CAPÍTULO 4. Dinámica de una partícula

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  • 7/28/2019 CAPTULO 4. Dinmica de una partcula

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    Godlo`a gm uda part`uha Eubj Lmgoda Buzld

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    @AYW]HJ 7$ GODLO@A GM ]DA YASW@]HA

    ODWSJG]@@ODMd mh `aptuhj adtmrojr mstugoaljs mh ljvolomdtjgm uda part`uha `jd rmspm`tj a ud sostmla gm

    rmfmrmd`oa sod prmbudtardjs sjcrm ha `ausa gmhljvolomdtj$ Hj gms`rocoljs solphmlmdtm md

    trlodjs gm hjs vm`tjrms

    r(

    v y

    a $Dumstra gos`usod fum bmjltro`a( md mstm`aptuhj gos`utormljs ha `ausa gmh ljvolomdtj$Pmbuormljs tratadgj a hjs `umrpjs `jlj

    part`uhas solphms$ Yjstmrojrlmdtm tratarmljssjcrm sostmlas gm part`uhas y `umrpjs rbogjs$

    MH JSOBMD GMH LJVOLOMDWJQu jroboda mh ljvolomdtj: Qu gmtomdm mh

    ljvolomdtj: Pm dm`msota `ausa para ljvmr has`jsas: Yjr qu ud jckmtj ah qum sm hm ga udmlpukd prjdtj sm gmtomdm: Yjr qu hjs phadmtasladtomdmd su ljvolomdtj ahrmgmgjr gmh sjh:Arosttmhms kjvmd fohsjfj brombj %sobhj OV a$`$/gm`a qum ud `umrpj pmrladm`omra mdljvolomdtj mra dm`msaroj mkmr`mr ahbuda a``odsjcrm h ya qum mh mstagj daturah ms mh rmpjsj$Mstj parm`m smr razjdachm( `uadgj gmkaljs gmmlpukar ud `umrpj( mstm prjdtj ah`adza mh rmpjsj$Yarm`m smr dm`msaroa uda a``od mxtmrojr j fumrzaapho`aga ah `umrpj para ladtmdmr mh ljvolomdtj$

    Pod mlcarbj( jcsmrvmljs msta sotua`od `jdlayjr gmtmdolomdtj$ Ha fobura sobuomdtm lumstraud chjqum gm lagmra sjcrm ud phadj$

    Apho`aljs uda fumrza pmquma ah chjqum( dj pasadaga$ Od`rmlmdtaljs ha fumrza y a ud vahjr

    parto`uhar mh chjqum sm lumvm$ Po smbuoljs

    od`rmlmdtadgj ha fumrza mlpukadgj j kahadgjls( mh jckmtj sm lumvm `jd layjr rapogmz(@uadgj gmkaljs gm mlpukar mh `umrpjrpogalmdtm vumhvm ah rmpjsj$ Pod mlcarbj so

    pjdmljs rumgas ah chjqum mh rmsuhtagj msgofmrmdtm( uda fumrza luy pmquma `ausa mhljvolomdtj$ Ha gofmrmd`oa sjd has rumgas gmcogja ha fro``od$Yara ea`mr ud mstugoj hocrm gm ha fro``od

    cusqumljs hhmbar `mr`adalmdtm a msta `jdgo`od(uda fjrla gm hjbrar mstj ms `jd uda lmsadmulto`a( sm sjpha aorm sjpha ea`oa arroca a

    travs gm pmqumjs abukmrjs ladtmdomdgj udgos`j suspmdgogj sjcrm ud `jh`ed gm aorm$ Qupasa `uadgj mlpukaljs ud jckmtj md ausmd`oa

    gm fro``od: Mstm sm ladtomdm md ljvolomdtj avmhj`ogag `jdstadtm$

    Md ausmd`oa gm uda fumrza rmsuhtadtm( mh jckmtj smladtomdm md ljvolomdtj `jd vmhj`ogag udofjrlmj pmrladm`m md rmpjsj$ Msta ms ha YSOLMSAHM_ GM DMRWJD GMH LJVOLOMDWJAejra pjgmljs pmdsar a`mr`a gm ha sotua`od

