{"id":34474,"date":"2024-09-16T13:00:49","date_gmt":"2024-09-16T13:00:49","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34474"},"modified":"2025-09-08T03:55:39","modified_gmt":"2025-09-08T03:55:39","slug":"leges-newtoni","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/la\/leges-newtoni\/","title":{"rendered":"Leges Newtoni"},"content":{"rendered":"<p><center><\/p>\n<h1>Leges Newtoni<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\n    Haec lectio tractat de legibus Newtoni earumque munere in dynamica corporum. Explicatur quomodo massa et velocitas momentum lineare determinent, et describuntur tres leges: inertia quae statum motus servat absente viribus externis, relatio inter vim et accelerationem, atque actio et reactio inter corpora. Per exempla sicut lapsus in planis et motus pendulorum illustratur applicatio harum legum, quae in exercitationibus practicis ad corroborandum discendi processum culminat.<\/em><\/p>\n<p><strong>Proposita Discendi:<\/strong><br \/>\n    Confecta hac lectione discipulus poterit:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Intellegere<\/strong> tres leges Newtoni earumque applicationem in dynamica corporum.<\/li>\n<li><strong>Applicare<\/strong> leges Newtoni ad analysin et solutionem problematum dynamicorum.<\/li>\n<li><strong>Agnoscere<\/strong> relationem inter massam, velocitatem et momentum lineare.<\/li>\n<li><strong>Analyzare<\/strong> momentum observatorum inertialium in studio dynamicae.<\/li>\n<li><strong>Explicare<\/strong> quomodo secunda lex Newtoni vim et accelerationem connectat.<\/li>\n<li><strong>Describere<\/strong> notionem massae inertialis et modum comparandi inter corpora diversa.<\/li>\n<\/ol>\n<p>    <strong>INDEX CONTENTORUM<\/strong><br \/>\n    <a href=\"#1\">Introductio<\/a><br \/>\n    <a href=\"#2\">Leges Newtoni de Dynamica corporum<\/a><br \/>\n    <a href=\"#3\">Quomodo uti legibus Newtoni?<\/a><br \/>\n    <a href=\"#4\">Solutio problematum legibus Newtoni utens<\/a><br \/>\n    <\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>Introductio<\/h2>\n<p style=\"text-align: justify;\">Si cinematica quam in <a href=\"https:\/\/toposuranos.com\/posicion-velocidad-y-aceleracion-cinematica\/\" rel=\"noopener\" target=\"_blank\">praeteritis lectionibus<\/a> recensuimus nobis permittit motum corporum describere, per leges Newtoni obtinemus dynamicam quae nobis concedit de causis motus (vel mutationum status motus) rationem reddere. Hic notiones positionis et temporis magni momenti sunt, quia per haec definimus velocitatem et accelerationem, sed his adiungitur aliud: massa.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=127s\" target=\"_blank\" rel=\"noopener\"><strong>Massa est magni momenti ad definiendum statum motus<\/strong><\/a> corporum, sive momentum lineare. Dicitur momentum lineare corporis, <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{p}<\/span>, esse productum massae per velocitatem.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Large \\vec{p}=m\\vec{v}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Status motus est idea praecipua quae subiacet legibus Newtoni.<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Tcp_1M1a7H0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"2\"><\/a><\/p>\n<h2>Leges Newtoni de Dynamica corporum<\/h2>\n<h4>Prima Lex (de Inertia):<\/h4>\n<p style=\"text-align: center; color: #000080; background-color: #80ff80;\"><strong>Absente causis externis, omnia corpora statum suum motus constantem servant.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=208s\" target=\"_blank\" rel=\"noopener\"><strong>Prima lex Newtoni<\/strong><\/a> ingeniose constituit duas quaestiones magni momenti pro physica. Prima et manifestior: momentum lineare tamquam magnitudinem conservatam statuit; secunda autem aeque gravis, sed implicitius posita, concedit nobis definire quid sit <strong>observator inertialis.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=322s\" target=\"_blank\" rel=\"noopener\"><strong>Multae sunt formae observatorem definire,<\/strong><\/a> sed inter omnes invenitur genus speciale quod observatorem inertialem appellamus. Differentia consistit in eo quod, ex prospectu observatoris inertialis, absente causa externa status motus corporum est magnitudo conservativa.