A Primer on Dimensions and Units

Transcription

1 1 Dienion v Unit A Prier on Dienion and Unit Glen Thornrot Mehanial Enineerin Departent Cal Poly State Univerity, San Lui Obipo Nearly every enineerin proble you will enounter will involve dienion: the lenth o a bea, the a o a onrete blok, the tie and veloity o an objet all, the ore o the air reitane on an airplane, and o orth We expre thee dienion uin peii unit: or exaple, lenth an be expreed in eet, a a kilora, tie a inute, veloity a ile per hour, and ore a newton The oal o thi paper i to explain the ue o dienion and unit in enineerin alulation, and to introdue a ew o the tandard yte o unit that are ued How Dienion Relate to Eah Other Dienion (a well a unit at jut like alebrai ybol in enineerin alulation For exaple, i an objet travel 4 eet in 10 eond, we an alulate it veloity Firt, alebraially: d v =, t where v i the ybol or veloity, d or ditane, t or tie Pluin in the atual value (and unit, (4 t t v = = 04 (10 Thu we an ee that veloity in thi ae ha the unit eet per eond (t/ We an onvert eet to whatever we like: eter, ile, et We an alo onvert eond to inute, hour, day, et But the dienion are alway the ae: [lenth] veloity = [tie] There are two kind o dienion: (1 priary dienion, like lenth and tie, and ( eondary dienion, like veloity, whih are obination o priary dienion Beaue any iven yte o unit we ue ha o any dierent eaureent, tandard unit have been developed to ake ouniation eaier We will explore three o thee tandard yte: the SI yte, the Britih Gravitational yte, and the Enlih Enineerin Syte There are ore! 1

2 3 The SI yte The SI (Sytèe International d Unitè yte i the oiial nae or the etri yte The yte i deribed a an MLtT yte, beaue it priary dienion are a (M, lenth (L, tie (t, and teperature (T The tandard unit are lited below Priary Dienion a (M lenth (L tie (t teperature (T Standard Unit kilora (k eter ( eond ( Kelvin (K Seondary unit are derived ro thee priary unit For exaple, veloity ha unit o /, aeleration i /, and ore ha unit o? How do we relate ore to the priary unit? Iaa Newton diovered that the ore on an objet i proportional to it a tie it aeleration: F a I we plu dienion into the above relation, we ee that [L] Fore [M] [t] Or, i we ue priary SI unit, we ee that Fore k In honor o Newton, it wa deided to ive thi partiular et o ter the nae newton (N It i deined a k 1N 1 So the unit o ore in the SI yte i the newton (N, deined a the ore required to aelerate a a o 1 k to an aeleration o 1 / Why not k? Or 10 /? Atually, the nuber i arbitrary, but the nuber 1 i hoen or onveniene (1 4 The Britih Gravitational Syte ( Slu Syte The Britih Gravitational yte o unit i reerred to a an FLtT yte, beaue the priary dienion are ore (F, lenth (L, tie (t, and teperature (T The tandard unit are:

3 Priary Dienion Standard Unit Fore (F pound-ore ( lenth (L oot (t tie (t eond ( teperature (T Rankine (R I ore i a priary dienion, how do we ind the unit o a? Ma i now a eondary dienion; we have to derive it Newton eond law alway hold: F a or, dienionally, [F] [L] [a] [t] I we ue priary unit, we ee that Rearranin the above, a t, a t We need a nae or the unit o a Let all it a lu! Then we ll deine it by 1 lu t 1 ( We an interpret the above by ayin, one pound-ore i the ore required to aelerate 1 lu to an aeleration o 1 t/ Aain, we ould have deined the lu a 10 /t, or 9361 /t, but or the ake o ipliity, we hooe one a the ontant 5 The Enlih Enineerin Syte ( Pound-Ma Syte In the Enlih Enineerin yte o unit, the priary dienion are are ore (F, a (M, lenth (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 Fore (F pound-ore ( a (M pound-a ( lenth (L oot (t tie (t eond ( teperature (T Rankine (R 3

