LECTURE 2 1. THE SPACE RELATED PROPRIETIES OF PHYSICAL QUANTITIES

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1 LECTURE. THE SPCE RELTED PROPRIETIES OF PHYSICL QUNTITIES Phss uses phsl prmeters. In ths urse ne wll del nl wth slr nd vetr prmeters. Slr prmeters d nt depend n the spe dretn. Vetr prmeters depend n spe dretns. E: n nset mves n pln.e. D spe;ts dsplement s vetr ut the trvelled dstne s slr.. VECTOR NOTTION ND OPERTIONS WITH VECTORS One drws vetr s dreted lne nd lels t sml letter vered n rrw lne. E.: The length f the lne s prprtnl t the vetr mgntude. The mgntude s nted vetr lel r ts slute vlue sgn e. υ r fr velt. Nte tht the mgntude tself s pstve slr. The dretn f lne shws the dretn f vetr n spe nd the rrw shws ts renttn sense. D Fgure Equl dsplement vetrs. CD Tw vetrs re equl f the hve; sme unts equl mgntudes nd sme renttns n spe. C Fgure 3. SIC OPERTIONS WITH VECTORS Multplng vetr slrwth r wthut dmensns -When multpleddvded pstve slr wthut dmensns nl the vetr s length hnges. m * m.5* m m 3 Fgure 3 sme dmensn -When multpleddvded negtve slr wthut dmensns the vetr s length hnges nd the renttn s nverted. Fgure 4 sme dmensns -If the vetr s multpleddvded slr wth dmensns the sme rules ppl fr mgntude nd renttn ut the new vetr hs dfferent unts euse t s nther phsl quntt fgure 5. [ m s] [ m] [ m s]*[ s ] Dspl.= vel. * tme F el [ N] tn Fgure nd 5 sutrtn dfferent dmensns f vetrs. E[ N C] [ N] [ C] El. feld = El. Fre negtve hrge Sme phsl prmeters lled tensrs depend n mre mplted w n dretn n spe.

2 ddtn nd sustrtn f tw vetrs These tw pertns re llwed nl f the vetrs represent the sme phsl qunttes sme dmensns. One nnt dd velt vetr t dsplement vetr. One uses the Tp Tl methd fr vetr ddtn; Shft ne f vetrs prllel t tself s tht ts tp fts t the tl f the ther ne. The vetr sum hs the tl lted t the free tl nd the tp lted t the free tp. Fgure mprng drwngs n fg.6. nd ne sees tht the vetr -T sutrt frm t frst ne multples - nd gets the vetr - fg7.. Net ne pples the rules fr ddtn f vetrs nd - Fgure Fgure mprng drwngs n fg.7.3 nd 7.4 ne sees tht the vetr Imprtnt nte: In generl the mgntute f vetr result s nt equl t the sum r dfferene f mgntudes f dded r sustrted vetrs. nd 3 Fgure Nte: The ddtn f vetrs s sstve: 4 Fg. 8. shws tht ne m ppl the tl-tp rule even fr set f severl vetrs.

3 Multplng tw vetrs. -The slr r dt * prdut f tw vetrs s slr quntt wth new dmensn. It s defned the epressn * 5 nd n e pstve r negtve dependng n the ngle etween the vetrs see fg φ π φ > φ> π φ < Fgure One n shw grphll tht the slr prdut s dstrutve: * * * 6 -The vetrrss prdut f tw vetrs s new vetr s C nted s C 7 Ths vetr hs mgntude C sn 8 φ- s the ngle frm vetr t vetr. Nte: Selet lws φ π euse the mgntude f new vetr n nt e negtve. C snφ> The dretn f C s perpendulr t the plne defned vetrs nd ; ts renttn s defned nventn usng the rght- φ hnd rule: Curl the fngers f the rght hnd frm t ver the smllest ngle nd eep the thum up. Thum s dretn gves the Fgure renttn f C. One m use ls the rule f srew; The rght-hnd srew rule : If rght-hnd srew turns frm t lng smllest ngle etween them the dretn f ts dvnement s tht f rss prdut vetr C. Fgure.. s seen frm the fgure.- the vetr prdut s nt mmuttve: 9 3

4 Unt vetrs rdntve sstems nd vetr mpnents. - Unt vetrs. unt vetr s dmensnless vetr wth unt mgntude tht serves nl t defne dretn n spe. It s ver useful fr presenttn f dfferent phsl vetrs. In ths urse we wll use rght-hnded Crtesn sstems see Fg. n whh O O O dretns re defned three perpendulr unt vetrs. usng the ddtn rule f vetrs ne n present the vetr s s vetr lng O e wth mgntude; m e pstve r negtve The dmensn f the vetr ges wth ts mpnents. The sme set f Fgure Fgure 3 unt vetrs n e used t present dfferent vetr phsl qunttes. - If s dsplement then [m] [m] [m]; f s velt [ms] [ms] [ms]. The three mpnents defne full the vetr n the sstem O. nwng the three mpnents ne m fnd the vetr mgntude v Pthgr s therem s - One m epress the sum r the dfferene f tw vetrs thrugh ther mpnents s fllws: If If If C then C ; C ; C C then C ; C ; C 3 C then C ; C ; C 4 The unt vetr  lng the dretn f hs the dretn f the mgntude ut n dmensn. S usng nd ne gets ˆ 5 E: If m m 3m then m m 3m 3 ˆ m 4 4