<\/p>\n<h5>Quid distinguit observatorem inertialem ab eo qui non est?<\/h5>\n<p style=\"text-align: justify;\">Differentia consistit in eo quod, ex prospectu observatoris inertialis, absente causa externa status motus corporum est magnitudo conservativa.<\/p>\n<h4>Secunda Lex (de Vi et Massa):<\/h4>\n<p style=\"text-align: center; color: #000080; background-color: #80ff80;\"><strong>Ex prospectu observatoris inertialis, vis ab agente externo corpori impressa aequivalet variationi status eius motus.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=396s\" target=\"_blank\" rel=\"noopener\"><strong>Aliter dicendo, si vis corpori applicatur<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}<\/span> tum habetur quod.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Large \\displaystyle \\vec{F}=\\frac{d\\vec{p}}{dt}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Relatio nota \u00abvis aequalis massae per accelerationem\u00bb, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}=m\\vec{a},<\/span><\/span> nihil aliud est quam consequentia Secundae Legis Newtoni, quae oritur ex proprietatibus derivationum et conservatione massae.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\vec{F} &amp; =\\displaystyle \\frac{d\\vec{p}}{dt} = \\frac{d}{dt}\\left(m\\vec{v} \\right) \\\\ \\\\\n\n&amp; =\\displaystyle \\underbrace{\\frac{dm}{dt}}_{= 0}\\vec{v} + m \\underbrace{\\frac{d\\vec{v}}{dt}}_{= \\vec{a}} = m\\vec{a}\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">In hoc ultimo passu consideratum est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">dm\/dt=0<\/span><\/span> quia ponitur nullam massam addi aut auferri, et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\vec{v}\/dt<\/span><\/span> est definitio accelerationis.<\/p>\n<h5>Massa inertialis<\/h5>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=572s\" target=\"_blank\" rel=\"noopener\"><strong>Secunda lex Newtoni<\/strong><\/a> etiam concedit accuratius tractare de notione massae. Hic apparet ut constans proportionalitatis inter vim et accelerationem. Quanto maior est massa, tanto maior vis impellenda est ad eandem accelerationem obtinendam; unde intellegitur massam esse mensuram inertiae corporum atque hinc nomen <strong>massa inertialis.<\/strong> Si in duo corpora in quiete relativa ad observatorem inertialem eadem vis (sine permutatione materiae) agit, tunc habetur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m_1 \\vec{a}_1 = \\vec{F} = m_2 \\vec{a}_2<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ex hoc possumus comparare massas corporum per rationem magnitudinum accelerationum<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{m_1}{m_2} = \\frac{\\|\\vec{a_2}\\|}{\\|\\vec{a_1}\\|}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quapropter, si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m_2<\/span><\/span> esset \u00abunum kilogramma exemplar\u00bb, sufficit observare rationem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{a}_2\\|\/\\|\\vec{a}_1\\|<\/span><\/span> ut sciamus quot kilogrammata contineat <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m_1.<\/span><\/span><\/p>\n<h4>Tertia Lex (de Actione et Reactione):<\/h4>\n<p style=\"text-align: center; color: #000080; background-color: #80ff80;\"><strong>Si corpus A vim \u00abactionem\u00bb in aliud B exercet, tum B vim \u00abreactionem\u00bb in A exercet aequalis magnitudinis sed contrariae directionis.<\/strong><\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=Tcp_1M1a7H0&amp;t=794s\" target=\"_blank\" rel=\"noopener\"><strong>Tertia lex Newtoni non solum concedit<\/strong><\/a> accuratius de viribus disserere, sed etiam expresse statuit agentes externos qui vim applicant esse quoque obiecta physica eisdem obnoxia:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Agens externus est obiectum physicum viribus affici posse.<\/li>\n<li>Vires numquam dantur seorsum, sed semper fiunt in paribus dictis \u00abparibus actionis-reactionis\u00bb, et summa vectorialis horum parium semper est nulla.<\/li>\n<li>Paria actionis-reactionis semper occurrunt in corporibus diversis, ita ut vis totalis in corpus non sit necessario nulla.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Cum paria actionis-reactionis semper in linea recta exerceantur, hoc secum fert quod postea videbimus, scilicet conservationem momenti angularis.