{"id":7336,"date":"2012-01-18T23:39:16","date_gmt":"2012-01-18T23:39:16","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=7336"},"modified":"2022-01-30T16:03:07","modified_gmt":"2022-01-30T16:03:07","slug":"a-escala-de-richter","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=7336","title":{"rendered":"A escala de Richter"},"content":{"rendered":"<p><ul id='GTTabs_ul_7336' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_7336' class='GTTabs_curr'><a  id=\"7336_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_7336' ><a  id=\"7336_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_7336'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<div id=\"attachment_7337\" style=\"width: 250px\" class=\"wp-caption alignright\"><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-7337\" data-attachment-id=\"7337\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=7337\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph.jpg\" data-orig-size=\"350,269\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Sism\u00f3grafo de Richter\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;Sism\u00f3grafo de Richter&lt;\/p&gt;\n\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph.jpg\" class=\"   wp-image-7337\" title=\"Sism\u00f3grafo de Richter\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph.jpg\" alt=\"\" width=\"240\" height=\"184\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph.jpg 350w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph-300x230.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/richterSeismograph-150x115.jpg 150w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a><p id=\"caption-attachment-7337\" class=\"wp-caption-text\">Charles Richter e o seu sism\u00f3grafo<\/p><\/div>\n<p>A <a href=\"http:\/\/pt.wikipedia.org\/wiki\/Escala_de_Richter\" target=\"_blank\" rel=\"noopener\">escala de Richter<\/a> permite converter a amplitude m\u00e1xima dos registos feitos por um sism\u00f3grafo num n\u00famero que nos permite estabelecer uma medida para a magnitude $M$ de um sismo.<\/p>\n<p>Naquela escala, <strong>um sismo de n\u00edvel zero<\/strong> \u00e9 aquele em que a amplitude\u00a0m\u00e1xima dos registos dos sism\u00f3grafos situados a $100$ km do epicentro \u00e9 $0,001$ mil\u00edmetros.<\/p>\n<p>A magnitude $M$ de um sismo em que o sism\u00f3grafo situado a $100$ km do epicentro regista amplitudes m\u00e1ximas de $x$ mil\u00edmetros \u00e9 dada, em fun\u00e7\u00e3o de $x$,\u00a0por:<\/p>\n<p>$$M(x)=\\log \\left( \\frac{x}{{{10}^{-3}}} \\right)$$<\/p>\n<ol>\n<li>\u00a0Mostre que $M(x)=3+\\log x$.<\/li>\n<li>Qual \u00e9, na escala de Richter, a magnitude de um terramoto que provoca um registo de $10$ mm de amplitude?<\/li>\n<li>Qual a amplitude m\u00e1xima do gr\u00e1fico desenhado por um sism\u00f3grafo situado a $100$ km do epicentro, se a magnitude do sismo for $6$?<\/li>\n<li>Se a amplitude m\u00e1xima do registo num sismo A for 10 vezes a amplitude m\u00e1xima registada num sismo B, o que podemos afirmar acerca das magnitudes destes sismos?<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_7336' onClick='GTTabs_show(1,7336)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_7336'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote><p>A magnitude $M$ de um sismo em que o sism\u00f3grafo situado a $100$ km do epicentro regista amplitudes m\u00e1ximas de $x$ mil\u00edmetros \u00e9 dada, em fun\u00e7\u00e3o de $x$,\u00a0por:<\/p>\n<p>$$M(x)=\\log \\left( \\frac{x}{{{10}^{-3}}} \\right)$$<\/p><\/blockquote>\n<p>\u00ad<\/p>\n<ol>\n<li>De facto, tem-se:<br \/>\n$$\\begin{array}{*{35}{l}}<br \/>\nM(x) &amp; = &amp; \\log \\left( \\frac{x}{{{10}^{-3}}} \\right)\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\log x-\\log {{10}^{-3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\log x-(-3)\u00a0 \\\\<br \/>\n{} &amp; = &amp; 3+\\log x\u00a0 \\\\<br \/>\n\\end{array}$$<br \/>\n\u00ad<\/li>\n<li>Ora, $M(10)=3+\\log 10=3+1=4$.<br \/>\nPortanto, na escala de Richter, \u00e9 de magnitude 4 um terramoto que provoca um registo de $10$ mm de amplitude.<br \/>\n\u00ad<\/li>\n<li>Ora, $$\\begin{array}{*{35}{l}}<br \/>\nM(x)=6 &amp; \\Leftrightarrow\u00a0 &amp; 3+\\log x=6\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\log x=3\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x={{10}^{3}}\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; x=1000\u00a0 \\\\<br \/>\n\\end{array}$$<br \/>\nSe a magnitude do sismo for $6$, \u00e9 de $1000$ mm a amplitude m\u00e1xima do gr\u00e1fico desenhado por um sism\u00f3grafo situado a $100$ km do epicentro.<br \/>\n\u00ad<\/li>\n<li>Tabelando os dados, temos:<br \/>\n<table class=\" aligncenter\" style=\"width: 80%;\" border=\"0\" align=\"center\">\n<tbody>\n<tr>\n<td style=\"border: #00008b 1px solid;\"><\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sismo B<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">Sismo A<\/td>\n<\/tr>\n<tr>\n<td style=\"border: #00008b 1px solid;\">Amplitude m\u00e1xima<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${{x}_{B}}$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${{x}_{A}}=10{{x}_{B}}$<\/td>\n<\/tr>\n<tr>\n<td style=\"border: #00008b 1px solid;\">Magnitude<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${{M}_{B}}=M({{x}_{B}})$<\/td>\n<td style=\"text-align: center; border: #00008b 1px solid;\">${{M}_{A}}=M({{x}_{A}})=M(10{{x}_{B}})$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Ora, ${{M}_{B}}=3+\\log {{x}_{B}}$.<\/p>\n<p>Por outro lado, tem-se: $${{M}_{A}}=3+\\log \\left( 10{{x}_{B}} \\right)=3+\\log 10+\\log {{x}_{B}}=3+1+\\log {{x}_{B}}=4+\\log {{x}_{B}}={{M}_{B}}+1$$<br \/>\nPortanto, a magnitude do sismo A tem mais uma unidade do que a do sismo B.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_7336' onClick='GTTabs_show(0,7336)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A escala de Richter permite converter a amplitude m\u00e1xima dos registos feitos por um sism\u00f3grafo num n\u00famero que nos permite estabelecer uma medida para a magnitude $M$ de um sismo. Naquela&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":21119,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,267],"tags":[427,275],"series":[],"class_list":["post-7336","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-exponenciais-e-logaritmicas","tag-12-o-ano","tag-funcao-logaritmica"],"views":2853,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2012\/01\/12V2Pag202-6_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7336","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=7336"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/7336\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/21119"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=7336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=7336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=7336"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=7336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}