5 5 Epressng the dt nd rss prdut use f vetr mpnents n Crtesne sstem O -Dt prdut * 6 One m fgure ut tht f α β γ re the ngles f vetr t es OOO then ne m lulte the mpnents f vetr lng these es s: ; _ ; _ 7 Nte tht the dt prdut etween eh tw unt vetrs f Crtesn frme O gves r r. * ; * ; * * ; * ; * 8 * ; * ; Cnsder tw vetrs nd nd ther mpnents n O. nd 9 One m lulte the dt prdut * usng the mpnents 9 n the sstem O nd pplng the results f dt prdut etween unt vetrs 8. S ne gets * * Imprtnt emple: Clulte the mgntude f ts mpnents. * = * nd * ** mprng * nd ** ne fnds tht nd -Crss prdut fter ntrdung the nept f unt vetr ne n defne the rss prdut epressn n sn where n s unt vetr perpendulr t the plne defned vetrs. ls ne m fnd the mpnents f vetr ther mpnents;.e. f nd 3 s sn = nd sn π= the epressns when ppled etween the unt vetrs gves: sn nd _ ; _ ; sn 4 Fgure 4 φ

6 Tng nt unt the reltns 4 ne fnds ut tht the epressn 3 trnsfrms t S f C C C C then C 5 ; C ; C 6 Nte: One m get qul thse mpnents f rss prdut thrugh the fllwng determnnt C C C C 7 The ppltn f vetr lger n trnsltnl equlrum Fg.5 - One ss tht fre F s ppled n n et f ths et underges n tn tht: ] tents t mve t lng gven dretn f the et s t rest; r ] tents t hnge ts velt vetr mgntude dretn r th f the et s n mvement; If the et velt des nt hnge even thugh ne r severl fres re tng n t ne ss tht t s n trnsltnl equlrum. We wll see tht n ths se the sum f ll fres tng n the et s F 8 Emple#: 5g r remns t restυ= n tle even thugh ne pples 5N fre dreted t 3 ve the hrntlfg.5. Fnd the fre f frtn nd the nrml fre n the r. Nte: The frtn fre s eerted the tle n the r lng ts surfe. It hs ppste dretn t the dretn f pssle l shft. Fgure 5 shws the set f ll fres tng n the r. F fr N F F g = 5N F=5N F Φ=3 There re fur fres eerted n the r: F g ; N; F; F fr. s the r s t rest F Fg N F Ffr 9 We selet es O O s shwn nd pret the vetr eq.9 n them. F 5N 3 _ nd _ F 5N sn3 O: O: F Ffr 53 Ffr Ffr N N Fg F N 5* 5sn 3 N 5 5sn N tng g =ms 6

7 Emple#:Let's nsder n et tht m rtte rund fed e pssng thrugh the et. Let's ssume tht the eerted tn s nt enugh t strt the rttn. S t remns t rest. One ss tht ths s n et n the stte f rttnl equlrum. smple emple s tht f tght lt under the effet f wrenh eng used t lsen t fg.6. Fgure 6 r F Frm ur eperene we nw tht the rttnl tn we ll t trque tn n the lt s lrger f ; - the mgntude f eerted fre F s gger. - the fre s ppled t gger dstne frm rttn e. - the fre s ppled t 9 ngle t wrenh e. S the rttn tn s ~ F ~ r ~ sn r. One defnes the trque vetr s r F 3 - The use f the rss prdut fr trque s mpletel ustfed; ts mgntude s r F sn r F nd t gves the renttn f rttnl tn t. In lt emple fg.6 the trque s dreted lng O nd rttnl tn s CCW. Imprtnt nte: The trque s defned lws wth respet t spef pnt; n e f pssle rttn psses ths pnt. When tlng ut the trque ne must prese the trque wth respet t O- pnt. In rttn prlems ne ples the rgn f rdnte sstem t O pnt. -s we wll see n rttnl equlrum the sum f ll trques tng n the et s equl t er. 3 pplng the ndtn 3 n the se f lt t rttnl equlrum we fnd tht the tn f the trque ppled wrenh s nelled tht f n nternl trque due t frtn f wll n the lt. These tw trques hve equl mgntude ut ppste dretn et nt nt et 3 Eemple: Fnd the mpnents f ppled trque n wrenh f r 3m. 3m nd the ppled fre 5N s perpendulr t wrenh. In O sstem see fg.6 the ppled fre hs the mpnents 5N nd the pstn r-vetr hs mpnents.3mmm. usng the rss prdut generl frmul n gets rf.3m 5N.3m.3m 5N 5N S ; ; 4. 5 the s reltn Nm. The sme result mes ut usng the defntn f mgntude usng r F sn r F.3m 5N sn9 4. 5Nm. One m get the sme mgntude f trque even pplng fre wth smller mgntude ut gger r gger lever rm. E: Fnd F mgntude tht pples the sme trque f r. 4m. 7

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