<\/p>\n<p style=\"text-align: justify;\">Praeter haec, tertia lex Newtoni etiam implicite dicit alia:<\/p>\n<ul style=\"text-align: justify;\">\n<li>Ut vis neta non-nulla corpori applicari possit, necesse est saltem alterum obiectum.<\/li>\n<li>Actio et reactio simul fiunt. Cum duo corpora possint a distantia inter se agere (per gravitationem vel electromagnetismum), sequitur ut necessario in mechanica Newtoniana adsit modus transferendi informationem ex uno puncto ad alterum celeritate infinita. Scimus tale esse impossibile, quia secundum relativitatem specialem celeritas maxima est celeritas lucis in vacuo; ideo dicimus hanc tertiam legem esse approximationem ad realitatem.<\/li>\n<\/ul>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/DuO-cvLNzwQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h2>Quomodo uti legibus Newtoni?<\/h2>\n<p style=\"text-align: justify;\">Ad intellegendum quomodo leges Newtoni adhibeantur ut earum significatio clara sit, optimum est ad exempla spectare quae in rebus concretis fundantur atque ad constructionem diagrammatum corporis liberi.<\/p>\n<h3>Diagrammata Corporis Liberi<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=96s\" target=\"_blank\" rel=\"noopener\"><strong>Diagramma corporis liberi est<\/strong><\/a> schema pictorium in quo vires quae in corpus agunt repraesentantur. Pro ratione eorum quae de pondere recensuimus, possumus construere sequentia exempla diagrammatum corporis liberi.<\/p>\n<h4>Corpus innixum in plano horizontali<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=180s\" target=\"_blank\" rel=\"noopener\"><strong>Propter gravitatem, omnia corpora massa praedita sentiunt vim<\/strong><\/a> ad terram directam. Per secundam Legem Newtoni observamus talem vim dari per productum massae et accelerationis gravitatis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{g}=-g\\hat{y},<\/span><\/span> ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">g=9,81[m\/s^2].<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{peso}=m\\vec{g} = -mg\\hat{y}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Quod intellegimus ut \u00abpondus\u00bb corporis re vera est magnitudo huius vis ponderis quam modo vidimus.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">{pondus}=\\|\\vec{F}_{peso}\\|= mg<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Cum quadrum in plano horizontali deponimus, apparet par virium actionis-reactionis: hae sunt vis ponderis et normalis. Taliae vires sunt aequales magnitudine sed oppositae directione ita ut summa vectorialis virium in corpus sit nulla et, propterea, status eius motus in tempore constans maneat.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ylYcrS1tJbw\/YVfZYBYCPjI\/AAAAAAAAFlU\/0mOIkMhF-Hwrysc_WB5o7MLvrQtxQJu2QCLcBGAsYHQ\/s0\/reposo-plano-horizontal.PNG\" width=\"497\" height=\"280\" alt=\"corpus innixum in plano horizontali\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ylYcrS1tJbw\/YVfZYBYCPjI\/AAAAAAAAFlU\/0mOIkMhF-Hwrysc_WB5o7MLvrQtxQJu2QCLcBGAsYHQ\/s0\/reposo-plano-horizontal.PNG\" width=\"497\" height=\"280\" alt=\"corpus innixum in plano horizontali\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<h4>Lapsus in plano horizontali<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=343s\" target=\"_blank\" rel=\"noopener\"><strong>Imaginamur nunc quadrum chorda alligari<\/strong><\/a> et trahi, ut ostenditur in sequenti diagrammate corporis liberi:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-a2GB2RJ7vTY\/YVgEny43ZkI\/AAAAAAAAFlk\/uZ9_XQ3JwVMiiWjZDNC5Y4ePVZ9GH1bngCLcBGAsYHQ\/s0\/fuerza-cuerpo-en-plano-horizontal.PNG\" width=\"793\" height=\"386\" alt=\"diagramma virium in obiecto in plano horizontali\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-a2GB2RJ7vTY\/YVgEny43ZkI\/AAAAAAAAFlk\/uZ9_XQ3JwVMiiWjZDNC5Y4ePVZ9GH1bngCLcBGAsYHQ\/s0\/fuerza-cuerpo-en-plano-horizontal.PNG\" width=\"793\" height=\"386\" alt=\"diagramma virium in obiecto in plano horizontali\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Hic cernimus apparentiam duorum parium actionis-reactionis: uno latere habemus paria cum viribus ponderis et normalibus corporum, tertium par actionis-reactionis cum extremis chordae quibus subiectum quadrum trahit, et ultimo par coniunctum vi impressa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\vec{F}_1<\/span><\/span> et vi attritus,<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{roce},<\/span><\/span> cuius valor maximus est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu\\|\\vec{F}_\\textnormal{normal}\\|.