1.3.3 Statistical (or precision) uncertainty Due to transient variations, spatial variations 100%

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1 1.1 Why eaure, and why tudy eaureent? 1. Introductory Exaple Exaple: Meauring your weight: The eaureent i not the thing. 1.3 Practical Source o Meaureent Uncertainty Exaple: Meauring T o roo Reading Uncertainty (oetie called iniu or reolution uncertainty) 1.3. Intruent (or calibration) uncertainty Statitical (or preciion) uncertainty Due to tranient variation, patial variation uu rrrrrrrrrrrrrr = ± 1 (rrrrrrrrrrrrrrrrrrrr) (1.1) Cobining uncertaintie: root-u-quare (uation in quadrature) uu tttttt = ±(uu 1 + uu + uu 3 + ) 1/ (1.) 1.4 Error, Accuracy, and Uncertainty Error: How ar o a eaureent i ro the true value. Error doe not ean itake. Abolute Error: dierence between eaured value and true value o a variable, ε = eaured value true value (1.3) Percent Error: eaured value true value % error = 100% (1.4) true value Note that the ign i retained, which convey inoration: a percent error o -%, or exaple, ean that the eaureent i % below the true value Bia and Preciion: Two general claiication o error. Bia: dierence between average eaured value and true value Preciion: rando variation in repeated eaureent Note: preciion ha econd coon ue, to decribe reolution! 1-1

2 1.4.3 Accuracy: A qualitative deinition. The degree to which the eaured value repreent the true value (i.e., high accuracy = low error, both preciion and bia) Percent Accuracy: % accuracy = 100% %error (1.5) Uncertainty: Norally the error, deined by Eq. (1.3), i not known, and can only be etiated. We reer to thi etiate o the error a the uncertainty in the eaureent, which i uually preented a a range o value about the noinal value: x = (noinal value) u x, (1.6) where u x i the uncertainty in x. Note that we oten ue the ter error and uncertainty interchangeably, a we did in Section NOTE: uncertainty and error are not the ae thing! The ter are NOT interchangeable, although even engineer ue error when they ean uncertainty. Never do thi. (What IS the dierence between error and uncertainty?) Abolute Uncertainty: The uncertainty o a eaureent expreed in the unit o the eaureent. (Exaple: 10 ± 3, or ore iply, 10 ± 3 ) Percent Uncertainty: The uncertainty expreed a a percentage o the noinal value. (Exaple: 10 ± (3 )/(10 ) 100% 10 ±.5%. Relative (or Fractional) Uncertainty: The uncertainty expreed a a raction o the noinal value. (Exaple: 10 ± (3 )/(10 ) 10 ± 0.05 /. Note that the uncertainty i unitle, but the unit (/) have been included anyway to ake ure the audience know what the value repreent.) 1.5 Other Deinition Variable: Any quantity that can be eaured (e.g., teperature, weight) or oberved (roll o a die, nuber o tudent) a. Dependent Variable: aected by change o one or ore variable (e.g., the teperature in a roo i aected by the tie o day; the teperature i thereore the dependent variable) b. Independent Variable: can be varied independent o other variable (e.g., the tie o day i independent o the teperature in a roo) c. Dicrete Variable: poible value can be enuerated (e.g, roll o a die, nuber o tudent) d. Continuou Variable: poible value are ininite (e.g., all phyical propertie, uch a teperature, denity, length) e. Controlled Variable: held at a precribed value during a eaureent (e.g., teperature were eaured every 10 econd: tie i thereore a controlled variable). Extraneou Variable: not controlled during an experient g. Rando Variable: contain rando catter 1.5. Paraeter: a unctional relationhip between variable (e.g., drag coeicient) 1-

