{"id":443,"date":"2021-12-06T11:34:11","date_gmt":"2021-12-06T17:34:11","guid":{"rendered":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/?page_id=443"},"modified":"2022-05-17T14:27:14","modified_gmt":"2022-05-17T20:27:14","slug":"cantidad-de-movimiento-e-impulso","status":"publish","type":"page","link":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/cantidad-de-movimiento-e-impulso\/","title":{"rendered":"Cantidad de movimiento e impulso"},"content":{"rendered":"\n<div class=\"wp-block-getwid-section alignfull alignfull\"><div class=\"wp-block-getwid-section__wrapper\"><div class=\"wp-block-getwid-section__inner-wrapper\"><div class=\"wp-block-getwid-section__background-holder\"><div class=\"wp-block-getwid-section__background\"><\/div><div class=\"wp-block-getwid-section__foreground\"><\/div><\/div><div class=\"wp-block-getwid-section__content\"><div class=\"wp-block-getwid-section__inner-content\">\n<p class=\"has-text-align-center\"> <object data=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2022\/05\/secc_cant-mov.svg\"><\/object> <\/p>\n\n\n\n<h1 class=\"has-text-align-center wp-block-heading\">Cantidad de movimiento e impulso<\/h1>\n<\/div><\/div><\/div><\/div><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">1. Sumario<\/h3>\n\n\n\n<ul class=\"wp-block-list\"><li>Cantidad de movimiento<\/li><li>Impulso<\/li><li>Segunda Ley de Newton y cantidad de movimiento<\/li><li>Colisiones<\/li><li>Conservaci\u00f3n del movimiento<\/li><li>Centro de masa<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">2. Objetivo<\/h3>\n\n\n\n<p>Al finalizar el estudio de este cap\u00edtulo, entre otras habilidades, usted ser\u00e1 capaz de:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Aplicar la ley de conservaci\u00f3n de la cantidad de movimiento para la soluci\u00f3n de problemas de la vida cotidiana.<\/li><\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">3. Introducci\u00f3n<\/h3>\n\n\n\n<p>A continuaci\u00f3n, se presentan una serie de sugerencias para cumplir con las experiencias de aprendizaje, que se proponen en el programa del curso de F\u00edsica I (teor\u00eda). Si bien son aplicables al curso, no son exhaustivas, por lo cual, cada estudiante puede aprovechar otros materiales recomendados en las referencias bibliogr\u00e1ficas.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">4. Gu\u00eda de lectura y recursos adicionales<\/h3>\n\n\n\n<ol type=\"a\" class=\"list-alpha wp-block-list\"><li>Laboratorio virtual de colisiones: <a href=\"https:\/\/www.walter-fendt.de\/html5\/phes\/collision_es.htm\" target=\"_blank\" rel=\"noreferrer noopener\">Choque El\u00e1stico e Inel\u00e1stico (walter-fendt.de)<\/a><\/li><li>Video sobre impulso y momento (Duraci\u00f3n 10min):<a href=\"https:\/\/www.youtube.com\/watch?v=f4F_Y6xsBhY\" target=\"_blank\" rel=\"noreferrer noopener\">(9) Impulso y momento &#8211; YouTube<\/a><\/li><\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">5. Esquemas y ejemplos<\/h3>\n\n\n\n<div class=\"wp-block-getwid-accordion has-icon-left\" data-active-element=\"none\">\n<div class=\"wp-block-getwid-accordion__header-wrapper\"><span class=\"wp-block-getwid-accordion__header\"><a href=\"#\"><span class=\"wp-block-getwid-accordion__header-title\"> Cuadro sin\u00f3ptico<\/span><span class=\"wp-block-getwid-accordion__icon is-active\"><i class=\"fas fa-plus\"><\/i><\/span><span class=\"wp-block-getwid-accordion__icon is-passive\"><i class=\"fas fa-minus\"><\/i><\/span><\/a><\/span><\/div><div class=\"wp-block-getwid-accordion__content-wrapper\"><div class=\"wp-block-getwid-accordion__content\">\n<p>A continuaci\u00f3n, se le muestra un ejemplo de esquema comparativo sobre las tres leyes de Newton.