<\/span><\/span><\/p>\n<h5>Coefficientia attritus et vires attritus<\/h5>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=458s\" target=\"_blank\" rel=\"noopener\"><strong>Hic <span class=\"katex-eq\" data-katex-display=\"false\">\\mu<\/span> est coefficientia attritus<\/strong><\/a> quae exprimit resistentiam lapsui inter duas superficies; coefficientia attritus duas habet formas: unam kineticam (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_c<\/span><\/span>) et alteram staticam (<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_e<\/span><\/span>). Attritus staticus apparet cum corpus in quiete manet, dum kineticus apparet cum corpus labi coepit.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{lcr}\\mu = \\left\\{\\begin{array}{lll} \\mu_e &amp; ;&amp; \\textnormal{Corpus in quiete} \\\\ \\\\ \\mu_c &amp; ;&amp; \\textnormal{Corpus in motu} \\end{array}\\right. &amp; ; &amp; \\textnormal{Ubi } \\mu_c \\leq \\mu_e\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Vis attritus motui corporis quod patitur opponitur et potest (simpliciter) exprimi per sequentem formulam<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n        \\vec{F}_\\textnormal{roce} ( \\vec{F}_1 ) = \\left\\{\n\n            \\begin{array}{lll}\n\n            - \\vec{F}_1 &amp; ; &amp; \\|\\vec{F}_1\\| \\leq \\mu_e \\|\\vec{F}_\\textnormal{normal}\\| \\\\ \\\\\n\n            -\\mu_c \\|\\vec{F}_\\textnormal{normal}\\|\\hat{x} &amp; ; &amp; \\mu_e \\|\\vec{F}_\\textnormal{normal}\\| \\lt \\|\\vec{F}\\|\n\n            \\end{array}\n\n    \\right.<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Cum vis impressa minor vel aequalis sit attritui statico maximo, corpus in quiete manet respectu terrae. Si vis impressa maior est quam attritus staticus, tum corpus incipit moveri et attritus fit kineticus; vis neta in corpus est igitur: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{neta} = \\vec{F}_1 - \\mu_c\\|\\vec{F}_\\textnormal{normal}\\|\\hat{x},<\/span><\/span> et propterea movetur cum acceleratione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{a} = \\vec{F}_{neta}\/M<\/span><\/span>. Si postquam corpus motum est vis impressa aequatur attritui kinetico, corpus igitur movetur cum velocitate constanti.<\/p>\n<h4>Lapsus in plano inclinato<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=753s\" target=\"_blank\" rel=\"noopener\"><strong>Cum obiectum labitur per planum inclinatum<\/strong><\/a> ad angulum <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> habetur sequens diagramma virium:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-vMApZqsSDs0\/YVjUkV187NI\/AAAAAAAAFls\/eZ2DPO6f54AKJDtB7-y1DP3TFjIMZjT7ACLcBGAsYHQ\/s0\/diagrama%2Bde%2Bfuerzas%2Ben%2Bplano%2Binclinado.PNG\" width=\"823\" height=\"554\" alt=\"Diagramma virium pro lapsu in plano inclinato\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-vMApZqsSDs0\/YVjUkV187NI\/AAAAAAAAFls\/eZ2DPO6f54AKJDtB7-y1DP3TFjIMZjT7ACLcBGAsYHQ\/s0\/diagrama%2Bde%2Bfuerzas%2Ben%2Bplano%2Binclinado.PNG\" width=\"823\" height=\"554\" alt=\"Diagramma virium pro lapsu in plano inclinato\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Hic, propter commoditatem, electum est systema referentiae ita dispositum ut coordinata horizontalis cum plano lapsus coniungatur. In hoc schemate, vis ponderis in duas partes dividitur: unam parallelum et alteram perpendicularum ad motum.