    `uadgj ud jckmtj mra mlpukagj sjcrm ud phadj$@uadgj ha fumrza mra pmquma dj eacaljvolomdtj( pmrj uda fumrza gmcmra `ausarljvolomdtj> ha `jd`husod ms qum gmcm eacmrjtra fumrza a`tuadgj sjcrm mh `umrpj ha `uah aduha

    kustalmdtm mh mfm`tj gm ha fumrza qum apho`aljs$Ah od`rmlmdtar dumstra fumrza( ha fumrza jpumstatalcod sm od`rmlmdta( easta qum md ahbd vahjr

    parto`uhar ha fumrza jpumsta tmrloda gmod`rmlmdtarsm y `jlomdza mh ljvolomdtj pjrqumeay uda fumrza rmsuhtadtm a`tuadgj sjcrm mhjckmtj$ Ha fumrza jpumsta ms ha fumrza gm Fro``od

    Q] MP F]MS[A:Md ha voga `jtogoada sm `jdsogmra fumrza a udasmdsa`od `jld asj`oaga `jd ha gofo`uhtag paraljvmr j hmvadtar ud `umrpj$ Md Fso`a smogmdtofo`a uda fumrza pjr mh mfm`tj qum prjgu`m$]dj gm hjs mfm`tjs gm uda fumrza ms `alcoar mhmstagj gm rmpjsj j gm ljvolomdtj gmh `umrpj(ls `jd`rmtalmdtm( uda fumrza `alcoa havmhj`ogag gm ud jckmtj( ms gm`or prjgu`m udaa`mhmra`od$ @uadgj sm apho`a uda fumrza sjcrmud `umrpj y dj sm prjgu`m ljvolomdtj( mdtjd`ms

    pumgm `alcoar su fjrla( ad so mh `umrpj ms luyrbogj$ Ha gmfjrla`od pumgm j dj smr

    pmrladmdtm$ Mdtjd`ms hjs mfm`tjs gm ha fumrzadmta sjd gjs; `alcoar mh mstagj gm ljvolomdtjgm ud `umrpj j prjgu`or uda gmfjrla`od( jalcas `jsas$

    Djrlahlmdtm sjcrm ud `umrpj pumgmd a`tuarvaroas fumrzas( mdtjd`ms mh `umrpj a`mhmrar`uadgj mh mfm`tj gm ha fumrza dmta qum a`tasjcrm h dj ms `mrj$Pm hhala fumrza dmta j fumrza rmsuhtadtm a ha

    sula gm tjgas has fumrzas qum a`tad sjcrm ud`umrpj$ Po ha fumrza dmta ms `mrj( ha a`mhmra`odms `mrj( mh ljvolomdtj ms `jd vmhj`ogag obuah a`mrj %`umrpj gmtmdogj/ j `jd vmhj`ogag

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    `jdstadtm$ @uadgj ud `umrpj mst md rmpjsj j smlumvm `jd vmhj`ogag `jdstadtm( sm go`m qum mstmd mquohocroj$Pm pumgmd gostodbuor gjs bradgms `hasms gmfumrzas; fumrzas gm `jdta`tj( rmprmsmdtad mhrmsuhtagj gmh `jdta`tj fso`j mdtrm mh `umrpj y susahrmgmgjrms( pjr mkmlphj ljvmr ud `arrj j mstorarud rmsjrtm> y fumrzas gm a``od a gostad`oa quma`tad a travs gmh mspa`oj sod qum eaya `jdta`tjfso`j mdtrm mh `umrpj y sus ahrmgmgjrms( pjrmkmlphj ha fumrza `jd qum ha Womrra atram a hjs`umrpjs qum `amd md `aga hocrm$ Wjgas hasgofmrmdtms fjrlas gm fumrzas sm md`umdtrad gmdtrjgm msas gjs bradgms `hasofo`a`ojdms$Yara gms`rocor mh ludgj( ha fso`a `jdtmlpjrdmarm`urrm a `uatrj odtmra``ojdms j fumrzasfudgalmdtahms$

    1$ Has fumrzas mhm`trjlabdto`a gm atra``od jrmpuhsod mdtrm part`uhas `arbagas md rmpjsj jmd ljvolomdtj( ms lu`ej ls odtmdsa qum hafumrza bravota`ojdah$3$ Has fumrzas du`hmarms odtmdsas mdtrm part`uhassucatlo`as( sjd has gm layjr labdotug %193 ,190 vm`ms ha fumrza mhm`trjlabdto`a/$0$ Has fumrzas du`hmarms gcohms gm `jrtjah`ad`m( sjd varojs rgmdms gm labdotug %1913/lmdjr qum ha fumrza mhm`trjlabdto`a$7$ Ha fumrza gm atra``od bravota`ojdah mdtrm`umrpjs gmcogj a sus lasas( ms ha fumrza gm

