Phys 231 General Physics I Lecture 2 1
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1 Phys 3 General Physcs I Lecure Fr. 9/ Velocy & Momenum RE.b Mon. 9/3 Tues. 9/4.-.3, (.9,.0) Momenum Prncple & Smple Eamples RE.a EP, HW: Ch Pr.98 Se up compuer or clckers wh se parcpans Dsrbue clckers Equpmen o Clckers o rrack & cars wh bouncng ends o Pp sldes o Seres o balls / pendulum o RoangVecor. Py nnouncemens o Te I ve now scanned he nd chaper o he e, so you re wang or an ordered e o arrve, emal me and I can send you a copy; I m arad s a b bulky (abou 4M). I have a second copy on order, and once arrves I ll pu on reserve n he lbrary. o Oce Hours Monday :30-3:50, W 4-5. u don worry neher o hose mes s your schedule any me I m no n class, you re welcome o drop by. I ve lnked my schedule o he Oce Hours lne on he webse. Quesons o ny quesons abou he readng, he readng workshee, class srucure, ec.? Remember I you look a he class schedule ha I gave you Wednesday, or a gven day, you ll see he readng ha s o be done and he Readng Eercse ha s o be begun beore class ha day slgh bonus or quesons correcly done and submed beore class, slgh dock or quesons done and submed aer class. Inro. Newon s s n objec acceleraes only o he een ha eperences a ne orce. Ths echoes some o he senmen n ha prncple. Fundamenal Prncple o Mechancs: Moon s neher creaed nor desroyed, bu ranserred va neracons. Demos: Here are wo eamples o he prncple s playng ou Two cars on ar rack Seres o balls bumpng no each oher The program: Over he ne ew days, we re workng o ranslae hs prncple rom he ambguous language o Englsh no he precse language o mahemacs so we can make precse predcons abou s ramcaons. Quanyng Moon. So, our ask s o land on a combnaon o measurable, quanable characerscs o an objec ha can reasonably represen moon and obey hs prncple. Momenum. We pu a name o ha combnaon o measurable momenum. Tool Prep Vecors Mahemacally represenng moon hrough 3-space, akes vecors. So, we spen a lo o las me meeng / geng amlar wh hem. Noaon, addon & subracon, magnude, drecon / un vecor Pp. sldes - 8
2 ẑ ŷ Phys 3 General Physcs I Lecure Today Oulne o lle more vecor mah o Uns o Dsplacemen & Velocy o Momenum.3. 3D Coordnaes & Poson I we re alkng moon we ve go o alk geng rom one poson o anoher. So we ve go o quany posons. Mos commonly we alk n erms o rec-lnear coordnaes, such as norh-souh, easwes, and up-down. Then gven a reerence pon, an orgn, such as he class room door, any locaon on campus, such as he peak o he chapel bell ower, can be dened by hree numbers: how many meers Norh, how many meers Eas, and how many meers Up s rom here. In more generc erms, nsead o Norh, Eas, and Up, we oen alk abou he, y, and z drecons whch may be anchored no jus o he magnec led o he Earh, bu o he corner o a able, or anyhng else convenen. ˆ Q: Wha knd o mahemacal objec can hold hese hree peces o normaon? o Vecors -D Eample: Snce I can draw n 3-D, we ll work n -D, bu everyhng we ll do wll generalze o 3-D. Say each block s (or a leas represens) dm wde and dm all. Say I asked you o pu a do 3 dm over and 6 dm up rom he orgn o my coordnae sysem. o Pp. slde 5 Raher han specyng n Englsh yeah ar over and yeah ar up, I could use he acceped vecor shorhand o Vecor = <-componen, y-componen, z-componen>.
3 Phys 3 General Physcs I Lecure 3 o <3,6,0>dm reads 3 dm n he drecon and 6 dm n he y drecon rom he orgn. Q: Wre down all s poson componens: <,, >m. Wha do you have? <-7,-,0>m. Graphcal Vecor. So, we ve dened he locaon. Q: Wha s he graphcal represenaon o hs poson vecor? o For a poson vecor, he sandard s an arrow srechng rom he orgn sragh o he desgnaed pon,.e., he hypoenuse o a rangle wh sdes o he gven lenghs. For ha reason, he shor-hand symbol used or a vecor s some meanngul leer wh a lle arrow over. For ball s poson, a sandard choce would be r.
4 Phys 3 General Physcs I Lecure 4 r r Separaon Vecor / Vecor Subracon. Oen we ll be neresed n a he separaon o wo locaons, ha s, he vecor beween he wo. Jus as we ve drawn a vecor rom he orgn o o represen he poson o relave o he orgn, we would draw a vecor rom o o represen he poson o relave o. On he grd above draw he relave poson vecor r r r, and label clearly. r r r Componens. Wha are hs vecor s componens? Graphcally, we jus coun how ar over and how ar up/down.
