measured value true value

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1 1.1 Why study measurement? 1. Introductory Examples Example: Measuring your weight: The measurement is not the thing. 1.3 Some Important Deinitions We ll start by deining some terms. Some o them will seem simple, but the ollowing terms are so conused and misused by engineers (and non-engineers) that we need to spend some time describing them. From now on, we will use these terms careully! Accuracy The most basic deinition o accuracy is: how close a measurement is to the true value. An accurate instrument, thereore, represents the thing you are trying to measure closely Error ways: Error is the dierence between the measured value and the true value. We can express error several Absolute Error: dierence between measured value and true value o a variable, or ε = measured value true value (1.1) Percent Error: measured value true value % error = 100% (1.) true value Note that the sign is retained, which conveys inormation: a percent error o -%, or example, means that the measurement is % below the true value. Percent Dierence: mmmmmmmmmmmmmmmmmmmmmm 1 mmmmmmmmmmmmmmmmmmmmmm % dddddddddddddddddddd = 100% (1.3) mmmmmmmmuuuuuuuuuuuuuu What s the dierence between percent dierence and percent error? How do you decide which measurement is measurement? Two Important Points Regarding Error First, error does not mean mistake. You might be thinking o the term human error, but this is not an engineering measurements term. Second, and perhaps more importantly, error is usually not known. Why? Because the true value is not usually known. I the true value were known, why are we bothering 1-1

2 to measure it? (There is a scenario when you would measure something even though you know the true value: when you are checking the calibration o your measuring device!) Bias and Precision Since we are talking about error, there are two general kinds o error. Some error is ixed, like i you weigh yoursel on a scale that you know measures ive pounds heavy. This ixed error is called bias. Technically, engineers reer to a measuring device as accurate i it has low bias error. But there is second kind o error. For example, i you weigh yoursel ten times in a row, and get a slightly dierent reading each time, the random variation in the measurement is reerred to as precision. (Side note: precision, or random error, could be seen as you weigh yoursel repeatedly with the same scale; this kind o precision is also reerred to as repeatability. But i you weigh yoursel with dierent scales, the degree to which these dierent scales agree is called reproducibility.) A graphical depiction o these terms is presented in Figure 1.1. precision measurement bias true value time Figure 1.1. Bias and precision as depicted in a plot. The key dierence between bias and precision is that bias can be corrected; i you know the scale reads ive pounds heavy, you can subtract that rom your reading. Or, perhaps you can adjust the scale, or recalibrate it. Precision, on the other hand, is random; trying to characterize the random variation in measurements requires a course in Statistics. Fortunately, that is what this course is about! Accuracy vs. Precision People oten misuse and conuse the terms accuracy and precision. First, remember that accuracy is related to the lack o bias (ixed) error in a measurement, while precision has to do with the randomness in repeated measurements. But we will go urther, and invoke a classic analogy rom nearly every textbook on measurement; we ll call it the target analogy. Consider the targets shown in Figure 1.. In this analogy, the bullseye represents the true value, and the holes represent our measurements. In the irst target, the spread o the bullet holes (what shooters call the pattern ) represents the precision o the measurement, and the spread is airly large. In addition, the hole pattern isn t centered over the bullseye; that distance is the bias, or lack o accuracy. So, the measurement is neither precise nor accurate. 1-

3 In the second target, the precision has improved, but because o the bias that remains, the measurement is still not accurate. The third target is the opposite, where the measurement is accurate but not precise. Finally, the best case is one with high precision and low bias, and is this accurate and precise. bias low precision (bad) high bias (bad) Not precise nor accurate high precision (good) high bias (bad) precise, but not accurate low precision (bad) low bias (good) accurate, but not precise accurate and precise Figure 1.. The target analogy or accuracy (lack o bias) and precision Resolution vs. Precision These two terms get conused perhaps the most o all measurement terminology. For example, consider the digital temperature gauge depicted in Figure 1.3. By merely looking at the gauge, you d be tempted to say that the gauge is accurate to one hundredth o a degree Fahrenheit (0.01 F), but that statement is alse! The truth is, when simply looking at a gauge, you can only tell its resolution. You don t know i it s accurate, unless you have its calibration inormation (or you calibrated it yoursel). By the way, gauges are requently less accurate than their display s resolution would imply! So, in short, you should never assume that a device is as accurate as its resolution implies. So, we ve decided that the statement that the gauge is accurate to 0.01 F is incorrect (or at least ignorant, or misleading). But can we say that the gauge is precise to 0.01 F? Again, the answer is no, this time because resolution and precision are not the same. We already know that precision reers to the randomness in repeated measurements. Resolution reers to the ineness o a measurement, meaning the small division or change in a measurement can be detected. So, the resolution o the instrument is 0.01 F. The act is, i we take repeated measurements, we might ind that the temperature luctuates by more than merely its resolution. However, these two terms are linked: i the random variation in temperature were smaller than the resolution o the instrument (or stated dierently, i the precision were higher), then the instrument s reading would never change. That is, you d not be able to detect precision better than the instrument s resolution. So, the limit o an instrument s precision is its resolution. 1-3