    lmdjr labdotug `jlparaga `jd has jtras$

    Yara qum mh `jd`mptj gm fumrza sma mxa`tj smgmcm mstachm`mr ud ltjgj para lmgorha$ ]dafumrza sm pumgm lmgor pjr mh mfm`tj qum prjgu`m$Yjr mkmlphj sm pumgm usar ha gmfjrla`od qumuda fumrza prjgu`m md ud rmsjrtm( `jlj md hafobura$ Po sm apho`a uda fumrza vmrto`ahlmdtm a udrmsjrtm y sm mstora uda udogag( hm asobdaljs a hafumrza uda labdotug udotaroaF$ Pm apho`a aejrajtra fumrza ah loslj rmsjrtm ejrozjdtahlmdtm(

    prjgu`odgjhm ud mstoralomdtj gm gjs udogagms(

    ha labdotug gm ha fumrza smr gm 3F$ Po sm apho`adsoluhtdmalmdtm has gjs fumrzas( mh rmsjrtm sm

    od`hoda( y sm mstora = vm`ms$ Ha fumrzamquovahmdtm qum prjgu`m msm mstoralomdtj gmhrmsjrtm ms ha sula vm`tjroah gmFy 3F$ Ms gm`or(ha fumrza ms ud vm`tjr$

    Mh odstrulmdtj para lmgor fumrzas sm hhalagodallmtrj( ms ud rmsjrtm qum sm mstora sjcrmuda ms`aha$ Po sm apho`a uda fumrza gm uda udogagsjcrm mh godallmtrj( mh rmsjrtm sm mstora eastaqum mkmr`m uda fumrza obuah y `jdtraroa a haapho`aga$ Md ha ms`aha sm logm mh aharbalomdtjgmh rmsjrtm y sm hm asobda uda udogag gm fumrza$Gm msa ladmra sm `ahocra mh godallmtrj y smusa para lmgor fumrzas( pjr mkmlphj sm apho`a udafumrza sjcrm mh godallmtrj y so sm mstora 3(=udogagms( mdtjd`ms ha fumrza apho`aga ms 3(=vm`ms ha udogag gm fumrza$Mstm prj`mgolomdtj ms vhogj para pmqumjsaharbalomdtjs gmh rmsjrtm( ya qum so ha fumrza msluy odtmdsa( sm pumgm gmfjrlar y dj vjhvmr a sufjrla jrobodah$

    @ALCOJ GM VMHJ@OGAGDumstrj sobuomdtm prjchmla ms md`jdtrar udarmha`od mdtrm ha fumrza y mh `alcoj md mhljvolomdtj prjgu`ogj pjr sta$Yara mstj dm`msotaljs hj sobuomdtm;1$ ]d `arrj luy hobmrj qum pumga ljvmrsm sodfro``od sjcrm uda supmrfo`om ejrozjdtah$3$ ]da fumrza `jdstadtm$ Msta pjgmrdjs jctmdmrhalmgoadtm ud rmsjrtm %Po ladtmdmljs ud rmsjrtmmstoragj uda losla hjdbotug( ha fumrza qum hamstora ms `jdstadtm/$

    0$ ]d rmbostragjr gm tomlpj$ Mh ljvolomdtj gmh`arrj pumgm mstugoarsm so uda `odta gm papmh atagaa stm pasa a travs gmh rmbostragjr qum prjgu`mlar`as md ha `odta a odtmrvahjs gm tomlpjrmbuharms$

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    Ha fobura sobuomdtm lumstra ha `odta gm papmhprjgu`oga pjr uda fumrza `jdstadtm$

    @jd hjs gatjs jctmdogjs md msta mxpmromd`oa smrmahoza mh brfo`j gostad`oa , tomlpj y sm jctomdmuda `urva$@jd hjs gatjs talcod sm pumgm jctmdmr havmhj`ogag lmgoa md `aga odtmrvahj gm tomlpj$ Mhbrfo`j vmhj`ogag , tomlpj ms uda hdma rm`ta qumodgo`a qum mh ljvolomdtj ms `jd a`mhmra`od`jdstadtm$ Gm aqu pjgmljs `jd`huor qum uda

    fumrza `jdstadtm prjgu`m uda a`mhmra`od`jdstadtm$

    Gupho`aljs ha fumrza usadgj gjs rmsjrtms obuahmsmstoragjs ha losla hjdbotug( `jlj sm lumstra mdha fobura$