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">COLISI\u00d3N<\/th><th class=\"has-text-align-center\" data-align=\"center\">EL\u00c1STICA<\/th><th class=\"has-text-align-center\" data-align=\"center\">INEL\u00c1STICA<\/th><\/tr><\/thead><tbody><tr><td>Cantidad de movimiento total<\/td><td class=\"has-text-align-center\" data-align=\"center\">Se conserva<\/td><td class=\"has-text-align-center\" data-align=\"center\">Se conserva<\/td><\/tr><tr><td>Energ\u00eda cin\u00e9tica total<\/td><td class=\"has-text-align-center\" data-align=\"center\">Se conserva<\/td><td class=\"has-text-align-center\" data-align=\"center\">No se conserva (es menor despu\u00e9s de la colisi\u00f3n)<\/td><\/tr><\/tbody><\/table><figcaption>Fuente: Elaboraci\u00f3n propia a partir de Young &amp; Freedman, 2018<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-getwid-accordion__header-wrapper\"><span class=\"wp-block-getwid-accordion__header\"><a href=\"#\"><span class=\"wp-block-getwid-accordion__header-title\">Ejercicios<\/span><span class=\"wp-block-getwid-accordion__icon is-active\"><i class=\"fas fa-plus\"><\/i><\/span><span class=\"wp-block-getwid-accordion__icon is-passive\"><i class=\"fas fa-minus\"><\/i><\/span><\/a><\/span><\/div><div class=\"wp-block-getwid-accordion__content-wrapper\"><div class=\"wp-block-getwid-accordion__content\">\n<p>A continuaci\u00f3n, se presentan cuatro ejercicios resueltos, sobre las colisiones.<\/p>\n\n\n\n<p><strong>Ejercicio No. 1. Colisi\u00f3n el\u00e1stica entre dos elementos esf\u00e9ricos.<\/strong><\/p>\n\n\n\n<p>Una bola de 0,440 kg de masa se mueve al este (direcci\u00f3n + x) con una rapidez de 3,30 m\/ s y choca frontalmente con una bola de 0,220 kg en reposo. Si la colisi\u00f3n es perfectamente el\u00e1stica, \u00bfCu\u00e1l ser\u00e1 la rapidez y direcci\u00f3n de cada objeto despu\u00e9s de la colisi\u00f3n?<\/p>\n\n\n\n<p>$$m_{A}= 0,440\\ kg$$<\/p>\n\n\n\n<p>$$ m_{B}= 0,220\\ kg $$<\/p>\n\n\n\n<p>$$ v_{A1}= 3,30\\ m\/s $$ <\/p>\n\n\n\n<p> $$ v_{B1}= 0\\ m\/s $$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$ m_{A} v_{A1}+ m_{B} v_{B1} =  m_{A} v_{A2}+ m_{B} v_{B2} $$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$ \\left(0,440\\ kg\\right)\\left(3,30\\frac{m}{s}\\right)+\\left(0,220\\ kg\\right)\\left(0\\frac{m}{s}\\right)=\\ m_Av_{A2}+m_Bv_{B2} $$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$1,452\\ kg\\ \\bullet m\/s=\\ m_Av_{A2}+m_Bv_{B2}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$V_{B2}-V_{A2\\ }=-(V_{B1}-V_{A1\\ })$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$V_{B2}-V_{A2\\ }=-(0-3,30\\frac{m}{s}\\ )$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_{B2}-v_{A2\\ }=3,30\\ m\/s$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$1,452\\ kg\\ \\bullet m\/s=\\ m_Av_{A2}+m_B\\left[v_{A2\\ }+3,30\\ m\/s\\right]$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$1,452\\ kg\\ \\bullet\\frac{m}{s}-\\ \\left(0,220\\ kg\\right)\\left(3,30\\frac{m}{s}\\right)=v_{A2}\\ \\left[0,220\\ kg+0,440\\ kg\\right]$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_{A2}=\\frac{1,452\\ kg\\ \\bullet\\frac{m}{s}-0,726kg\\ \\bullet\\frac{m}{s}\\ }{\\left[0,220\\ kg+0,440\\ kg\\right]}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_{A2}= 1,1\\ m\/s$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_{B2}=3,30\\frac{m}{s}+\\ v_{A2\\ }= 4,4\\ m\/s$$<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Ejercicio No. 2. Colisiones inel\u00e1sticas en dos dimensiones<\/strong>.