4 Figure 1.3. Digital temperature gauge with a resolution o 0.01 F. Without knowing how well it was calibrated, you could never say that the instrument is accurate or precise to 0.01 F Uncertainty As we said earlier, normally the error, deined by Eq. (1.1), is not known, and can only be estimated. We reer to this estimate o the error as the uncertainty in the measurement, which is usually presented as a range o values about the nominal value: x = (nominal value) U x, (1.4) where U x is the uncertainty in x. Now, uncertainty can take on several ormats: Absolute Uncertainty: The uncertainty o a measurement expressed in the units o the measurement. (Example: 10 lbm ± 3 lbm, or more simply, 10 ± 3 lbm) Percent Uncertainty: The uncertainty expressed as a percentage o the nominal value. (Example: 10 lbm ± (3 lbm)/(10 lbm) 100% 10 lbm ±.5%. Relative (or Fractional) Uncertainty: The uncertainty expressed as a raction o the nominal value. (Example: 10 lbm ± (3 lbm)/(10 lbm) 10 lbm ± 0.05 lbm/lbm. Note that the uncertainty is unitless, but the units (lbm/lbm) have been included anyway to make sure the audience knows what the value represents.) NOTE: uncertainty and error are not the same thing! The terms are NOT interchangeable, although even engineers use error when they mean uncertainty. Never do this. (What IS the dierence between error and uncertainty?) Practical Sources o Measurement Uncertainty Example: Measuring the temperature o this room. Reading Uncertainty (sometimes called minimum or resolution uncertainty) uu rrrrrrrrrrrrrr = ± 1 (rrrrrrrrrrrrrrrrrrrr) (1.5) Instrument (or calibration) uncertainty 1-4

5 Statistical (or precision) uncertainty Due to transient variations, spatial variations Combining uncertainties: root-sum-square (summation in quadrature) uu tttttt = ±(uu 1 + uu + uu 3 + ) 1/ (1.6) 1.4 Other Deinitions Variable: Any quantity that can be measured (e.g., temperature, weight) or observed (roll o a die, number o students) a. Dependent Variable: aected by changes o one or more variables (e.g., the temperature in a room is aected by the time o day; the temperature is thereore the dependent variable) b. Independent Variable: can be varied independent o other variables (e.g., the time o day is independent o the temperature in a room) c. Discrete Variable: possible values can be enumerated (e.g, roll o a die, number o students) d. Continuous Variable: possible values are ininite (e.g., all physical properties, such as temperature, density, length) e. Controlled Variable: held at a prescribed value during a measurement (e.g., temperatures were measured every 10 seconds: time is thereore a controlled variable). Extraneous Variable: not controlled during an experiment g. Random Variable: contains random scatter 1.4. Parameter: a unctional relationship between variables (e.g., drag coeicient) Noise: variation in a measured signal due to random luctuations o extraneous variables Intererence: variation in a measured signal due to deterministic variation in extraneous variables (e.g., electrical intererence) Sequential Test: experiment where the controlled variable is varied in order Random Test: experiment where the controlled variable is varied randomly Repetition: repeated measurements made rom the experiment, to examine the variability in observations or a single condition Replication: duplication o the experiment under similar operating conditions (e.g. same test, dierent day) Concomitant Methods: dierent methods or measuring the same variable; best i measurement technique is based on dierent physical properties Static Measurement: one in which input variable is constant with time Dynamic Measurement: one in which input variable changes with time 1-5

6 1.5 A Primer on Dimensions and Units Dimensions versus Units Nearly every engineering problem you will encounter will involve dimensions: the length o a beam, the mass o a concrete block, the time and velocity o an object s all, the orce o the air resistance on an airplane, and so orth. We express these dimensions using speciic units: or example, length can be expressed in eet, mass as kilograms, time as minutes, velocity as miles per hour, and orce as newtons 1. The goal o this section is to explain the use o dimensions and units in engineering calculations, and to introduce a ew o the standard systems o units that are used How Dimensions Relate to Each Other Dimensions (as well as units) act just like algebraic symbols in engineering calculations. For example, i an object travels 4 eet in 10 seconds, we can calculate its (average) velocity. First, algebraically: d V average =, t where v is the symbol or velocity, d or distance, t or time. Plugging in the actual values (and units), (4 t) t V average = = 0.4. (10 s) s Thus we can see that velocity in this case has the units eet per second (t/s). We can convert eet to whatever we like: meters, miles, etc. We can also convert seconds to minutes, hours, days, etc. But the dimensions are always the same: [length] velocity =. [time] There are two kinds o dimensions: (1) primary dimensions, like length and time, and () secondary dimensions, like velocity, which are combinations o primary dimensions. Because any given system o units we use has so many dierent measurements, standard units have been developed to make communication (and science and commerce) easier. We will explore three o these standard systems: the SI system, the British Gravitational system, and the English Engineering System. There are more! The SI system The SI (Système International d Unitès) system is the oicial name or the metric system. The system is described as an MLtT system, because its primary dimensions are mass (M), length (L), time (t), and temperature (T). The standard units are listed below. Primary Dimension mass (M) length (L) time (t) temperature (T) Standard Unit kilogram (kg) meter (m) second (s) Kelvin (K) 1 Notice that newtons is not capitalized. It is standard not to capitalize the name o the unit, even though the unit abbreviation is capitalized (i.e., N). It s a conusing rule. 1-6