5 Phys 3 General Physcs I Lecure 5 Mahemacally, he subracon s done componen wse; he proon s - componen subracs he elecron s -componen, ec. r r r. r., r. y r. y, r. z r. z ( 7) 3,( ) 6,0 0 dm 0, 8, 0 dm Ths essenally says, you re a ball, go down 0 meers and le 8 meers o arrve a ball. Q. Okay, you do one on your own: Wha s < 0, 0, -5 >dm - < 5, -8, 7 >dm? <5,8,->dm Magnude and Drecon. Que oen we wll be neresed n eher he magnude or he drecon o a vecor. Magnude. For eample, he srengh o he gravaonal orce depends upon he dsance beween he wo masses,.e., he magnude o he separaon vecor. Q: Wha s he magnude o r, hs s he dsance beween he wo balls? r Smply employ Pyhagorean s heorem. r = hs s jus lke beng asked wha s he lengh o he hypoenuse o a rgh rangle wh hese legs m 00 64m. 8m Pp. Q..5.d
6 Phys 3 General Physcs I Lecure 6 Pp. Q..5.e Drecon. The drecon o he gravaonal orce s sragh rom one ball o he oher,.e. he drecon o hs relave poson vecor. Ths s mos concsely gven by a vecor, ponng he same drecon as our separaon vecor, bu o magnude = (so aecs drecon, bu no magnude). Q: How would we calculae he un vecor rˆ (r-ha) Ths s he vecor o magnude bu ponng n he drecon rom o. Ths s he drecon rom ball o ball. Show your work. I you dvde our vecor by s magnude, hen he new vecor wll have a magnude o. Dvdng Vecor by Scalar. Dvdng or mulplyng by a scalar apples o each componen. r 0, 8,0 m rˆ = 0.78, 0.65, 0 noe: unless. r.8m Magnude & ngle / componens Thnk o he wo deren ways ha specyng somehng s locaon: o Componens: go 4 paces Eas and 7 paces Norh r = 4 paces, r y = 7 paces o Dsance & drecon: go 9 paces NE R = 9 paces, = 45 o I s oen useul o ranslae beween hese wo knds o descrpons. To do ha, we use a lle rgonomery. r r y r
7 Phys 3 General Physcs I Lecure 7 r r y rz cos, cos y, cos z r r r Rearrangng: r r cos r r cos Then agan, r r y rz rˆ,, cos r r r r cos, y y, z z r,cos y, cos z.6 SI Uns There are wo mporan hngs o know abou uns o lways use hem! o lways use he rgh ones SI uns! lways use Uns o In physcs, our numbers are quancaons o he naural world. We are very seldom alkng abou raw numbers or her own sakes, s 6, hen s 6 kg, 6m, 6s, 6m/s, 6 somehngs a mass, dsance, me, speed, Jus or proprey sake, you should ndcae ha n your work. o Usng uns caches errors. I you re solvng or a dsance and your work leads you o he ollowng b o mah 900m + 300s you know you ve done somehng wrong; whereas you jus wroe , you mghn noce your error. lways use he Rgh Uns SI uns o I s mporan o work n a conssen se o uns: no o add meers and yards ogeher o ge meer-yards, or even o add km and m ogeher o ge km-m s. o The conssen se o uns ha we wll be usng (unless oherwse saed) s he SI sysem. Meers, m, or dsance Seconds, s, or me Klograms, kg, or mass We ll come across some oher uns as we go. Conversons. I you re gven values n oher uns, you ll wan o conver uns. o I s good pracce o conver your uns beore you really ge gong n a problem, so You don have o remember o do laer Uns wll oen smply as you do your work You don make a huge mess by converng uns aer you ve begun buldng your equaons. o Pracce The eases way o conver uns s o mulply by. E.: Conver 4 nches o meers:.54cm The back o he book ells me ha nch. 54cm so, nch m o course here are 00cm n m, so 00 cm m 00cm
8 Phys 3 General Physcs I Lecure 8.7 Velocy verage Velocy.54cm m So, 4 nches 4nches 4nches 0. 0m nch 00cm (no worryng abou sg gs) o Denon : Change n poson per change n me. o Use denon n easy way Q..7 a Consder a bee lyng n a sragh lne wh consan speed. me = 5 s aer 9:00 M, he bee s poson vecor was <, 4, 0 > m. me = 5.5 s aer 9:00 M, he bee s poson vecor was < 3, 3.5, 0 > m. Wha s he velocy o he bee? r r r 3,3.5,0 m,4,0 m, 0.5,0 m v,,0 m / s 5.5s 5s 0.5s o o Rearrange denon o updae poson / Smulaons I. Say nsead o knowng he nal poson, you know he velocy, hen you can urn hs relaon around and use o predc he nal poson. r r v ave r r vave * We ll oen use n hs orm n our compuer smulaons. Even when he velocy s changng, we use a small enough me nerval, we can updae he posons and veloces eravely o map he objec s rajecory. o Updang Posons Q..7 c o verage vs. Insananeous -D moon represened n, plane o v ave r r r. nal nal 0 o average speed = slope o lne n, plane s ave = The same as he consan speed a whch hey could have gone, and goen here n he same amoun o me. Somemes he do moves orward, somemes backward o Calculus: Insananeous Speed
