Unformatted text preview: PHYSICS 120 PRINCIPLES OF PHYSICS I 1 LAB EXPERIMENT: MEASUREMENTS AND UNCERTAINTIES Physics is an experimental science. No matter how elegant and compelling a theory may appear, its acceptance or rejection by the scientific community ultimately depends on one thing, namely whether the predictions of the theory agree with observations and measurements. But no measurement is ever exact: errors of various sorts can creep into the data and there may be nothing the investigator can do to eliminate some of the sources of these errors. Thus, it is possible for two investigators to measure a fundamental quantity such as the speed of light and come up with two different numbers, yet agree that their results are the same. This consensus is possible only because both investigators agree that the sources of uncertainty in their experimental measurements are due to variations in the data from the accumulation of random, uncontrollable events. The main educational goals of this lab are to acquaint you with the different types of errors encountered in laboratory measurements and to teach you some uncertainty analysis skills. ERRORS AND STATISTICS We can identify three general types of experimental errors: (1) mistakes or human errors, (2) systematic errors due to measurement or equipment problems and (3) inherent uncertainties. It is commonly believed that both mistakes and systematic errors can be eliminated completely if the person taking data is extremely careful and uses the best measuring equipment. (A systematic error could be caused, for example, by a stopwatch that runs slow, so that the stopwatch reading is always smaller than the actual elapsed time.) However, inherent uncertainties do not result from mistakes or errors. Instead they can be attributed in part to the impossibility of building measuring equipment that is precise to an infinite number of significant figures. A ruler, for example, can be made better and better but it always has an ultimate limit of precision. Another cause of inherent uncertainties is the large number of random variations affecting any phenomenon being studied. For instance, if you repeatedly drop a ball from a certain height and measure the time of each fall with a stopwatch, the measurements will probably not all be the same. Even if the stopwatch was started and stopped electronically so as to be as precise as possible, there would be small fluctuations in the flow of currents through the circuits as a result of random thermal motion of the atoms and molecules that make up the wires and circuit elements. This could change the stopwatch reading from measurement to measurement. Or, slight air currents in the room _________ 1 This lab writeup is based on materials in the Workshop Physics Activity Guide by Priscilla Laws (Wiley, New York, 1997) as well as elements of the Six Ideas That Shaped Physics lab manual by Thomas Moore and the Pomona College Physics Department. Physics 120 Lab Measurements & Uncertainties page 2 could change the ball's time of fall the ball may stick occasionally to your hand as you release it and so on. Random errors, unlike systematic errors, can be quantified by statistical analysis. To begin, imagine that we have a group of students independently measuring the mass of a hockey puck using an ordinary beam balance. While we would not expect all the measurements to yield the same value, we would expect that most of the measurements are clustered around the true value (170 grams, say). Some results will be greater than 170 grams, some will be less, some may be right on, and a few may be dramatically different that the true value. An example set of mass measurements taken by ten students is shown below. Trial 1 2 3 4 5 6 7 8 9 10 Measured mass in grams 168.2 168.5 170.7 165.8 171.4 170.5 173.0 171.2 167.9 171.6 The average value, or mean, of a set of data is a good way of getting at the "true" value of a set of measurements if the variation in the data is truly due to random effects. When you calculate a mean, you add up all the measured values and then divide by N, the number of measurements. Because random errors are just as likely to be positive (leading to measurements above the true value) as negative, when the mean of a set of data is calculated the random errors will tend to cancel out. Assuming there are no systematic errors in the measurements, the resulting mean should be close to the true value, especially if N is large. Although the mean may be our best guess at the true value of a quantity, we can't expect the random errors in the data to cancel completely, and so we would like to quantify the uncertainty in the mean. This can be done with the aid of a mathematical quantity which you may be familiar with, the standard deviation. The standard deviation of a set of N measurements is defined by the formula s N 1 1 ( x i  < x > ) 2 ( N  1 ) i = (Equation 1) th where