{"id":194838,"date":"2025-02-25T07:26:14","date_gmt":"2025-02-25T07:26:14","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=194838"},"modified":"2025-02-25T07:26:16","modified_gmt":"2025-02-25T07:26:16","slug":"consider-the-physical-quantities-s-v-a-and-t-with-dimensions-s-l-v-lt-a-lt-tt","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/25\/consider-the-physical-quantities-s-v-a-and-t-with-dimensions-s-l-v-lt-a-lt-tt\/","title":{"rendered":"Consider the physical quantities s, v, a, and t with dimensions [s] = L, [v] =LT, [a] =LT, [t]=T"},"content":{"rendered":"\n<p>Consider the physical quantities s, v, a, and t with dimensions [s] = L, [v] =LT, [a] =LT, [t]=T.<br>Determine whether each of the following equations is dimensionally consistent:<br>1) svt+0.5at<br>2) svt+0.5at<br>3) vsin (at\/s)<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine whether the equations are dimensionally consistent, we need to ensure that every term in each equation has the same dimensions.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Equation 1: ( svt + 0.5at )<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( s ): L (length)<\/li>\n\n\n\n<li>( v ): L\/T (length per time)<\/li>\n\n\n\n<li>( t ): T (time)<\/li>\n\n\n\n<li>( a ): L\/T\u00b2 (length per time squared)<\/li>\n<\/ul>\n\n\n\n<p>Checking the dimensions of each term:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( svt ): L \u00d7 (L\/T) \u00d7 T = ( L^2 )<\/li>\n\n\n\n<li>( 0.5at ): 0.5 \u00d7 (L\/T\u00b2) \u00d7 T = ( L\/T )<\/li>\n<\/ul>\n\n\n\n<p>Here, the first term has the dimension ( L^2 ) and the second term has ( L\/T ), which are not the same. Therefore, the equation ( svt + 0.5at ) is dimensionally inconsistent.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Equation 2: ( svt + 0.5at )<\/h3>\n\n\n\n<p>This equation is the same as Equation 1 and therefore also dimensionally inconsistent.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Equation 3: ( v \\sin(at\/s) )<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( v ): L\/T (length per time)<\/li>\n\n\n\n<li>( a ): L\/T\u00b2 (length per time squared)<\/li>\n\n\n\n<li>( s ): L (length)<\/li>\n\n\n\n<li>( t ): T (time)<\/li>\n<\/ul>\n\n\n\n<p>Checking the dimensions of ( \\sin(at\/s) ):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( at\/s ): (L\/T\u00b2) \u00d7 T \/ L = 1\/T\u00b2 \u00d7 T \/ L = 1\/T, which is dimensionless since ( \\sin ) function requires a dimensionless argument.<\/li>\n<\/ul>\n\n\n\n<p>The term ( v \\sin(at\/s) ) itself:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( v \\sin(at\/s) ): (L\/T) \u00d7 dimensionless = L\/T<\/li>\n<\/ul>\n\n\n\n<p>Since the term ( v \\sin(at\/s) ) has consistent dimensions of L\/T, the equation is dimensionally consistent.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion<\/h3>\n\n\n\n<p>Equations 1 and 2 are dimensionally inconsistent because the terms within each equation do not share the same dimensions. Equation 3 is dimensionally consistent as the function ( \\sin ) correctly takes a dimensionless input, and the remaining terms ( v ) maintain consistent dimensions throughout.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Generating an Image<\/h3>\n\n\n\n<p>Let&#8217;s create an image that visualizes this concept of dimensional analysis, featuring symbolic representations of each equation and their dimensional evaluations to visually indicate consistency and inconsistency.<\/p>\n\n\n\n<p>Here is the image prompt for visualization:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A diagram showing three equations. Each equation is paired with symbols representing their dimensional forms. Equation 1 and 2 will show mismatched dimensions, indicated by different shapes or colors, highlighting their inconsistency. Equation 3 will show harmonious symbols, indicating dimensional consistency. The background is educational, with a subtle grid, and each equation is clearly labeled for clarity.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Consider the physical quantities s, v, a, and t with dimensions [s] = L, [v] =LT, [a] =LT, [t]=T.Determine whether each of the following equations is dimensionally consistent:1) svt+0.5at2) svt+0.5at3) vsin (at\/s) The Correct Answer and Explanation is : To determine whether the equations are dimensionally consistent, we need to ensure that every term in [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-194838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=194838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/194838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=194838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=194838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=194838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}