9 Phys 3 General Physcs I Lecure 9 s ave s( ) lm o ( ( ) ) ( ) ( ) ( ) d d ( ( ) ) ( ) Insananeous Velocy Smlarly, he nsananeous velocy s he speed and drecon o moon a a gven nsan. r( ) r( ) dr v( ) lm o ( ) d Q.7 d Q.7 e.8 Momenum o For more gudance on how o quany moon, we urn o wha I lke o call The undamenal Prncple o Moon (we ll mee hs agan) Moon s neher creaed nor desroyed hrough me or space, bu ranserred va neracons. Demos: o Row o balls ha swng no each oher Moon beore = Moon aer o r rack collson. Frs, wo cars o same mass see perec ranser o moon Moon beore = Moon aer Moon v?: v. beore v. beore v. aer v. aer? Then wo cars o deren mass. Tha worked prey well, bu als when he objecs have deren masses. pparenly ha was jus a specal case. Imporance o Mass o moon. Noe ha he objec wh he bgger mass underwen he smaller change n velocy. Wndsheld eample: When a bug hs a ruck s wndsheld, he massve ruck mpercepbly slows whle he puny bug drascally changes s velocy. Ths leads o proposng more generally: Moon mv?: v m v m v m v? m. beore. beore. aer. aer Ths works even when he objecs have deren masses, and reduces o our specal case when hey have he same. Ths works well enough n mos cases, bu jus lke he prevous verson, shows a law under specc condons. When he objecs are movng very quckly, becomes evden ha hs relaon sn rgh. I you measure he masses and he speeds and plug hem n, you don acually ge an equaly. Ths sn a perec mahemacal represenaon o moon eher.
10 Phys 3 General Physcs I Lecure 0 mv Momenum: moon p where c = speed o lgh n a vacuum = v c m/s. I urns ou ha hs s he epresson ha sases our noon o moon. o Vecor: hs s a vecor quany, wh s drecon beng ha o he velocy, ha n whch he objec s movng, and s magnude dependng on how massve and how as he objec s. o Uns: he square roo conrbues no uns snce s go a rao o lke hngs, so s jus uns o mass mes uns o velocy. In he common scenc un sysem, ha s kg m / s. E. Calculae he acor m/s) o so, p mv Q..8 b Calculae he acor v c v c he speed s 0.00c. (.e., around mv he speed s c. ) square o ).0000 Probably used v = m/s nsead o v = c 3).4 Used v = 0.999c 4).44e- Ths s he nverse o he acor 5) 70.7 Correc ppromaely mv. You ll noce ha he denomnaor s a rao o speed o he speed o lgh. For normal speeds lke, say 60 mph or even 000 mph, hs rao s realy, really small. So he denomnaor s really, really close o. When analyzng a suaon / workng a problem, v<<c, hen jus use mv. Noe on Physcs: Fundamenal Vs. Good Enough o Ths s an eample o a general aspec o Physcs. o Fundamenal Truh: On he one hand, s he scence ha eplores he absoluely undamenal laws o he unverse. Physcss work very hard o uncover unversal ruhs, such as moon beng neher creaed nor desroyed, bu ranserred va mv neracons we dene moon, or momenum as p v c o Good Enough: On he oher hand, once we uncover hese grea ruhs, we wan o do somehng praccal wh hem, so we re also very pragmac and we d raher use he
11 Phys 3 General Physcs I Lecure appromaon p mv n our mah when s good enough or our ends. For eample, I wan o ell you how as he ar car wll be gong aer a collson wh anoher ar car, and I wan an accuracy down o a ew sg gs, hen hs shorer, easer o use, appromaon, s good enough or my ends. o Updang Poson /Smulaons II: Non-relavsc poson updaes. p dr o In hs case, p mv v m d o So gven a small enough me nerval or momenum and p velocy o be consan, r r m Ne me: Ch.. The Momenum Prncple Quanyng he eec o neracons on moon.
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