7 Secondary units are derived rom these primary units. For example, velocity has units o m/s, acceleration is m/s, and orce has units o? How do we relate orce to the primary units? Isaac Newton discovered that the orce on an object is proportional to its mass times its acceleration: F ma. I we plug dimensions into the above relation, we see that Force [L] [M] [t]. Or, i we use primary SI units, we see that Force kg m. s In honor o Newton, it was decided to give this particular set o terms the name newton (N). It is deined as kg m 1N 1. (1.7) s So the unit o orce in the SI system is the newton (N), deined as the orce required to accelerate a mass o 1 kg to an acceleration o 1 m/s. Why not kg? Or 10 m/s? Actually, the number is arbitrary, but the number 1 is chosen or convenience. Example 1-1. An object has a mass o 80 kg. I the acceleration o gravity is 9.81 m/s, what is its weight? Solution: The weight o an object is the orce o gravity on the object, which is given by W = mg. Plugging in values (and units) or m and g, W. (a) = (80 kg)(9.81 m/s As you can see, the result o the above calculation does give us the correct dimensions and units or orce. But or convenience, we know by deinition that 1N = 1kg m/s. ) get Notice that we can manipulate the above equation slightly: I we divide both sides by 1 kg m/s, we 1 N 1 kg m/s =

8 Thus, i we multiply the right-hand-side o Equation (a) by the ratio above, we are merely multiplying by one and a unitless value o one which doesn t change anything: 1 N (80 kg)(9.81 m/s ) 1 kg m/s W. = Note that all the units cancel except or N, which yields Comments: W = N. 1. Note that we just used the deinition o a newton as a kind o conversion actor to convert the answer above into a more convenient orm. To be honest, it s not necessary to use newtons, and in act some engineers leave the units o orce as kg m/s sometimes, because they know the units will cancel later. But just remember that you want to express your inal answer in as relatable units as possible, or your audience s understanding.. Recall that we determined the gravitational orce by the equation W = mg. Why didn t we use Newton s second law, F = ma, where a = g? Isn t that the same? Absolutely not! GRAVITY IS NOT ACCELERATION. IT IS A FORCE (PER UNIT MASS). It only looks like acceleration because it has units like that o acceleration (In act, dimensionally, acceleration and orce per unit mass are the same). Think about this. What is the orce o gravity acting on your body right now? Are you in motion right now? I you are sitting still, you are not accelerating (relative to the ground). Then a=0! So is the orce on your body zero? No! Remember that in stating Newton s second law, F is the net orce acting on the mass m. I the mass is stationary, the net orce is zero. That is, the orce o gravity on your body is exactly balanced by the orce o the ground pushing up on you. You are in equilibrium, and thereore your acceleration is zero The British Gravitational System ( Slug System) The British Gravitational system o units is reerred to as an FLtT system, because the primary dimensions are orce (F), length (L), time (t), and temperature (T). The standard units are: Primary Dimension Standard Unit Force (F) pound-orce (lb ) length (L) oot (t) time (t) second (s) temperature (T) Rankine (R) I orce is a primary dimension, how do we ind the unit o mass? Mass is now a secondary dimension; we have to derive it. Newton s second law always holds: or, dimensionally, F ma. [L] [F] [mass]. [t] 1-8

9 I we use primary units, we see that Rearranging the above, lb mass mass t, s lb s. t We need a name or the unit o mass. Let s call it a slug! Then we ll deine it by 1 lb slug t 1. s (1.8) We can interpret the above by saying, one pound-orce is the orce required to accelerate 1 slug to an acceleration o 1 t/s. Again, we could have deined the slug as 10 lb s /t, or lb s /t, but or the sake o simplicity, we choose 1 as the constant. Example 1-. An object has a mass o 5.59 slugs. What is its weight in Earth s gravity? Solution: As in Example 1, the weight o the object can be determined by W = mg. Substituting the mass and the value o standard Earth gravity, t/s, into the above, W = (5.59 slug)(3.174 t/s ) The units above are not useul as units o orce. But we know by deinition that 1 slug =1lb s /t, or 1 lb s /t = 1 1 slug. Multiplying the weight by the above gives W = (5.59 slug)(3.174 = lb. t/s 1 lb s /t ) 1 slug We see that the units in the above relation cancel, leaving the more convenient units o orce. 1-9

