which problem you are responding. 1. Describe some possible sources of error that contributed to the uncertainty associated with measuring your reaction time.. Using the error propagation rule for functions of a single variable, derive a general expression for the fractional error, q/q, where q(x) = x n and n is an integer. Explain your answer in terms of n, x, and x. 3. You have a ruler and a ream (500 sheets) of paper, and you are asked to measure the thickness of a single piece of paper. Realizing that the paper thickness is much less than the precision of your ruler, you decide to measure the thickness of the entire ream of paper to estimate the paper thickness. If you measure the ream to be cm thick, what is the thickness of one piece of paper? Write an algebraic expression for the uncertainty in the sheet thickness in terms of the uncertainty of the ream thickness. Then determine a numeric value for the uncertainty in the sheet thickness. Discuss why this is a good method for determining the thickness of a single sheet of paper rather than trying to measure the thickness of a single sheet directly. 4. Based on Archimedes Principle, describe an experiment to determine the amount of gold in a crown. This technique must be non-destructive. (Hint: assume you know how much gold was supposed to go into making the crown and describe how you would determine if a less expensive metal like silver had been substituted for some of the gold by a dishonest goldsmith) 8

9 Appendix 1: Rules for Propagation of Uncertainties ADDITION & SUBTRACTION: If several quantities x,,w are measured with uncertainties x,,w, and the measured values are used to compute q = x + + z (u + + w), then the uncertainty in the computed value of q is the quadratic sum of the original uncertainties: q (x)... (z) (u)... (w) MULTIPLICATION & DIVISION: If several quantities x,,w are measured with uncertainties x,,w, and the measured values are used to compute x q u z w then the uncertainty in the computed value of q is the quadratic sum of the fractional uncertainties in x,,w:, q q x x q... w w where q/i is the partial derivative of q with respect to the i th measured variable. FUNCTIONS OF ONE VARIABLE: If the quantity x is measured with uncertainty x, and the measured value is used to compute q(x), then the uncertainty in the value of q(x) is given by q dq dx x 9