4 In thi yte, ore and a are priary dienion They ut till be related by Newton eond law: or, dienionally, F a [L] [F] [a] [t] I we ue the priary Enlih unit, we ee that t, We don t need to deine a new unit, but we need to deterine a ontant in order to ake the above relation exat Let ue 3174! Then the relationhip between pound-ore and pound-a i a ollow: t (3 So in word, one pound-ore i the ore required to aelerate one pound-a to 3174 t/ Why 3174? Beaue that jut happen to be the value or the aeleration o ravity, = 3174 t/ Thi value wa hoen o that i an objet ha a a o 10, it weiht on the Earth will alo be 10 Thi onveniene will beoe apparent later in one o the exaple whih ollow One inal note: I we opare Equation (3 with Equation (, we ee that lu and pound-a are related by 1 lu = Exaple Exaple 1 An objet ha a a o 80 k I the aeleration o ravity i 981 /, what i it weiht? Solution: The weiht o an objet i the ore o ravity on the objet, whih i iven by W = Pluin in value (and unit or and, W = (80 k(981/ (a A you an ee, the reult o the above alulation will not ive u the orret unit or ore But we know by deinition that 1N k / 4

5 I we divide both ide by 1 k /, we et 1 N 1 k / = 1 Thu, i we ultiply the riht-hand-ide o Equation (a by the ratio above, we are erely ultiplyin by one, whih doen t hane anythin: 1 N (80 k(981 / 1 k / W = Note that all the unit anel exept or N, whih yield Coent: W = 7848 N 1 Note that we jut ued the deinition o a newton a a onverion ator to onvert the anwer above into a ueul or Reall that we deterined the ravitational ore by the equation W = Why didn t we ue Newton eond law,? In t that the ae? Abolutely not! GRAVITY IS NOT ACCELERATION IT IS A FORCE (PER UNIT MASS It only look like aeleration beaue it ha unit like that o aeleration (In at, dienionally, aeleration and ore per unit a are the ae Think about thi What i the ore o ravity atin on your body riht now? Are you in otion riht now? I you are ittin till, you are not aeleratin (relative to the round Then a=0! So i the ore on your body zero? No! F = a, where a = Reeber that in tatin Newton eond law, F i the net ore atin on the a I the a i tationary, the net ore i zero That i, the ore o ravity on your body i exatly balaned by the ore o the round puhin up on you You are in equilibriu, and thereore your aeleration i zero Exaple An objet ha a a o 559 lu What i it weiht in Earth ravity? Solution: A in Exaple 1, the weiht o the objet an be deterined by W = Subtitutin the a and the value o tandard Earth ravity, 3174 t/, into the above, W = (559 lu(3174 t/ The unit above are not ueul a unit o ore But we know by deinition that 1 lu =1 /t, or 5

6 1 /t 1 lu Multiplyin the weiht by the above ive W = (559 lu( t/ 1 /t 1 lu We ee that the unit in the above relation anel, leavin the appropriate unit o ore Exaple 3 An objet ha a a o 180 What i it weiht in Earth ravity? Solution: Aain, the weiht i iven by W =, whih beoe W = (180 (3174 t/ To onvert the unit in the above equation into ueul ore unit, we note that by deinition, 1 =3174 -t/ Or, t/ Multiplyin thi ontant with the weiht ive W = ( (3174 t/ t/ Coent: Note that in Earth ravity, and the pound-a yte, the value o a and weiht are the ae! In at, that how the relationhip between and wa deined Reeber, thouh, that the unit repreent dierent dienion: repreent ore, while repreent a 7 The Proportionality Contant A a inal note, i you haven t yet heard o ( ub in your tudie, you will oon enouh What i, and what do we do with it? Did you notie that, in every exaple above, we had to ultiply the weiht we alulated by a onverion ator to ake the unit oe out riht? Well, what oe people do i jut eploy a ator, alled, diretly in the equation they are uin For exaple, Newton eond law ould be written a 6

7 a F = Siilarly, the ravitational ore ould be written a W = Coparin in the equation above with the onverion ator we ued in the exaple, you an how that k / (SI yte, N and lu t/ ( lu yte, t/ = 3174 ( pound-a yte You ay hooe not to ue the approah in your alulation A you an ee ro the exaple proble, we inored entirely A lon a you ALWAYS keep trak o ALL your unit, you will know when you need to peror unit onverion 8 Suary o Unit Syte Syte SI Britih Gravitational Enlih Enineerin ( Metri yte ( lu yte ( pound-a yte Priary Di MLtT FLtT FMLtT Ma k lu Lenth t t Fore N Tie Teperature K R R lu t t Fore-Ma k N 1 Relationhip 1 lu = 3174 k / N lu t/ = 3174 t/ 7