10 1.5.5 The English Engineering System ( Pound-Mass System ) In the English Engineering system o units, the primary dimensions are are orce (F), mass (M), length (L), time (t), and temperature (T). Thereore this system is reerred to as a FMLtT system. The standard units are shown below: Primary Dimension Standard Unit Force (F) pound-orce (lb ) mass (M) pound-mass (lb m) length (L) oot (t) time (t) second (s) temperature (T) Rankine (R) In this system, orce and mass are primary dimensions. They must still be related by Newton s second law: or, dimensionally, F ma. [L] [F] [mass]. [t] I we use the primary English units, we see that lb lb m t, s We don t need to deine a new unit, but we need to determine a constant in order to make the above relation exact. Let s use 3.174! Then the relationship between pound-orce and pound-mass is as ollows: 1 lb lb m t s (1.9) So in words, one pound-orce is the orce required to accelerate one pound-mass to t/s. Why 3.174? Because that just happens to be the value or the acceleration o gravity, g = t/s. This value was chosen so that i an object has a mass o 10 lb m, its weight on the Earth will also be 10 lb. This convenience will become apparent later in one o the examples which ollow. One inal note: I we compare Equation (3) with Equation (), we see that slugs and pounds-mass are related by 1 slug = lb m. (1.10) Example 1-3. An object has a mass o 180 lb m. What is its weight in Earth s gravity? Solution: Again, the weight is given by W = mg, which becomes W = (180 lb )(3.174 t/s ). m 1-10

11 To convert the units in the above equation into useul orce units, we note that by deinition, 1 lb =3.174 lb m-t/s. Or, 1 lb lb m t/s Multiplying this constant with the weight gives = 1. W = (180 lb = 180 lb m )( t/s 1 lb ) lb m t/s Comment: Note that in Earth s gravity, and the pound-mass system, the values o mass and weight are the same! In act, that s how the relationship between lb and lb m was deined. Remember, though, that the units represent dierent dimensions: lb represents orce, while lb m represents mass. So it is NEVER acceptable to write 1 lb = 1 lbm. This is not dimensionally correct; it is like saying that 1 apple = 1 orange The Proportionality Constant g c As a inal note, i you haven t yet heard o g c ( g sub c ) in your studies, you might. It s sometimes reerred to as the gravitational constant, and it is a less-common (some may say it s obsolete, or oldashioned) way to deal with the orce-mass units relationship. DON T USE THIS. But i you run across this term, how does it work? Did you notice that, in every example above, we had to multiply the weight we calculated by a conversion actor to make the units come out right? Well, what some people do is just employ a actor, called g c, directly in the equation they are using. For example, Newton s second law could be written as ma F =. Similarly, the gravitational orce could be written as mg W =. g c Comparing g c in the equation above with the conversion actors we used in the examples, you can show that kg m/s g c = 1 (SI system), N g c and slug t/s g c = 1 ( slug system), lb lb t/s m g c = ( pound-mass system). lb 1-11

12 Again, DON T USE the g c approach in your calculations. This technique can be conusing because you have to remember when you have to include g c in your general equation. But, as you can see rom the example problems, we ignored g c entirely; as long as you ALWAYS keep track o ALL your units, you will know when you need to perorm unit conversions in order to cancel certain units. Think o the deinitions (1.7), (1.8), and (1.9) as a wild card that you can insert into a calculation when you need to simpliy the units. A summary o the basic unit systems is presented in Table 1.1. Memorize the orce-mass relationships, and always use them explicitly in your calculations. I can t over-emphasize this point: NEVER DO UNIT CONVERSIONS IN YOUR HEAD. Always show them, no matter how trivial. Incorrect units are a leading cause o mistakes in calculations, sometimes leading to tragic results. Being explicit with your unit calculations will help you catch your own mistakes, and help your audience understand your calculations (and convince them that you know what you are doing). Table 1.1. Summary o Unit Systems System SI British Gravitational English Engineering ( Metric system) ( slug system) ( pound-mass system) Primary Dim s MLtT FLtT FMLtT Mass kg slug lb m Length m t t Force N lb lb Time s s s Temperature K R R slug t lb m t Force-Mass kg m 1 lb 1 1 lb N 1 Relationship s s s 1 slug = lb m g c g c kg m/s = 1 N g c slug t/s = 1 lb g c lb = m t/s lb 1-1

