{"id":2365,"date":"2020-03-11T05:53:31","date_gmt":"2020-03-11T05:53:31","guid":{"rendered":"http:\/\/lash.utrng.edu.mx\/?p=2365"},"modified":"2020-03-11T13:00:45","modified_gmt":"2020-03-11T13:00:45","slug":"sobre-la-transformada-de-laplace","status":"publish","type":"post","link":"https:\/\/lash.utrng.edu.mx\/?p=2365","title":{"rendered":"Sobre la Transformada de Laplace"},"content":{"rendered":"\n<p><strong>Sistema Masa-Resorte<\/strong><\/p>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><a href=\"blob:http:\/\/lash.utrng.edu.mx\/7a29742a-97c2-4004-a5a7-d073a740cbba\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01-1024x576.png\" alt=\"\" class=\"wp-image-2366\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01-1024x576.png 1024w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01-300x169.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01-768x432.png 768w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01-1536x864.png 1536w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte01.png 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure><\/div>\n\n\n\n<ul class=\"wp-block-list\"><li>m: masa<\/li><li>y: posici\u00f3n del objeto<\/li><li>k: constante del Resorte<\/li><\/ul>\n\n\n\n<!--more-->\n\n\n\n<p>En la posici\u00f3n de equilibrio tenemos que <strong>w=Fr<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-style-default\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02-1024x576.png\" alt=\"\" class=\"wp-image-2367\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02-1024x576.png 1024w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02-300x169.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02-768x432.png 768w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02-1536x864.png 1536w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte02.png 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>Considerando la Ley de Hooke:<\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size\"><strong>Fr = -kx<\/strong><\/p>\n\n\n\n<p>entonces:<\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size\"><strong>w = Fr<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size\"><strong>w = kd<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-large is-style-default\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"576\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03-1024x576.png\" alt=\"\" class=\"wp-image-2368\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03-1024x576.png 1024w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03-300x169.png 300w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03-768x432.png 768w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03-1536x864.png 1536w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/resorte03.png 1920w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p>El resorte se estira una distancia<strong> d + y<\/strong>, desde el reposo, si <strong>y &gt; -d<\/strong>, entonces:<\/p>\n\n\n\n<p class=\"has-text-align-center has-medium-font-size\"><strong>Fr = k(y + d)<\/strong><\/p>\n\n\n\n<p>Ahora se aplica la 2da Ley de Newton: <strong>Suma de Fzas = masa X aceleraci\u00f3n<\/strong>, en este caso:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li><strong>w &#8211; Fr = ma<\/strong><\/li><li><strong>kd &#8211; k(y + d) = ma<\/strong><\/li><li><strong>kd &#8211; ky &#8211; kd = ma<\/strong><\/li><li><strong>0 = ma + ky<\/strong><\/li><li><strong>ma + ky  = 0<\/strong>  **a ser\u00e1 la segunda derivada de y<\/li><li>my\u00bb + ky = 0  **obtenemos una ec. de 2do orden<\/li><\/ul>\n\n\n\n<p>Ejemplo:<\/p>\n\n\n\n<p class=\"has-text-color has-text-align-center has-medium-font-size has-vivid-red-color\"><strong> my\u00bb + ky = 0 <\/strong><\/p>\n\n\n\n<p>Se cuelga un objeto de <strong>1.5kg<\/strong> de un resorte, y se observa un estiramiento de 61cm, hasta el punto de equilibrio.<\/p>\n\n\n\n<ol class=\"wp-block-list\"><li>\u00bfcu\u00e1l es la constante del resorte?<\/li><li>si se estira el objeto hasta 40cm, abajo del punto de equilibrio y se suelta, \u00a1cu\u00e1l es la posici\u00f3n del objeto, 10 segundos despu\u00e9s?<\/li><li>\u00bfcu\u00e1l es su velocidad en ese momento?<\/li><\/ol>\n\n\n\n<p>* El equilibrio se alcanza cuando <strong>w = Fr<\/strong>, entonces hay que recordar que el peso(w) = a la masa por la gravedad(mg), mientras que la Fuerza del resorte, se calcula con la ley de Hooke  = constante del resorte(k), por la Distancia del estiramiento(y)<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>w = Fr<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>mg = ky<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\">1.5kg(9.8m\/s<sup>2<\/sup>) = k(0.61m)    **61cm a metros<\/p>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large is-resized\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-21.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-21.png\" alt=\"\" class=\"wp-image-2375\" width=\"388\" height=\"96\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-21.png 531w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-21-300x74.png 300w\" sizes=\"auto, (max-width: 388px) 100vw, 388px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"has-medium-font-size\"><strong>1)<\/strong> La fuerza del resorte ser\u00e1 de <strong>24 Newton\/metro<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"alignleft size-large is-resized\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-23.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-23.png\" alt=\"\" class=\"wp-image-2377\" width=\"590\" height=\"34\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-23.png 734w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-23-300x17.png 300w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/a><\/figure><\/div>\n\n\n\n<p><\/p>\n\n\n\n<p><\/p>\n\n\n\n<p><strong>1.5y &#8216; &#8216; + 24y =0<\/strong>     **esta ecuaci\u00f3n es la que se va a resolver con la transformada de Laplace<\/p>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-24.png\"><img loading=\"lazy\" decoding=\"async\" width=\"323\" height=\"257\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-24.png\" alt=\"\" class=\"wp-image-2378\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-24.png 323w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-24-300x239.png 300w\" sizes=\"auto, (max-width: 323px) 100vw, 323px\" \/><\/a><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large is-resized\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-25.png\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-25.png\" alt=\"\" class=\"wp-image-2380\" width=\"531\" height=\"680\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-25.png 504w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-25-234x300.png 234w\" sizes=\"auto, (max-width: 531px) 100vw, 531px\" \/><\/a><\/figure><\/div>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-26.png\"><img loading=\"lazy\" decoding=\"async\" width=\"337\" height=\"204\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-26.png\" alt=\"\" class=\"wp-image-2382\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-26.png 337w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-26-300x182.png 300w\" sizes=\"auto, (max-width: 337px) 100vw, 337px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"has-medium-font-size\"><strong>2) <\/strong>27 cent\u00edmetros arriba del punto de equilibrio<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>Para el inciso 3), hay que derivar la expresi\u00f3n <strong>y=0.4cos(4t)<\/strong>:<\/p>\n\n\n\n<div class=\"wp-block-image is-style-default\"><figure class=\"aligncenter size-large\"><a href=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-27.png\"><img loading=\"lazy\" decoding=\"async\" width=\"325\" height=\"347\" src=\"http:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-27.png\" alt=\"\" class=\"wp-image-2385\" srcset=\"https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-27.png 325w, https:\/\/lash.utrng.edu.mx\/wp-content\/uploads\/2020\/03\/image-27-281x300.png 281w\" sizes=\"auto, (max-width: 325px) 100vw, 325px\" \/><\/a><\/figure><\/div>\n\n\n\n<p class=\"has-medium-font-size\"><strong>3)<\/strong> El objeto se mueve hacia arriba a una velocidad de <strong>1.18m\/s<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sistema Masa-Resorte m: masa y: posici\u00f3n del objeto k: constante del Resorte<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[53],"tags":[],"class_list":["post-2365","post","type-post","status-publish","format-standard","hentry","category-tercerparcialmatepaing"],"_links":{"self":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/2365","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2365"}],"version-history":[{"count":8,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/2365\/revisions"}],"predecessor-version":[{"id":2388,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=\/wp\/v2\/posts\/2365\/revisions\/2388"}],"wp:attachment":[{"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2365"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2365"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lash.utrng.edu.mx\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2365"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}