    Gupho`a ha fumrza y prjgu`m mh gjchm gma`mhmra`od$ Po tropho`aljs ha fumrza sm jctomdmuda a`mhmra`od gm vahjr trophm$@jd`huoljs qum ha a`mhmra`od a gmh `umrpj msgorm`talmdtm prjpjr`ojdah a ha fumrza$

    Fa Yjgmljs ms`rocor mstj `jlj laF< ( gjdgm lms ha `jdstadtm gm prjpjr`ojdahogag$ A msta

    `jdstadtm ha hhalarmljs LAPA$Yara uda gmtmrlodaga fumrza a layjr `jdstadtml ha a`mhmra`od ms lmdjr$ A layjr vahjr gm ha`jdstadtm ms ls gof`oh a`mhmrar mh `umrpj$Yara `jdj`mr qu fa`tjrms `alcoad msta `jdstadtmrmaho`mljs mh sobuomdtm mxpmrolmdtj; md hubar gmusar ud sjhj `arrj kahagj pjr mh rmsjrtm mstoragjusmljs gjs `arrjs udj sjcrm jtrj y humbj trms`arrjs `jlj sm lumstra md ha fobura

    Ha a`mhmra`od qum sm jctomdm `jd hjs `arrjs msobuah a ha lotag y `jd trms ms obuah a ud tmr`oj$@jlj mh vahjr gmFms obuah md tjgjs hjs `asjs(quomrm gm`or qum ha `jdstadtm `jd gjs `arrjs msobuah a 3l y `jd trms `arrjs ms 0l$@jlj ha a`mhmra`od ms uda `adtogag vm`tjroah hafumrza talcod hj ms y tomdm ha losla gorm``odqum ha a`mhmra`od( pmrj ud lguhj l vm`mslayjr( gm ljgj qum ha rmha`od adtmrojr pumgmms`rocorsm md ha fjrla

    < alF Fumrza < lasa x a`mhmra`od$Msta mxprmsod `jdstotuym ha PMB]DGA HM_GM DMRWJD GMH LJVOLOMDWJ$Ha fumrza qum a`ta sjcrm ud `umrpj ms obuah ah

    prjgu`tj gm ha lasa gmh `umrpj pjr ha a`mhmra`odqum hm olprolm$

    ]DOGAGMP GM F]MS[A _ LAPAHa rmha`od laF< djs ga uda rmha`od mdtrmfumrza( lasa y a`mhmra`od$ Md mh sostmlaodtmrda`ojdah %P$O$/ ha udogag gm a`mhmra`od msl*s$ @uahms sjd has udogagms gm fumrza y gmlasa: @jlj sjd gjs `adtogagms qum smrmha`ojdad shj tmdmrdjs qum mspm`ofo`ar udmstdgar para uda gm mhhas$Mh sostmla odtmrda`ojdah agjpta `jrdj udogag uda

    pomza gm latmroah hhalagj NOHJBSALJ( `uyj

    slcjhj ms nb$ Mh nohjbralj ms ha lasa udprjtjtopj gm phatodj orogoagj sad`ojdagj pjr ha@jdfmrmd`oa Bmdmrah gm Ymsas y Lmgogasrmahozaga md Yars md 155? y gmpjsotagj md mh

    pacmhhd gm Crmhmuoh md Pmvrms$Ha udogag gm fumrza ms mh dmwtjd( `uyj slcjhjms D y sm gmfodm as;Mh dmwtjd ha fumrza qum prjgu`m uda a`mhmra`odgm ud lmtrj pjr smbudgj ah `uagragj a uda lasagm ud nohjbralj$

    3s

    nblD <

    Jtrjs sostmlas;LNP; obuah ah P$O$@BP; Lasa bralj %b/( h b < 19,0 nb

    A`mhmra`od `l*s3Fumrza goda < b$`l*s3

    Odbhs t`do`j; Md mstm sostmla ha udogagfudgalmdtah ms ha udogag gm fumrza$

    Fumrza hocra %hc/( 1 hc < 7(7= DA`mhmra`od pom*s3Lasa shub < hc s3*pom

    YMPJ GM ]D @]MSYJ$ Mh pmsj gm ud `umrpjms ha fumrza gm atra``od qum mkmr`m ha Womrrasjcrm mh `umrpj$ ]d `umrpj gm lasa l sjlmtogj

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    Godlo`a gm uda part`uha Eubj Lmgoda Buzld

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    a `omrta fumrza `am `jd ha a`mhmra`od gm habravmgag b( mh pmsjYgm mstm `umrpj ms