13 Unit examples (solutions on the next page): Example 1-4. The pressure acting on a 1.5 in test specimen equals 15 MPa. What is the orce (in N) acting on the specimen? Answer Example 1-5. The weight o a large steel cylinder is to be computed rom measurements o its diameter and length. Let its length L be equal to 3.3 m and its diameter d equal to m. Suppose that the density o the steel equals 7835 kg/m 3. Calculate the weight o the cylinder (in N) and report your result in a clear and unambiguous orm. Answer 1-13

14 Solutions Example 1-4. The pressure acting on a 1.5 in test specimen equals 15 MPa. What is the orce (in N) acting on the specimen? Example 1-5. The weight o a large steel cylinder is to be computed rom measurements o its diameter and length. Let its length L be equal to 3.3 m and its diameter d equal to m. Suppose that the density o the steel equals 7835 kg/m 3. Calculate the weight o the cylinder (in N) and report your result in a clear and unambiguous orm. 1-14

15 1.6 Signiicant Figures What Are Signiicant Figures, and Why Do We Care About Them? Signiicant igures is a way to express real measurements or calculated values as precisely as the values deserve. For example, i someone tells you that the mass o an object is 3.4 kg, they presumably decided that reporting the value to any more decimals, as in kg, overstates or misrepresents the precision o the actual measurement. Perhaps they know that the mass scale, no matter many decimals it displays, is only accurate to a tenth o a kilogram. Now, you may not have seen the topic o signiicant igures in your coursework. In act, most o your science and engineering courses practically ignore the topic. Why? Because those courses are ocused more on developing theory, which is hard enough; the last thing they want to do is add to it the complexity o dealing with real numbers with measurement uncertainties. And i you have seen signiicant igures beore, it was likely introduced in your Physics or Chemistry laboratories, because these laboratories work with real data. We can t overstate the importance o being honest and accurate when reporting experimental data. In act, one o the irst questions you ask every time you report data or calculations is, Are the signiicant igures correct? Or am I reporting too many digits, and thereore lying about the precision o my results? So in summary, there are two reasons why we care about signiicant igures: 1. Signiicant igures are used because we don t want to overstate or misrepresent the precision o the measurements or calculations that we report to an audience.. The number o signiicant igures implies (at least roughly) the uncertainty o a measurement when the uncertainty isn t explicitly stated What igures are signiicant? Identiying the number o signiicant igures is not quite as simple as merely counting the digits, because not all digits have something to do with the precision or the uncertainty o a measurement. So, what ollows are the rules or identiying which igures are signiicant in reported data: Rules or Identiying Signiicant Figures in Reported Data 1. All non-zero numbers are signiicant. Examples: a. The value 37 has three signiicant igures. b. The value has ive signiicant igures.. Zeros between non-zero numbers are signiicant. Examples: a. The value has our signiicant igures. b. The value has eight signiicant igures 3. Zeros used to locate the decimal point are NOT signiicant, because they deine magnitude, not precision. 1-15

16 Examples: a. The value has three signiicant igures. (The zeros are merely placeholders that deine the order o magnitude.) b. The value has three signiicant igures. (And it s the same value as in the previous example, just in scientiic notation. Note that the exponent -5 only deines order o magnitude.) 4. Zeros to the right o non-zero numbers are usually signiicant. Examples: Comment: a. The value has three signiicant igures. b. The value 1.00 x 10 3 has our signiicant igures. c. The value 100 has either one, two, or three signiicant igures. The value 100 is ambiguous: we don t know whether the zeroes are signiicant or merely placeholders. This problem can be avoided by reporting the value several ways, rom worst to best: Some texts suggest underlining the least signiicant digit, as in 100, which implies two signiicant igures. Since your audience probably doesn t know that rule, avoid using it. Some texts suggest that including the decimal, i.e implies that the all zeros are signiicant. But this only works i all zeroes are indeed signiicant. Also, it relies on your audience to know that rule. So avoid it. Using scientiic notation: (it works every time, although scientiic notation can be awkward to read and should be used only or very small or very large numbers). Using words: 100 to two signiicant igures. Clear and direct. Finally, i you do know the uncertainty, report it explicitly and remove all doubt as to how precise the measurement is. Example: 100±5. 5. Some values are exact, and thereore not analyzed or signiicant igures. Examples: a. Some conversion actors are exact. The conversion 1 in =.54 cm is exact. b. There are 1 eggs in a dozen. This value is exact. Example 1-6. Identiy the number o signiicant igures in the ollowing values: (a) 13.1, (b) 5.00, (c) (d) 1900 (e) Solution: a. The value 13.1 has our signiicant igures, by Rule 1. b. The value 5.00 has three signiicant igures, by Rule 4. c. The value has our signiicant igures, by Rule 4. d. The value 1900 is ambiguous; it has either two, three, or our signiicant igures. To eliminate this ambiguity and imply three signiicant igures, this value could be written one o two ways: to three signiicant igures e. The value has our signiicant igures, by Rules and