3 1.5.3 Noie: variation in a eaured ignal due to rando luctuation o extraneou variable Intererence: variation in a eaured ignal due to deterinitic variation in extraneou variable (e.g., electrical intererence) Sequential Tet: experient where the controlled variable i varied in order Rando Tet: experient where the controlled variable i varied randoly Repetition: repeated eaureent ade ro the experient, to exaine the variability in obervation or a ingle condition Replication: duplication o the experient under iilar operating condition (e.g. ae tet, dierent day) Concoitant Method: dierent ethod or eauring the ae variable; bet i eaureent technique i baed on dierent phyical propertie Static Meaureent: one in which input variable i contant with tie Dynaic Meaureent: one in which input variable change with tie 1.6 A Prier on Dienion and Unit Dienion v. Unit Nearly every engineering proble you will encounter will involve dienion: the length o a bea, the a o a concrete block, the tie and velocity o an object all, the orce o the air reitance on an airplane, and o orth. We expre thee dienion uing peciic unit: or exaple, length can be expreed in eet, a a kilogra, tie a inute, velocity a ile per hour, and orce a newton 1. The goal o thi ection i to explain the ue o dienion and unit in engineering calculation, and to introduce a ew o the tandard yte o unit that are ued How Dienion Relate to Each Other Dienion (a well a unit) act jut like algebraic ybol in engineering calculation. For exaple, i an object travel 4 eet in 10 econd, we can calculate it velocity. Firt, algebraically: d v =, t where v i the ybol or velocity, d or ditance, t or tie. Plugging in the actual value (and unit), (4 t) t v = = 0.4. (10 ) 1 Notice that newton i not capitalized. It i tandard not to capitalize the nae o the unit, even though the unit abbreviation i capitalized (i.e., N). It a conuing rule. 1-3

4 Thu we can ee that velocity in thi cae ha the unit eet per econd (t/). We can convert eet to whatever we like: eter, ile, etc. We can alo convert econd to inute, hour, day, etc. But the dienion are alway the ae: [length] velocity =. [tie] There are two kind o dienion: (1) priary dienion, like length and tie, and () econdary dienion, like velocity, which are cobination o priary dienion. Becaue any given yte o unit we ue ha o any dierent eaureent, tandard unit have been developed to ake counication (and cience and coerce) eaier. We will explore three o thee tandard yte: the SI yte, the Britih Gravitational yte, and the Englih Engineering Syte. There are ore! The SI yte The SI (Sytèe International d Unitè) yte i the oicial nae or the etric yte. The yte i decribed a an MLtT yte, becaue it priary dienion are a (M), length (L), tie (t), and teperature (T). The tandard unit are lited below. Priary Dienion Standard Unit a (M) length (L) tie (t) teperature (T) kilogra (kg) eter () econd () Kelvin (K) Secondary unit are derived ro thee priary unit. For exaple, velocity ha unit o /, acceleration i /, and orce ha unit o? How do we relate orce to the priary unit? Iaac Newton dicovered that the orce on an object i proportional to it a tie it acceleration: F a. I we plug dienion into the above relation, we ee that Force [L] [M] [t]. Or, i we ue priary SI unit, we ee that Force kg. In honor o Newton, it wa decided to give thi particular et o ter the nae newton (N). It i deined a kg 1N (1.7)

5 So the unit o orce in the SI yte i the newton (N), deined a the orce required to accelerate a a o 1 kg to an acceleration o 1 /. Why not kg? Or 10 /? Actually, the nuber i arbitrary, but the nuber 1 i choen or convenience. Exaple 1. Solution: An object ha a a o 80 kg. I the acceleration o gravity i 9.81 /, what i it weight? The weight o an object i the orce o gravity on the object, which i given by W = g. Plugging in value (and unit) or and g, W = (80 kg)(9.81 / ). (a) A you can ee, the reult o the above calculation doe give u the correct dienion and unit or orce. But or convenience, we know by deinition that 1N 1kg / =. get Notice that we can anipulate the above equation lightly: I we divide both ide by 1 kg /, we 1 N 1 kg / = 1. Thu, i we ultiply the right-hand-ide o Equation (a) by the ratio above, we are erely ultiplying by one and a unitle value o one which doen t change anything: 1 N (80 kg)(9.81 / ) 1 kg / W. = Note that all the unit cancel except or N, which yield W = N. Coent: 1. Note that we jut ued the deinition o a newton a a kind o converion actor to convert the anwer above into a ore convenient or. To be honet, it not neceary to ue newton, and in act oe engineer leave the unit o orce a kg / oetie, becaue they know the unit will cancel later. But jut reeber that you want to expre your inal anwer in a relatable unit a poible, or your audience undertanding.. Recall that we deterined the gravitational orce by the equation W = g. 1-5