<\/p>\n<ul>\n<li><strong>Componente parallela:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{\\textnormal{peso},x}=mg\\sin(\\alpha)\\hat{x}<\/span><\/span><\/li>\n<li><strong>Componente perpendicularis:<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}_{peso,y}=-mg\\cos(\\alpha)\\hat{y}<\/span><\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Vis attritus apparet ut reactio ad partem parallelum motui virium ponderis, et vis normalis ut reactio ad partem perpendicularem virium ponderis. Si pars horizontalis virium ponderis superat attritum staticum maximum, tum mutabitur status motus quadrati cum acceleratione<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{a} = mg\\left(\\frac{\\sin(\\alpha) - \\mu_c \\cos(\\alpha)}{m}\\right)\\hat{x}<\/span><\/span><\/p>\n<h4>Massa pendens<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=1085s\" target=\"_blank\" rel=\"noopener\"><strong>Massa quae e chorda ad laquearia adligata pendet<\/strong><\/a> et quae in quiete manet habet sequentem diagramma corporis liberi:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-jWSXd7lQhiw\/YVjpzr8z-pI\/AAAAAAAAFl0\/sZ955vFZz3Y-MtKXQQK8UFPRAaHhgxyPgCLcBGAsYHQ\/s0\/masa-colgante.PNG\" width=\"555\" height=\"521\" alt=\"Diagramma corporis liberi massae pendentis\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-jWSXd7lQhiw\/YVjpzr8z-pI\/AAAAAAAAFl0\/sZ955vFZz3Y-MtKXQQK8UFPRAaHhgxyPgCLcBGAsYHQ\/s0\/masa-colgante.PNG\" width=\"555\" height=\"521\" alt=\"Diagramma corporis liberi massae pendentis\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Massa quae e chorda ad laquearia adligata pendet et quae in quiete manet habet sequentem diagramma corporis liberi:<\/p>\n<p style=\"text-align: justify;\">In chorda sunt par virium quas \u00abtensiones\u00bb appellamus; si chorda est inextensibilis, hae vires sunt aequales et oppositae. In quadrum etiam agit par virium: pondus et tensio chordae. Si quadrum pendet et quiescit, pondus et tensio oppositae sunt et aequalis magnitudinis. Est quarta vis quae hic non ostenditur, quae chordam laqueari coniungit; coniunctio harum quattuor virium constituit duo paria actionis-reactionis.<\/p>\n<h4>Motus penduli simplicis<\/h4>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=DuO-cvLNzwQ&amp;t=1158s\" target=\"_blank\" rel=\"noopener\"><strong>Massa ad chordam inextensibilem adligata, et haec ad laquearia, quae oscillat<\/strong><\/a> circa positionem aequilibrii propter pondus suum est quod pendulum simplex appellamus. Infra habemus eius diagramma corporis liberi.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-KEfWZhZZNl8\/YVkYdDhxLcI\/AAAAAAAAFl8\/CXQTZSVYbxIDw9G_JaVtV9VaG-ruwqHewCLcBGAsYHQ\/s0\/pendulo%2Bsimple.PNG\" width=\"366\" height=\"452\" alt=\"Diagramma corporis liberi penduli simplicis\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-KEfWZhZZNl8\/YVkYdDhxLcI\/AAAAAAAAFl8\/CXQTZSVYbxIDw9G_JaVtV9VaG-ruwqHewCLcBGAsYHQ\/s0\/pendulo%2Bsimple.PNG\" width=\"366\" height=\"452\" alt=\"Diagramma corporis liberi penduli simplicis\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify;\">Cum chorda sit inextensibilis, habemus accelerationem radialem esse nullam et, consequenter:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_{p,\\parallel} + T = ma_{\\parallel}(t) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Alterum autem, pro componente perpendiculare ad chordam habebitur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_{p,\\bot}=-mg\\sin(\\theta) = ma_{\\bot}(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">Ex hac ultima expressione possibile est inferre aequationem differentialem quae nobis permittet modelare positionem angularis <span class=\"katex-eq\" data-katex-display=\"false\">\\theta<\/span> penduli simplicis in tempore<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{d^2\\theta(t)}{dt^2} + \\frac{g}{l}\\sin(\\theta) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify;\">At obtentio huius aequationis et consequentiae quas ex ea deducere possumus sunt res quas postea diligentius videbimus.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Solutio problematum legibus Newtoni utens<\/h2>\n<p style=\"text-align: justify;\">Utere legibus Newtoni ad solvenda sequentia problemata:<\/p>\n<ol style=\"text-align: justify;\">\n<li>Quadrum <span class=\"katex-eq\" data-katex-display=\"false\">15[kg]<\/span> in superficie horizontali ponitur. Inter quadrum et superficiem est attritus staticus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_e=0,55<\/span><\/span> et attritus kineticus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_c=0,31<\/span><\/span>\n<ol>\n<li type=\"a\">Quae erit vis minima necessaria ut quadrum moveri incipiat?<\/li>\n<li type=\"a\">Computa accelerationem quadri cum hoc incipit moveri propter vim in priori inciso obtentam.