xi is the i element of the data set and <x> is the mean of the data. Generally, if the standard deviation of a set of measurements is small, this means there is not much variation in the data about the mean. Note that the standard deviation is always a positive number and has the same units as the measured values. Physics 120 Lab Measurements & Uncertainties page 3 Doing a hand calculation of the standard deviation of a set of numbers is tedious but straightforward. The basic steps are: (1) find the average <x> of the N measurements (2) subtract this average from each of the N measurements (i.e., x1 <x>, x2 <x>, etc.) to obtain N residuals (3) square each of these N residuals and add them up (4) divide this result by (N 1) and take the square root. Once we know the standard deviation, we can then quantify the uncertainty of the mean Um of a set of measurements. Um is defined as that value such that we are 95% confident that <x>Um encloses the "true" value of the measurement. Mathematicians tell us that, if our set of measurements is randomly distributed in a nice "bell curve" (discussed in the timeoffall exercise below), the uncertainty of the mean can be estimated by the formula U m t s N (Equation 2) where t is a quantity known as the student tfactor, a number that depends on the number of measurements N. A table of tvalues is shown below. Notice that, for large N, t has a value very close to 2.00, but that for small N (N < 10), the value of t varies rapidly with N. Table of Student tValues N tvalue N tvalue 2 12.7 10 2.26 3 4.3 12 2.2 4 3.2 15 2.15 5 2.8 20 2.09 6 2.6 30 2.05 7 2.5 50 2.01 8 2.4 100 1.98 9 2.3 1.97 As an example, suppose we have made repeated, independent measurements of the length of a pencil and find, after some calculation, that the mean value of the measurements is 15.82 cm with an uncertainty in the mean of 0.07 cm. We would report our result as 15.82 0.07 cm, indicating that we are 95% confident that the length of the pencil is somewhere between 15.75 cm and 15.89 cm. (This is a compact way of saying that if, say, 100 measurements are made, we expect 95 of them to fall within the indicated range.) Physics 120 Lab Measurements & Uncertainties page 4 A PreLab Exercise For the hockey puck data on the previous page, calculate the mean, standard deviation, and the uncertainty of the mean. What value (with error) should the students report for the mass of the puck? PROPAGATION OF UNCERTAINTY In laboratory work, we are often confronted with situations in which the quantity of interest let's call it f is not measured directly but instead is calculated from two or more different quantities which we have measured. (For example, we might want to know the speed of a bullet, which is determined by measuring the distance it travels and the time of flight.) In this case, it is important to know how to calculate the uncertainty in f in terms of the uncertainty in the measured quantities. This can be done by using a bit of calculus (okay, take a deep breath before proceeding). Suppose x, y, and z are the measured quantities and the associated uncertainties are x, y, and z. (Note: if these quantities have been determined from numerous trials, then x, y, and z would correspond to the mean of these quantities and the uncertainties would be the uncertainties in the means.) Now, if we know the desired function f (x,y,z) of the measured quantities, then according to differential calculus we can write the uncertainty of f, f, as follows: f f f Dx + Dy + Dz (Equation 3) x y z The derivatives in the above expression are written with the strange symbol to indicate that they are "partial" derivatives. A partial derivative tells us how a function of several variables changes when we make an infinitesimal change in one variable, holding the others constant. It is calculated 2 3 just like a normal derivative. To take a simple example, if f(x,y) = x y , then D = f f = 2xy 3 (treat y as a constant, find the derivative of f with respect to x) x and f = 3 2 y 2 (treat x as a constant, find the derivative of f with respect to y) x y It turns out that equation 3 is not exactly the uncertainty in f. Assuming that the uncertainties x, y, and z are due to random errors, statistical theory shows that the uncertainty in f is more accurately given by the formula f f f D = Dx + Dy + Dz f x y z 2 2 2 (Equation 4) Physics 120 Lab Measurements & Uncertainties page 5 Equation 4 tells us how the uncertainties in measured quantities "propagate" to determine the uncertainty in a derived quantity. In the following, we look at how equation 4 is applied to two special, and important, cases. Propagation of Uncertainties: Addition and Subtraction Consider the function f(x,y,z) = ax + by + cz, where a, b, and c are known constants and x, y, and z are measured quantities with uncertainties x , y, and z. From equation 4, the uncertainty in f is given by D = ( Dx 2 + ( Dy 2 + ( Dy 2 f a ) b ) c ) (Equation 5) For example, suppose that f(x,y,z) = x + 2y + z, where x, y, and z are lengths and have been measured as follows: x x = 23.6 cm 0.1 cm y y = 17.8 cm 0.2 cm z z = 33.9 cm 0.2 cm Then, from equation 5, the uncertainty in f is D = ( Dx 2 + ( Dy 2 + ( Dz ) 2 = 0 5 cm f 1 ) 2 ) 1 . The result is reported as f f = 93.1 0.5 cm. Propagation of Uncertainties: PowerLaw Relations Many important calculated quantities f that arise in physics experiments can be expressed in terms of a powerlaw relation of the form f ( x y z ) = kx a y b z c , , (Equation 6) where k is a constant and a,b, and c are exponents which may be positive or negative. (Note that equation 6 can be easily extended to include functions of more than three variables.) An example 2 1 of such a function would be the volume V of a cylinder: V = r h , where r is the radius of the cylinder and h is its height. If the calculated quantity f can be written in the powerlaw form of equation 6, then our general formula for propagation of uncertainties (equation 4) tells us that the uncertainty in f is given by Physics 120 Lab Measurements & Uncertainties page 6 D f = ( kax a  1 y b z c D x ) 2 + ( kbx a y b  1 z c D y ) 2 + ( kcx a y b z c  1 D z ) 2 If we divide the above equation by f, we get the simpler formula D f D D D a x b y c z = + y + z f x where f/f is called the relative error in f. Consider the example of the cylinder cited above. Referring to equation 6, here we would have k = , x = r, a = 2, y = h, b = 1, and c = 0. Now, suppose our radius measurement is r = 1.42 0.02 cm and the height measurement is h = 5.77 0.03 cm. Then, by equation 7, the relative error in the volume V is D V . . 2 0 02 1 0 03 = . + = 0 03 V . . 1 42 5 77 3 3 and so V = (0.03)V = (.03)(36.6 cm ) = 1.1 cm . The volume of the cylinder would be reported 3 as 37 1 cm . The Density of PlayDoh 2 2 2 2 2 (Equation 7) This lab exercise will give us an opportunity to practice the propagation of uncertainty analysis described in the section above. Here, the objective of the lab is to determine the mass density of PlayDoh and to compare the results of different lab groups. (A) Obtain a goodsized chunk of PlayDoh (pick your favorite color) and form the PlayDoh into a sphere. Find the mass of the PlayDoh using a beam balance. Be sure to include an estimate of the uncertainty in the mass reading. (B) To determine the density of the PlayDoh, you need to know the volume of the sphere, and so you'll need to measure the diameter of the sphere. To do this, you'll be using a vernier caliper. In case you haven't used one before, we briefly explain its operation. Examine the figure below, which shows a sample reading of the caliper. To read the caliper, first look to see where, on the main scale, the zero line of the movable scale lies. In general, it will not line up with any line on the main scale but will exceed some line by some fraction of a division (each division on the main scale is separated from its neighbors by 1 mm). In the example below, the zero line is between 9 and 10 mm, so the number of whole millimeters Physics 120 Lab Measurements & Uncertainties page 7 is 9. The purpose of the movable, or "vernier", scale is to determine the reading to within 1/20 mm. To do this, you count the tick marks of the vernier scale, starting from the left, until you come to the mark that is best aligned with a line on the main scale. In the example below, it looks like the tenth tick mark (out of twenty) on the vernier scale lines up with the line below, so the final reading is 9.50 mm. Since the reading error is typically one tick mark out of twenty (corresponding to 0.05 mm), we would report the reading as 9.50 mm 0.05 mm. (C) Make at least a halfdozen measurements of the sphere's diameter taken at different points on the sphere. When writing down caliper readings, don't forget units and instrument uncertainties. (D) Calculate the average value of the diameter and determine the uncertainty in the mean. Use these results to find the volume of the sphere along with the uncertainty in the volume using propagation of uncertainties. (E) Finally, calculate the mass density from the relation = mass/volume. Again, to find the uncertainty in the density, you'll need to make use of the propagation of uncertainty formula. Report your density result to the lab instructor, who will post it on the blackboard along with the results of the other lab teams. When uncertainties are taken into account, are the density results in agreement with each other? In order to receive full credit for this lab, you must have the following materials ready before meeting with your lab instructor for the exit interview: (1) calculations and result for the "PreLab Exercise" on page 4 (2) result for the density of PlayDoh. ...
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 Spring '08
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