17 1.6.3 How Do Signiicant Figures Relate to Uncertainty? We said that signiicant igures imply the uncertainty in a measurement or calculated value. Now we will attempt to assign a value to that uncertainty. Let s consider the example we began with: a reported mass o 3.4 kg. I we assume that the raw value contained more digits, it seems reasonable to assume that they rounded the raw value to obtain the reported value. That is, the true value is likely between and 3.450, because any number between these limits would get rounded to 3.4 kg. In other words, the true value could be anywhere ±0.05 kg o the reported value o 3.4 kg. In summary, we obtained the ollowing estimate or the uncertainty to be: 3.4 kg implies 3.4 ± 0.05 kg. By the way, this approximation is the same as when we estimate the uncertainty in a digital measurement. Now, remember that we don t know the actual uncertainty in the measurement. This technique merely estimates the uncertainty. Admittedly, i the engineer had reported the actual uncertainty along with the measurement, e.g., 3.4 ± 0.03 kg, we wouldn t need to estimate it! That said, this technique seems to be a reasonable approximation when uncertainties are not explicitly given. To summarize this rule: Rule or estimating the uncertainty in a value when none is given: Treat the number as i it had been rounded to the least signiicant digit, so the uncertainty is one-hal the resolution. This is the same rule we ollow or estimating the uncertainty in a digital readout How Do Signiicant Figures Work in Calculations? We now know how to identiy the signiicant igures in a measurement, and roughly estimate the uncertainty they imply. But what happens when we perorm calculations with real measurements? For example, take the calculation 10.0/3.0 = Given the precision o the values 10.0 and 3.0, it makes no sense to report all the digits that our calculator supplies. It turns out that analyzing how measurement uncertainties propagate through a ormula or series o calculations is not easy, and requires a calculus-based approach that we will learn later in the course. However, or very simple mathematic expressions, there are some rules that you would have seen in your Physics or Chemistry Labs. It s worth reviewing these rules, since they give us some mathematical insight. But in Chapter 6 we will develop a more rigorous technique that applies to any mathematical calculation and will render the ollowing rules largely obsolete. Rules or Signiicant Figures in Calculations 1. Addition or subtraction: The resulting value should be reported to the same number o decimal places as that o the term with the least number o decimal places. Examples: a = ( rounded to one decimal place) b = ( rounded to two decimal places) Why? When numbers are added or subtracted, it seems reasonable that their uncertainties would approximately add (even i the nominal values are subtracted, the 1-17

18 uncertainties would add why?). Considering the irst example, i the irst number, 5.6, is only known to the tenths place, does it matter that the other value is known to the thousandths place? The uncertainty in the value 5.6 would overshadow the precision o the other number. In eect, the least accurate value wins.. Multiplication or division: The resulting value should be reported to the same number o signiicant igures as that o the term with the least number o signiicant igures. Examples: a = 36 ( rounded to two signiicant igures) b / 19.5 = 54. ( rounded to three signiicant igures) Why? Consider the multiplication example. I two values are multiplied, it stands to reason that in some way their uncertainties also multiply. For example, i two numbers a and b are multiplied, but a is then increased by 10 percent, the result would be F = (0.1a)b = 0.1(ab). In other words, a percentage change in one o the values gives the same percentage change in the calculated result. How does that aect how report signiicant igures? You can probably imagine that the value with the smaller number o signiicant igures has the largest percentage uncertainty, and that thereore its uncertainty would dominate in the calculated result. I that explanation isn t perectly clear, don t worry; ocus on remembering the rule or now, and we ll explore how it works when we get to Chapter Averages and standard deviations: It is customary to report them to one more decimal place than the original data. Example: The average o 34.3, 0.0, 77.8, and 16.6 is reported as Why? It seems reasonable that i you averaged the values 1 and, that reporting the average as 1.5 instead o would be more inormative! 4. Keep an extra digit or two during intermediate calculations. Why? To avoid accumulating round-o error. For example, consider the expression: I we ollow the rules o signiicant igures at each step, we would get: Now, had we not rounded at all, the calculator would report , which, i we then reported to the same signiicant igures, would yield a value o 6, not 5. Why didn t the answers match? Remember that signiicant igures is an approximate way to estimate the uncertainty in a measurement. So, when we round 1-18