<\/p>\n\n\n\n<p>Un disco de hockey en movimiento choca de refil\u00f3n con otro estacionario de la misma masa, como se muestra en la figura adjunta al problema. Si la fricci\u00f3n es insignificante, \u00bfQu\u00e9 rapidez tendr\u00e1n los discos despu\u00e9s del choque? <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"787\" height=\"409\" src=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio2_movimiento-impulso.png\" alt=\"\" class=\"wp-image-481\" srcset=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio2_movimiento-impulso.png 787w, https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio2_movimiento-impulso-300x156.png 300w, https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio2_movimiento-impulso-768x399.png 768w\" sizes=\"auto, (max-width: 787px) 100vw, 787px\" \/><figcaption>Ilustraci\u00f3n 5. Colisi\u00f3n en dos dimensiones<\/figcaption><\/figure><\/div>\n\n\n\n<p>En el eje x<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$mv_{1&#215;0}=mv_{1x}+mv_{2x}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_{10}=v_1\\ cos\\ 50\u00b0+v_2\\ cos\\ 40\u00b0 [1]$$<\/p>\n\n\n\n<p> En el eje y<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$0=\\ mv_{1y}-mv_{2y}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$0 =\\ v_1\\ sin\\ 50\u00b0- v_2\\ sin\\ 40\u00b0 [2]$$<\/p>\n\n\n\n<p>De la ecuaci\u00f3n 2 despejamos v<sub>1<\/sub><\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_1=\\ \\frac{v2\\ sin\\ 40\u00b0}{sin\\ 50\u00b0}$$<\/p>\n\n\n\n<p>Sustituyendo en 1<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$0,95\\ m\/s= \\left[\\frac{v2\\ sin\\ 40\u00b0}{sin\\ 50\u00b0}\\right]cos\\ 50\u00b0+ v2\\ cos\\ 40\u00b0$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_2=\\ \\frac{0,95\\ m\/s}{\\frac{sin\\ 40\u00b0}{sin\\ 50\u00b0}cos\\ 50\u00b0+cos\\ 40\u00b0} 0,73\\ m\/s$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$v_1=\\ \\frac{(0,73\\ m\/s)\\ sin\\ 40\u00b0}{sin\\ 50\u00b0} = 0,61\\ m\/s$$<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>Ejercicio No. 3. <\/strong>&nbsp;<strong>Colisiones y conservaci\u00f3n de la energ\u00eda combinados.<\/strong><\/p>\n\n\n\n<p>Un proyectil se dispara horizontalmente contra la pesa de un p\u00e9ndulo en la cual se incrusta, como se muestra en la ilustraci\u00f3n 6. El p\u00e9ndulo oscila hasta cierta altura <em>h<\/em>, la cual se mide. Se conocen las masas del p\u00e9ndulo y la bala. Utilizando los principios de conservaci\u00f3n de la cantidad de movimiento y de la energ\u00eda, demuestre que la velocidad inicial del proyectil est\u00e1 dada por:<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p class=\"has-text-align-center\">$$V_0=\\ \\left[\\frac{M+m}{m}\\right]\\bullet\\sqrt{2gh}$$ <\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"657\" height=\"375\" src=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio3_movimiento-impulso.png\" alt=\"\" class=\"wp-image-489\" srcset=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio3_movimiento-impulso.png 657w, https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio3_movimiento-impulso-300x171.png 300w\" sizes=\"auto, (max-width: 657px) 100vw, 657px\" \/><figcaption>Ilustraci\u00f3n 6. Sistema de bala incrust\u00e1ndose en un bloque<\/figcaption><\/figure><\/div>\n\n\n\n<p>Se tomar\u00e1 como punto 0, el momento antes de la colisi\u00f3n entre la bala y el bloque, el punto 1 ser\u00e1 el momento justo despu\u00e9s de la colisi\u00f3n y como punto 2, aquel donde el bloque se eleva una altura h.<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$K_0+U_0=K_1+U_1$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$\\frac{1}{2}m{V_0}^2=\\ \\frac{1}{2}(M+m){V_1}^2$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$V_1=\\ \\left[\\frac{m}{m+M}\\ \\right]V_0$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$K_1+U_1=K_2+U_2$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$\\frac{1}{2}\\left(M+m\\right){V_1}^2=(m+M)gh$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$V_1=\\sqrt{2gh}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$\\sqrt{2gh}=\\ \\left[\\frac{m}{m+M}\\ \\right]V_0$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$${\\therefore V}_0=\\ \\left[\\frac{M+m}{m}\\right]\\bullet\\sqrt{2gh}$$<\/p>\n\n\n\n<p><\/p>\n\n\n\n<p> <strong>Ejercicio No. 4. Centro de masas.