    < blY

    Pu gorm``od ms ea`oa acakj %ea`oa mh `mdtrj gm haWomrra/$ @jlj mh pmsj ms uda fumrza gmcmlmgorsm md Dmwtjds$Gmcogj a qum ha a`mhmra`od gm ha bravmgag vara

    gm ud hubar a jtrj gm ha Womrra( mh pmsj gm ud`umrpj ms gofmrmdtm md hubarms gostodtjs( sodmlcarbj ha lasa gm ud `umrpj ms ha `adtogag fokaqum dj gmpmdgm gmh hubar gjdgm mst sotuagj mh`umrpj(Audqum mh pmsj gm ud jckmtj vara gm ud sotoj ajtrj( msta varoa`od ms gmlasoagj pmquma parasmr jcsmrvaga md ha layjr partm gm hasapho`a`ojdms pr`to`as( pjr mstj( mh pmsj gm ud`umrpj parm`m smr uda `ara`tmrsto`a `jdstadtm ahobuah qum su lasa$ Mstm em`ej ea `jdgu`ogj ahmlphmj jrgodaroj gm jtras gjs lmgogas;

    NOHJBSALJ F]MS[A( ms mh pmsj gm udnohjbralj lasa$1 nbf < ?(5 DHOCSA LAPA( ms ha lasa gm ud `umrpj qum

    pmsa uda hocra$1 hocra lasa < 9(7=7 nb$Mstas udogagms sjd pr`to`as pmrj od`jrrm`tas ydj gmcmd smr usagas md Fso`a$

    A@@OD _ SMA@@OD$Eabaljs uda jcsmrva`od ls gmtahhaga `uadgj

    kahaljs mh `arrj `jd ud rmsjrtm mstoragj udagmtmrlodaga hjdbotug$

    Yara qum mh rmsjrtm mst mstoragj ms dm`msarojkaharhj pjr hjs gjs hagjs$ Pm dm`msotad fumrzas mdsmdtogjs jpumstas y md `aga mxtrmlj gmh rmsjrtm$@uadgj kahaljs mh `arrj( uda fumrza a`ta sjcrmmh `arrj y uda fumrza md smdtogj jpumstj a`ta

    sjcrm dumstra ladj$ @uhms sjd has labdotugmsgm mstas fumrzas:

    @jd mh jckmtj gm gar rmspumsta a msta prmbudtapjdbaljs gjs rmsjrtms obuahms ah prolmrj ykahmljs gm tah ladmra qum mh `arrj agquomra halosla a`mhmra`od qum adtms( mstj quomrm gm`or(

    pjr ha smbudga hmy gm dmwtjd qum somdgj halosla lasa l mstaljs apho`adgj ha loslafumrza %F< la/ qum adtms y jcsmrvaljs qum hjsrmsjrtms mstorad ha losla hjdbotug( hj qum quomrmgm`or qum ha fumrza sjcrm ha ladj ms obuah a hafumrza sjcrm mh `arrj$

    Mstj `jdstotuym ha WMS@MSA HM_ GMDMRWJD GMH LJVOLOMDWJ$Po ud `umrpj mkmr`m uda fumrza sjcrm ud smbudgj(stm mkmr`m uda fumrza obuah y jpumsta sjcrm mh

    prolmrj$ Ha fumrza mkmr`oga pjr mh prolmr `umrpjsjcrm mh smbudgj ms ha A@@OD( ha fumrza obuah yjpumsta a`tuadgj sjcrm mh prolmrj ms haSMA@@OD$Mxprmsagj md slcjhjs( ms;

    3agmcogj1sjcrm1agmcogj3sjcrm

    < FF Fumrza gm `jdta`tj gm ud `umrpj a jtrj `jd ud`alcoj gm gorm``od j sod hA `jdtodua`od prmsmdtardjs ahbudjs `asjs topjgm ha apho`a`od gm has hmyms gm Dmwtjd$

    AYHO@A@OJDMP GM HAP HM_MP GMDMRWJD@uadgj mstugoaljs @odmlto`a( md`jdtrardjs hasrmha`ojdms mdtrm gmsphazalomdtj( a`mhmra`od ytomlpj$ Yjr mkmlphj( `jdj`omdgj ha a`mhmra`od

    a has `jdgo`ojdms tahms `jlj pjso`od odo`oah(vmhj`ogag odo`oah( ms gm`or ha pjso`od y havmhj`ogag md mh tomlpj qum hhalaljs odo`oah %t