Units, Physical Quantities and Vectors

Transcription

1 What s Phscs? Unts, Phscal Quanttes and Vectos Natual Phlosoph scence of matte and eneg fundamental pncples of engneeng and technolog an epemental scence: theo epement smplfed (dealed) models ange of valdt Quantum Feld Theo Quantum Mechancs Temnolog let: Theo se s not an unpoven concept s an eplanaton of phenomena s based on obsevaton and accepted fundamental pncples Relatvstc Mechancs Classcal Mechanc s speed Phs211C1 p1 Quantfng pedctons and obsevatons phscal quanttes: numbes used to descbe phscal phenomena heght, weght e.g. opeatonal defnton: a quantt defned n tems of how t s measued standad unts: Intenatonal Sstem (SI aa Metc) defned unts establshed n tems of a phscal quantt deved unts establshed as algebac combnatons of othe unts Quantt Length Tme Mass Tempeatue Electc Cuent Unt mete (m) second (s) logam (g) elvn (K) ampee () Phs211C1 p2

2 Common pefes (now these!) Pef femto pco nano mco mll cent lo mega gga bbevaton f p n µ m c M G Powe of Ten /1,000,000,000,000,000 1/1,000,000,000,000 1/1,000,000,000 1/1,000,000 1/1,000 1/100 1,000 1,000,000 1,000,000,000 Phs211C1 p4

3 Dmensonal nalss: consstenc of unts lgebac equatons must alwas be dmensonall consstent. You can t add apples and oanges! d vt m 10 m 2 s/ ( 5s) Ca unts wth numbes though calculaton povdes chec on calculatons povdes coect unts fo answe also see google calculato! dstance speed tme / Phs211C1 p5 convetng unts teat unts as algebac quanttes multplng o dvdng a quantt b 1 does not affect ts value cm 1nch 2.540cm nch 12nches cm 1ft 1 ft cm ft nch Poblem Solvng Stateg (ISEE) Identf elevant concepts Set up the poblem Eecute the soluton Evaluate ou answe Phs211C1 p6

4 Unts Conveson Eamples Eample The wold speed ecod, set n 1997 s m/h. Epess ths speed n m/s Eample how man cubc nches ae thee n a lte engne? Phs211C1 p7 Sgnfcant Fgues and Uncetant Eve measuement of a phscal quantt nvolves some eo eo uncetant, not mstae andom eo aveages out small andom eo pecse measuement sstematc eo does not aveage out small sstematc eo accuate measuement Pecse and accuate Pecse ccuate less accuate 15 less pecse numbe numbe numbe Phs211C1 p8

5 Indcatng the accuac of a numbe: ± o ± δ sometmes (δ) eample: 20.3±.5 cm o 20.3(.5)cm. measued length of 20.3 cm ±.5 cm means that the actual length s epected to le between 19.8 cm and 20.8 cm. nomnal value: the ndcated esult of the measuement, 20.3 cm n the eample numecal uncetant: how much the actual value mght be epected to dffe fom the nomnal value,.5 cm n the eample sometmes called the numecal eo factonal uncetant: the facton of the nomnal value coespondng to the numecal uncetant.5cm cm pecentage uncetant: the pecentage of the nomnal value coespondng to the numecal uncetant.5cm 100% 100% 2.5% 20.3cm 20.3cm± 2.5% Phs211C1 p9 Sgnfcant Fgues: common wa of mplctl ndcatng uncetant numbe s onl epessed usng meanngful dgts (sg. fgs.) last dgt (the least sgnfcant dgt lsd) s uncetan 3 one dgt 3.0 two dgts (two sgnfcant fgues 2 sg. fgs.) 3.00 thee dgts,etc. (300 how man dgts?) Combnng numbes wth sgnfcant dgts ddton and Subtacton: least sgnfcant dgt detemned b decmal places (esult s ounded) Multplcaton and Dvson: numbe of sgnfcant fgues s the numbe of sg. fgs. of the facto wth the fewest sg. fgs / Intege factos and geometc factos (such as π) have nfnte pecson π Phs211C1 p10

6 Estmates and Ode of magntude calculatons an ode of magntude s a (ounded) 1 sg fg calculaton, whose answe s epessed as the neaest powe of 10. Estmates should be done n ou head chec aganst calculato mstaes! Compang Two numbes: Pecent Dffeence % dffeence 100% sometmes % dffeence 100% Phs211C1 p11 Vectos Scalas: a phscal quantt descbed b a sngle numbe Vecto: a phscal quantt whch has a magntude (se) and decton. Eamples: veloct, acceleaton, foce, dsplacement. vecto quantt s ndcated b bold face and/o an aow. notaton a o a o a o a etc The magntude of a vecto s the length o se (n appopate unts). a a (the magntude of a) The magntude of a vecto s alwas postve. The negatve of a vecto s a vecto of the same magntude put opposte decton (.e. antpaallel) Phs211C1 p12