<\/strong><\/p>\n\n\n\n<p>Se le presenta el siguiente sistema constituido por tres masas, determine la posici\u00f3n del centro de masa de dicho sistema&nbsp;e indique&nbsp; la posici\u00f3n que deber\u00eda tener una cuarta masa que cause que el centro de masa cambie a la posici\u00f3n (0,0) para el caso de una cuarta masa de 5 kg:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"414\" height=\"330\" src=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio4_movimiento-impulso.png\" alt=\"\" class=\"wp-image-493\" srcset=\"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio4_movimiento-impulso.png 414w, https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-content\/uploads\/sites\/6\/2021\/12\/ejercicio4_movimiento-impulso-300x239.png 300w\" sizes=\"auto, (max-width: 414px) 100vw, 414px\" \/><figcaption>Ilustraci\u00f3n 7. Sistema de masas en dos dimensiones.<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"has-text-align-center\">$$X_{cm}=\\ \\frac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}=\\ \\frac{m_2x_2}{m_1+m_2+m_3}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$X_{cm}=\\ \\frac{m_2x_2}{m_1+m_2+m_3}=\\ \\frac{(2\\ kg)(3m)}{1\\ kg+2\\ kg+2\\ kg}= 1,2{m}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$Y_{cm}=\\ \\frac{m_1y_1+m_2y_2+m_3y_3}{m_1+m_2+m_3}=\\ \\frac{m_3y_3}{m_1+m_2+m_3}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$Y_{cm}=\\ \\frac{m_3y_3}{m_1+m_2+m_3}=\\ \\frac{(2\\ kg)(4m)}{1\\ kg+2\\ kg+2\\ kg}= 1,6{m}$$<\/p>\n\n\n\n<p>Caso donde agregamos una cuarta masa<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$X_{cm}=\\ \\frac{\\left(2\\ kg\\right)\\left(3m\\right)+(5\\ kg)(x_4)}{1\\ kg+2\\ kg+2\\ kg}=0$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$\\left(2\\ kg\\right)\\left(3m\\right)=-(5\\ kg)(x_4)$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$x_4=\\ -\\frac{\\left(2\\ kg\\right)\\left(3m\\right)}{5\\ kg}= -1,2{m}$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$Y_{cm}=\\ \\frac{\\left(2\\ kg\\right)\\left(4m\\right)+(5\\ kg)(y_4)}{1\\ kg+2\\ kg+2\\ kg}=0$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$\\left(2\\ kg\\right)\\left(4m\\right)=-(5\\ kg)(y_4)$$<\/p>\n\n\n\n<p class=\"has-text-align-center\">$$y_4=\\ -\\frac{\\left(2\\ kg\\right)\\left(4m\\right)}{5\\ kg}= -1,6{m}$$<\/p>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\">6. Fuentes<\/h3>\n\n\n\n<p class=\"sang-frances\">Young, H. &amp; Freedman, R. (2018). <em>F\u00edsica universitaria con f\u00edsica moderna 1<\/em>. M\u00e9xico: Pearson<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. Sumario Cantidad de movimiento Impulso Segunda Ley de Newton y cantidad de movimiento Colisiones Conservaci\u00f3n del movimiento Centro de masa 2. Objetivo Al finalizar [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-443","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/pages\/443","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/comments?post=443"}],"version-history":[{"count":42,"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/pages\/443\/revisions"}],"predecessor-version":[{"id":706,"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/pages\/443\/revisions\/706"}],"wp:attachment":[{"href":"https:\/\/multimedia.uned.ac.cr\/pem\/wp\/fisica-general-1\/wp-json\/wp\/v2\/media?parent=443"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}