6 Why didn t we ue Newton econd law, F = a, where a = g? In t that the ae? Abolutely not! GRAVITY IS NOT ACCELERATION. IT IS A FORCE (PER UNIT MASS). It only look like acceleration becaue it ha unit like that o acceleration (In act, dienionally, acceleration and orce per unit a are the ae). Think about thi. What i the orce o gravity acting on your body right now? Are you in otion right now? I you are itting till, you are not accelerating (relative to the ground). Then a=0! So i the orce on your body zero? No! Reeber that in tating Newton econd law, F i the net orce acting on the a. I the a i tationary, the net orce i zero. That i, the orce o gravity on your body i exactly balanced by the orce o the ground puhing up on you. You are in equilibriu, and thereore your acceleration i zero The Britih Gravitational Syte ( Slug Syte) The Britih Gravitational yte o unit i reerred to a an FLtT yte, becaue the priary dienion are orce (F), length (L), tie (t), and teperature (T). The tandard unit are: Priary Dienion Standard Unit Force (F) pound-orce ( ) length (L) oot (t) tie (t) econd () teperature (T) Rankine (R) I orce i a priary dienion, how do we ind the unit o a? Ma i now a econdary dienion; we have to derive it. Newton econd law alway hold: or, dienionally, I we ue priary unit, we ee that F a. [L] [F] [a]. [t] Rearranging the above, a a t,. t We need a nae or the unit o a. Let call it a lug! Then we ll deine it by 1 lug t 1. (1.8) We can interpret the above by aying, one pound-orce i the orce required to accelerate 1 lug to an acceleration o 1 t/. Again, we could have deined the lug a 10 /t, or /t, but or the ake o iplicity, we chooe 1 a the contant. 1-6

7 Exaple. Solution: An object ha a a o 5.59 lug. What i it weight in Earth gravity? A in Exaple 1, the weight o the object can be deterined by W = g. Subtituting the a and the value o tandard Earth gravity, t/, into the above, W = (5.59 lug)(3.174 t/ ) The unit above are not ueul a unit o orce. But we know by deinition that 1 lug =1 /t, or 1 /t = 1 1 lug. Multiplying the weight by the above give W = (5.59 lug)(3.174 = t/ 1 /t ) 1 lug We ee that the unit in the above relation cancel, leaving the ore convenient unit o orce The Englih Engineering Syte ( Pound-Ma Syte ) In the Englih Engineering yte o unit, the priary dienion are are orce (F), a (M), length (L), tie (t), and teperature (T). Thereore thi yte i reerred to a a FMLtT yte. The tandard unit are hown below: Priary Dienion Standard Unit Force (F) pound-orce ( ) a (M) pound-a ( ) length (L) oot (t) tie (t) econd () teperature (T) Rankine (R) In thi yte, orce and a are priary dienion. They ut till be related by Newton econd law: F a. 1-7