7 Combnng scalas and vectos scalas and vectos cannot be added o subtacted. the poduct of a vecto b a scala s a vecto c a c a (note combnaton of unts) f c s postve, s paallel to a f c s negatve, s antpaallel to a Pottsvlle s about 5 mles noth Facvlle s about 3 tmes futhe,n the same decton Hambug s about 3 tmes futhe, n the opposte decton Phs211C1 p13 Vecto addton most easl vsualed n tems of dsplacements Let X C gaphcal addton: place and tp to tal; X s dawn fom the tal of the fst to the tp of the last X X Phs211C1 p14

8 Vecto ddton: Gaphcal Method of R Shft paallel to tself untl ts tal s at the head of, etanng ts ognal length and decton. Daw R (the esultant) fom the tal of to the head of. R the ode of addton of seveal vectos does not matte C C D C D D Phs211C1 p15 Vecto Subtacton: the negatve of a vecto ponts n the opposte decton, but etans ts se (magntude) ( Β) R Phs211C1 p16

9 Resolvng a Vecto (2-d) eplacng a vecto wth two o moe (mutuall pependcula) vectos > components dectons of components detemned b coodnates o geomet. -component -component θ 2 tanθ 2 cosθ snθ e caeful n 3 d, 4 th quadants when usng nvese tg functons to fnd θ. Component dectons do not have to be hoontalvetcal! θ Phs211C1 p17 Vecto ddton b components R C Resolve vectos nto components(, etc. ) dd le components C R C R The magntude and decton of the esultant R can be detemned fom ts components. n geneal R C Phs211C1 p18

10 Eample: coss-count se ses 1.00 m noth and then 2.00 m east on a hoontal snowfeld. How fa and n what decton s she fom he statng pont? Eample: dd the thee dsplacements: 72.4 m, 32.0 east of noth 57.3 m, 36.0 south of west 72.4 m, staght south Phs211C1 p19 Unt Vectos a unt vecto s a vecto wth magntude equal to 1 (unt-less and hence dmensonless) n the Catesan coodnates: î unt vecto n the decton ( hat ) ĵ unt vecto n the decton ( hat ) unt vecto n the decton ( hat ) Rght Hand Rule fo elatve dectons: thumb, ponte, mddle fo î, ĵ, Epess an vecto n tems of ts components: î ĵ Phs211C1 p20

11 Poducts of vectos (how to multpl a vecto b a vecto) Scala Poduct (aa the Dot Poduct) cosφ cosφ φ s the angle between the vectos 0 φ cos φ s the poton of along tmes the magntude of cos φ s the poton of along tmes the magntude of φ cosφ note: the dot poduct between pependcula vectos s eo Phs211C1 p21 Eample: Detemne the components of, and the scala poduct between (4.00m, 53.0 ) and (5.00m, ) Phs211C1 p22

12 Phs211C1 p23 Poducts of vectos (how to multpl a vecto b a vecto) Vecto Poduct (aa the Coss Poduct) 3-D alwas! φs the angle between the vectos Rght hand ule: C thumb ponte C mddle Catesan Unt vectos snφ C C φ Phs211C1 p24 Wte vectos n tems of components to calculate coss poduct ) ( ) ( ) ( ) ( ) ( C sn φ sn φ s the pat of pependcula tmes sn φ s the pat of pependcula tmes snφ C C φ snφ

13 Eample: s along the -as wth a magntude of 6.00 unts, s n the - plane, 30 fom the -as wth a magntude of 4.00 unts. Calculate the coss poduct of the two vectos. Phs211C1 p25 Vectos computes and calculatos: TI 89 epesentatons of 5î1 ĵ3 and 2 î7 ĵ 1 [5,1,3] dsplas as [ ] [2,7,-1] dsplas as [ ] [5,1,3][2,7,-1] poduces [ ] dotp( [5,1,3],[2,7,-1]) poduces 14. CossP( [5,1,3],[2,7,-1]) poduces [ ] Vectos n 2-D: pola coodnates, ectangula coodnates and comple numbes (watch degees vs. adans mode n calculato, mode->comple->ectangula o pola) R20 θ 37 can be nput as (20 37) whch poduces (ect. mode) nput nput becomes ( ) (pola mode) WTCH FOR HOW NGLES RE SPECIFIED IN ECH PROLEM Othe notons: <5,1,3> (POVRa), vaables n aas [] 0,1,2 etc Lean technques, not ust calculato shotcuts! Phs211C1 p26