19 a value, we introduce even more error, since we know the values o the decimals we re removing. The lesson here is that, i we strictly ollow signiicant igures rules at every step o a calculation, we introduce error that could have been avoided. The solution is to carry (and write down!) an extra digit or two at every step, then reduce the inal result (the reported answer) to the correct number o signiicant igures that you would have chosen had you ollowed the rules at each step How to Round Numbers It s hard to believe that a college-level textbook would contain a section on how to round numbers. But as it turns out with so many things we learned earlier in lie, the real world introduces complexities that we as engineers have to deal with. Rounding is a perect example. The rule we learned was easy: i you want to round a value to a particular decimal place, you simply look at the very next decimal. I that digit is 0, 1,, 3, or 4, you round UP. I the digit is 5 or greater (including i the value is 5 but ollowed by any non-zero digits), you round DOWN (truncate, in other words). But there s a problem with that rule: a special case. I that next digit is exactly 5 (meaning no other digits ollow other than zero), the unrounded number is exactly halway between the rounded and truncated value. Take, or example, the number I we wanted to round to the tenths place, the answer could be either 13.4 or 13.5, because again, the original value is exactly halway between them. So what should we do? We could simply ollow the rule we stated above, and simply round the value UP, to But you shouldn t always round up, because that introduces a bias. Over many, many calculations (especially in computer code involving millions o computations), this bias can build up. In act, there are examples where the accumulation o this rounding error has caused catastrophic ailure. So again, what should we do? Ideally, you should, on average, round up hal the time, and round down hal the time. And the simplest rule to apply is to look at the value o the decimal place you plan to round to: i that digit is even, leave it unchanged, and i the digit is odd, round it up. This technique eliminates the bias over many calculations. Finally, we can summarize the rounding rules as ollows. 1-19

20 Rounding Rules 1. I the number to the right o the least signiicant digit is LESS THAN 5, round the least signiicant digit DOWN. Examples: rounded to three signiicant igures is rounded to three signiicant igures is rounded to three signiicant igures is I the number to the right o the least signiicant digit is GREATER THAN 5 (including 5, with any non-zero digits to the right o it), round the least signiicant digit UP. Examples: rounded to our signiicant igures is (because the 5 th digit isn t exactly 5, it s greater than 5 because o the non-zero digit to the right) rounded to our signiicant igures is rounded to our signiicant igures is SPECIAL CASE: I the number to the right o the least signiicant digit is EXACTLY 5 (no nonzero digits aterward), round the least signiicant digit UP i it is an ODD number; i it is EVEN, leave it unchanged. Examples: rounded to ive signiicant igures is rounded to ive signiicant igures is rounded to ive signiicant igures is Example 1-7. Round the value to each successive decimal place, one at a time. Solution: Beginning with the largest decimal place, the rounded values are as ollows: (The value 3 was ollowed by exactly 5; since 3 is odd, it gets rounded up) (The value 8 gets rounded up, by Rule 1. Rule 3 does not apply because the irst 5 is ollowed by nonzero digits)

21 Extra: How Precise Can We Report Earth s gravity? Glen Thorncrot, ME Department, Cal Poly We use gravity (g = 9.81 m/s or 3. t/s ) oten in engineering calculations, but since this course deals with accuracy, have you ever stopped to consider how accurate the value is? You might be surprised to read this, but the value you choose or gravity may not always be correct, depending on the accuracy you need or your application! First o all, gravity is not a constant value. It varies along the surace o the earth, with elevation above the earth, and even with time. For the sake o simplicity, we usually use standard gravity, deined as m/s, or t/s. This value is not your local value instead, it represents the mean gravity at 45º latitude, at mean sea level. This is the value we typically use in engineering calculations, although we usually round it down to one or two decimal places. The gravity on the surace o the Earth varies due to a number o eects. The major eects are: 1. The rotation o the Earth. The rotation o the Earth reduces the orce you eel at your eet (you would thereore eel lighter at the equator than at the poles o the Earth). Also, the rotation causes the Earth to be shaped like an oblate spheroid (lattened sphere latter at the poles), which means that the radial distance you are rom the center o the Earth (and hence gravity) varies depending on where you are on the surace. Both o these eects vary with latitude. This variation is predictable, and is given by an equation known as the International Gravity Formula: s 6λ i9 g = , s 0λ i6 where λ is the geographic latitude and g 0 is called normal gravity. The variation due to Earth s rotation is on the order o ± 0.03 m/s.. Elevation above sea level. At higher elevations, you are urther rom the center o the Earth, so the Earth s pull is less. How big is this eect? At an elevation o about 1000 m, gravity reduces by about m/s. 3. Variation in mass. Gravity is a unction o mass, and since the mass o the earth is not uniorm, the gravity varies as well. The most accurate measurement o local gravity variation was perormed by GRACE (Gravity Recovery and Climate Experiment), a satellite-based measurement. These measurements show that the local variation o gravity (relative to normal gravity) is on the order o about ± m/s (See Figure 1). Established by the 1901 International Conerence on Weights and Measures (in French, the acronym is GCPM). 3 Established by the World Geodetic System There are earlier versions o this ormula. 1-1