8 or, dienionally, [L] [F] [a]. [t] I we ue the priary Englih unit, we ee that t, We don t need to deine a new unit, but we need to deterine a contant in order to ake the above relation exact. Let ue 3.174! Then the relationhip between pound-orce and pound-a i a ollow: 1 t (1.9) So in word, one pound-orce i the orce required to accelerate one pound-a to t/. Why 3.174? Becaue that jut happen to be the value or the acceleration o gravity, g = t/. Thi value wa choen o that i an object ha a a o 10, it weight on the Earth will alo be 10. Thi convenience will becoe apparent later in one o the exaple which ollow. One inal note: I we copare Equation (3) with Equation (), we ee that lug and pound-a are related by 1 lug = (1.10) Exaple 3. Solution: An object ha a a o 180. What i it weight in Earth gravity? Again, the weight i given by W = g, which becoe W = (180 )(3.174 t/ ). To convert the unit in the above equation into ueul orce unit, we note that by deinition, 1 = t/. Or, t/ Multiplying thi contant with the weight give = 1. W = (180 = 180 )( t/ 1 ) t/ Coent: Note that in Earth gravity, and the pound-a yte, the value o a and weight are the ae! In act, that how the relationhip between and wa deined. Reeber, though, that the unit 1-8

9 repreent dierent dienion: repreent orce, while repreent a. So it i NEVER acceptable to write 1 = 1. Thi i not dienionally correct; it i like aying that 1 apple = 1 orange The Proportionality Contant g c A a inal note, i you haven t yet heard o g c ( g ub c ) in your tudie, you ight. It oetie reerred to a the gravitational contant, and it i a le-coon (oe ay ay it obolete, or oldahioned) way to deal with the orce-a unit relationhip. So i you run acro thi ter, how doe it work? Did you notice that, in every exaple above, we had to ultiply the weight we calculated by a converion actor to ake the unit coe out right? Well, what oe people do i jut eploy a actor, called g c, directly in the equation they are uing. For exaple, Newton econd law could be written a a F =. Siilarly, the gravitational orce could be written a g c g W =. g c Coparing g c in the equation above with the converion actor we ued in the exaple, you can how that kg / g c = 1 (SI yte), N and lug t/ g c = 1 ( lug yte), t/ g c = ( pound-a yte). I advie you not to ue the g c approach in your calculation. Thi technique can be conuing becaue you have to reeber when you have to include g c in your general equation. But, a you can ee ro the exaple proble, we ignored g c entirely; a long a you ALWAYS keep track o ALL your unit, you will know when you need to peror unit converion in order to cancel certain unit. Think o the deinition (1.7), (1.8), and (1.9) a a wild card that you can inert into a calculation when you need to ipliy the unit. A uary o the baic unit yte i preented in Table 1.1. Meorize the orce-a relationhip, and alway ue the explicitly in your calculation. 1-9

10 I can t over-ephaize thi point: NEVER DO UNIT CONVERSIONS IN YOUR HEAD. Alway how the, no atter how trivial. Incorrect unit are a leading caue o itake in calculation, oetie leading to tragic reult. Being explicit with your unit calculation will help you catch your own itake, and help your audience undertand your calculation (and convince the that you know what you are doing). Table 1.1. Suary o Unit Syte Syte SI Britih Gravitational Englih Engineering ( Metric yte) ( lug yte) ( pound-a yte) Priary Di MLtT FLtT FMLtT Ma kg lug Length t t Force N Tie Teperature K R R lug t t Force-Ma kg N 1 Relationhip 1 lug = g c g c kg / = 1 N g c lug t/ = 1 g c = t/ 1-10

11 Unit exaple Exaple 1.1. The preure acting on a 1.5 in tet pecien equal 15 MPa. What i the orce (in N) acting on the pecien? Anwer Exaple 1. The weight o a large teel cylinder i to be coputed ro eaureent o it diaeter and length. Let it length L be equal to 3.3 and it diaeter d equal to Suppoe that the denity o the teel equal 7835 kg/ 3. Calculate the weight o the cylinder (in N) and report your reult in a clear and unabiguou or. Anwer 1-11