<\/li>\n<\/ol>\n<\/li>\n<li>Quadrum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">12[kg]<\/span><\/span> in plano inclinabili ponitur. Si coefficientia attritus statici est <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_e=0,03,<\/span><\/span> determina angulum inclinationis maximum quo quadrum in quiete manebit.<\/li>\n<li>Quadrum <span class=\"katex-eq\" data-katex-display=\"false\">75[kg]<\/span> ascendit cum velocitate constanti per planum inclinatum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">30^o<\/span><\/span> respectu horizontalis propter vim quae in eum horizontaliter applicatur. Si inter quadrum et superficiem plani est coefficientia attritus kinetici <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mu_c=0,21,<\/span><\/span> determina magnitudinem huius vis applicatae.<\/li>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/kQaZdNm-iDQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<li>Considera duas massas <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m_1<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m_2<\/span><\/span> chordae inextensibili et sine massa conexas quae per trochleam transit, ut in figura monstratur. Computa accelerationem amborum massarum.\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEg6YZ3Pc3UlsxceJu3isovO7by4X1DHEtV-82DhHMl7FLepZDp8YQt4DAGxlQGSCemJE1Ai-SS4OdSMoJUc66U2cEKTuSbCbM_MhX-8jTElj2kwGK589fba-JoIcF9fDw_v36cKBD9OPrPJ6ZaXp4tKhK7qvNftjVQDoIyQDgGUlKXHLj_T3LOgT-rNyw\" width=\"298\" height=\"345\" alt=\"Machina Atwood\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEg6YZ3Pc3UlsxceJu3isovO7by4X1DHEtV-82DhHMl7FLepZDp8YQt4DAGxlQGSCemJE1Ai-SS4OdSMoJUc66U2cEKTuSbCbM_MhX-8jTElj2kwGK589fba-JoIcF9fDw_v36cKBD9OPrPJ6ZaXp4tKhK7qvNftjVQDoIyQDgGUlKXHLj_T3LOgT-rNyw\" width=\"298\" height=\"345\" alt=\"Machina Atwood\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>\n<\/li>\n<li>Chorda flexibilis massae <span class=\"katex-eq\" data-katex-display=\"false\">M<\/span> inter duas parietes pendet angulum <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha<\/span> in punctis coniunctionis formans. Computa tensionem chordae in puncto infimo.\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEh3rmFCklJ9Z7Hb6M5Nb_lREohLFKZNhujGlrx1m8aUKGvr4RnoB0w3H2wQj1VHo6or-UgVj7_CtuBiL-mIs17CGJT4FF9gVPXPEKR34_6sRLW50L8q8bb5vby23Lby4xLzX92nfDvUnWnJsxtuTzehTRw-N3NJq9R91n-UFXzCgksrGooUWFGX8D1VLA\" width=\"343\" height=\"281\" alt=\"Chorda pendens\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEh3rmFCklJ9Z7Hb6M5Nb_lREohLFKZNhujGlrx1m8aUKGvr4RnoB0w3H2wQj1VHo6or-UgVj7_CtuBiL-mIs17CGJT4FF9gVPXPEKR34_6sRLW50L8q8bb5vby23Lby4xLzX92nfDvUnWnJsxtuTzehTRw-N3NJq9R91n-UFXzCgksrGooUWFGX8D1VLA\" width=\"343\" height=\"281\" alt=\"Chorda pendens\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>\n<\/li>\n<li>Corpus massae <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span> circulos facit in plano <span class=\"katex-eq\" data-katex-display=\"false\">x,y<\/span> cum radio <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span> et velocitate angulari <span class=\"katex-eq\" data-katex-display=\"false\">\\omega<\/span> constanti. Computa vim in massam applicatam.<\/li>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/D3fnVM-HKJ4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Leges Newtoni Summarium: Haec lectio tractat de legibus Newtoni earumque munere in dynamica corporum. Explicatur quomodo massa et velocitas momentum lineare determinent, et describuntur tres leges: inertia quae statum motus servat absente viribus externis, relatio inter vim et accelerationem, atque actio et reactio inter corpora. Per exempla sicut lapsus in planis et motus pendulorum illustratur [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28536,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":54,"footnotes":""},"categories":[1262,1250],"tags":[],"class_list":["post-34474","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fundamenta-mechanicae","category-physica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Leges Newtoni - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Leges Newtoni describunt motum corporum et vires quae in ea agunt. Prima lex statuit corpus in quiete aut in motu constanti permanere nisi vires externae adsint. 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