22 Gravity Anomaly (mgal) (1 mgal = m /s) Figure 1. Local variation in gravity (relative to normal gravity), as measured by GRACE (Gravity Recovery and Climate Experiment). {Note: Original plot is in color. This example illustrates one o the pitalls with color igures. See instructor i you would like a color copy.} 4. Tides. Tidal variation (due to the gravitational pull o the sun and moon) contributes to a variation o about ± m/s. These eects are summarized below, illustrating the relative eect o these actors: g = 9.XXXXXX m/s Tides ± m/s Elevation (below 1000 m the correction is less than approximately m/s ) Local gravity variation ± m/s Latitude ± 0.03 m/s Bottom line: I you do not account or latitude, a value or g o 9.8 m/s or 9.81 m/s is as accurate as you can claim. Correcting or latitude allows you to claim perhaps three decimal place accuracy. Beyond this, you would have to get better inormation on your local gravity variation. 1-

23 Reerences 1. Figliola, R.S. and Beasley, D.E., Theory and Design or Mechanical Measurements, 3 rd Edition, John Wiley and Sons, Inc., New York, Tayor, J.R., An Introduction to Error Analysis, nd Edition, University Science Books, Homework Use engineering paper, and show all work. Work all problems in the unit system given (i.e., do not simply convert to SI, solve the problem in SI, and then convert back to the original units). You may need to review some topics in your Physics textbook to solve some o these problems. 1-1 You measure the mass o 10 M&M candies, and ind the average mass to be g with a statistical uncertainty o 0.05 g. The resolution o the scale is g, and variation o temperature in the room adds an error o ± g to the scale. a. Find the total error (i.e., uncertainty) o the measurement o average mass. b. Express the complete measurement (nominal value and uncertainty). Do this three ways: with absolute uncertainty, percent uncertainty, and relative uncertainty. 1- To check the accuracy o a mass scale, you place calibration mass o g on the scale (or the purposes o this problem, the calibration mass is exact). Your scale reads g. Determine the (absolute) error, percent error, and percent accuracy o the device at this reading. 1-3 Answer the ollowing questions. Assume standard gravity in each case. a. How much does 3.0 kg weigh on the Earth, in N? b. How much does a person with a mass o 183 lbm weigh on the moon (lb)? (The moon s gravity is exactly one-sixth the Earth s) c. A gallon o water weighs about 8 lb on Earth. What is its mass in slugs? 1-4 Convert the ollowing, without using any special unctions in your calculator. a C to degrees Fahrenheit b. 103 F to degrees Rankine c. 37 K to degrees Celsius d. A temperature dierence o 10 degrees Celsius to degrees Kelvin. e. A temperature dierence o 10 degrees Rankine to degrees Fahrenheit.. A temperature dierence o 10 C to F. 1-5 What is the pressure o water on the bottom o a 6.0 t deep pool, (a) relative to the pressure at the surace (lb/in ), and (b) absolute, i the pressure at the surace is atmospheric? Use the principles o luid statics that you learned in Physics. Assume the density o water to be ρ w = 6.4 lbm/t A barrel o water is weighed on a scale to be 150 lb. a. Neglecting the weight o the barrel, and assuming the water to have a density o ρ w = 6.4 lbm/t 3, estimate the volume o the water. 1-3

24 b. To what extent does the buoyancy o the surrounding air aect the weight measurement? (Hint: estimate it using Archimedes Principle. Assume the air density to be ρ air = lbm/t 3 ). 1-7 You may recall rom Physics that the heat capacity, C, o a substance is the energy gained or a given temperature rise (units o Btu/ F, kj/k, etc.). Speciic heat, c, is the heat capacity per unit mass (Btu/lbm F, kj/kg K, etc.). The ollowing experiment has been designed to measure the speciic heat, c, o water: Water with a mass o 1.00 lbm is heated by an electric heater that delivers heat at a rate o 300 Btu/hr (to signiicant igures). Over a period o 15 minutes, the temperature o the water rises 75 F. What is the speciic heat o the water? 1-8 For the sake o conservation, you decide to measure how ast the water evaporates rom your swimming pool. You do this by recording the level o the pool every day or a month, and calculating rom it and the surace area the water evaporated. Answer the ollowing questions: a. Name as many variables that may inluence this measurement. b. Identiy any dependent and independent variables c. Identiy any discrete and continuous variables d. Identiy any controlled and extraneous variables 1-9 Explain three concomitant methods by which you could determine the diameter o a roughly -inch diameter steel sphere. In terms o accuracy, what are the advantages and disadvantages o each method? 1-4