12 Solution Exaple 1.1. The preure acting on a 1.5 in tet pecien equal 15 MPa. What i the orce (in N) acting on the pecien? Exaple 1. The weight o a large teel cylinder i to be coputed ro eaureent o it diaeter and length. Let it length L be equal to 3.3 and it diaeter d equal to Suppoe that the denity o the teel equal 7835 kg/ 3. Calculate the weight o the cylinder (in N) and report your reult in a clear and unabiguou or. 1-1

13 Reerence 1. Figliola, R.S. and Bealey, D.E., Theory and Deign or Mechanical Meaureent, 3 rd Edition, John Wiley and Son, Inc., New York, Tayor, J.R., An Introduction to Error Analyi, nd Edition, Univerity Science Book, Hoework Ue engineering paper, and how all work. Work all proble in the unit yte given (i.e., do not iply convert to SI, olve the proble in SI, and then convert back to the original unit). You ay need to review oe topic in your Phyic textbook to olve oe o thee proble. 1-1 You eaure the a o 10 M&M candie, and ind the average a to be g with a tatitical uncertainty o 0.05 g. The reolution o the cale i g, and variation o teperature in the roo add an error o ± g to the cale. a. Find the total error (i.e., uncertainty) o the eaureent o average a. b. Expre the coplete eaureent (noinal value and uncertainty). Do thi three way: with abolute uncertainty, percent uncertainty, and relative uncertainty. 1- To check the accuracy o a a cale, you place calibration a o g on the cale (or the purpoe o thi proble, the calibration a i exact). Your cale read g. Deterine the (abolute) error, percent error, and percent accuracy o the device at thi reading. 1-3 Anwer the ollowing quetion. Aue tandard gravity in each cae. a. How uch doe 3.0 kg weigh on the Earth, in N? b. How uch doe a peron with a a o 183 weigh on the oon ()? (The oon gravity i exactly one-ixth the Earth ) c. A gallon o water weigh about 8 on Earth. What i it a in lug? 1-4 Convert the ollowing, without uing any pecial unction in your calculator. a C to degree Fahrenheit b. 103 F to degree Rankine c. 37 K to degree Celiu d. A teperature dierence o 10 degree Celiu to degree Kelvin. e. A teperature dierence o 10 degree Rankine to degree Fahrenheit.. A teperature dierence o 10 C to F. 1-5 What i the preure o water on the botto o a 6.0 t deep pool, (a) relative to the preure at the urace (/in ), and (b) abolute, i the preure at the urace i atopheric? Ue the principle o luid tatic that you learned in Phyic. Aue the denity o water to be ρ w = 6.4 /t A barrel o water i weighed on a cale to be 150. a. Neglecting the weight o the barrel, and auing the water to have a denity o ρ w = 6.4 /t 3, etiate the volue o the water. 1-13

14 b. To what extent doe the buoyancy o the urrounding air aect the weight eaureent? (Hint: etiate it uing Archiede Principle. Aue the air denity to be ρ air = /t 3 ). 1-7 You ay recall ro Phyic that the heat capacity, C, o a ubtance i the energy gained or a given teperature rie (unit o Btu/ F, kj/k, etc.). Speciic heat, c, i the heat capacity per unit a (Btu/ F, kj/kg K, etc.). The ollowing experient ha been deigned to eaure the peciic heat, c, o water: Water with a a o 1.00 i heated by an electric heater that deliver heat at a rate o 300 Btu/hr (to igniicant igure). Over a period o 15 inute, the teperature o the water rie 75 F. What i the peciic heat o the water? 1-8 For the ake o conervation, you decide to eaure how at the water evaporate ro your wiing pool. You do thi by recording the level o the pool every day or a onth, and calculating ro it and the urace area the water evaporated. Anwer the ollowing quetion: a. Nae a any variable that ay inluence thi eaureent. b. Identiy any dependent and independent variable c. Identiy any dicrete and continuou variable d. Identiy any controlled and extraneou variable 1-9 Explain three concoitant ethod by which you could deterine the diaeter o a roughly -inch diaeter teel phere. In ter o accuracy, what are the advantage and diadvantage o each ethod? 1-14