{"id":240778,"date":"2025-07-03T11:48:57","date_gmt":"2025-07-03T11:48:57","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=240778"},"modified":"2025-07-03T11:48:59","modified_gmt":"2025-07-03T11:48:59","slug":"a-horizontal-pipe-carries-a-smoothly-flowing-liquid-of-density-1290-kg-m3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/03\/a-horizontal-pipe-carries-a-smoothly-flowing-liquid-of-density-1290-kg-m3\/","title":{"rendered":"A horizontal pipe carries a smoothly flowing liquid of density 1290 kg\/m^3."},"content":{"rendered":"\n<p>A horizontal pipe carries a smoothly flowing liquid of density 1290 kg\/m^3. At Locations 1 and 2 along the pipe, the diameters are d1 = 5.75 cm and d2 = 2.17 cm, respectively. Side view of pipe with two diameters. At location 1, the diameter of the pipe is d subscript 1. At location 2, the diameter of the pipe is d subscript 2, and is less than d subscript 1. Is the flow speed at Location 2 higher or lower than the flow speed at Location 1? higher cannot tell from the data lower. The flow speed at Location 1 is 1.45 m\/s. What is the pressure difference \u00ce\u201dP between Location 2 and Location 1 (including its sign)? Ignore viscosity. \u00ce\u201dP = Pa. Is the pressure at Location 2 higher or lower than the pressure at Location 1? cannot tell higher lower.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0);color:#0693e3\" class=\"has-inline-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>The flow speed at Location 2 is <strong>higher<\/strong> than at Location 1. Here&#8217;s why:<\/p>\n\n\n\n<p>To understand the relationship between flow speed and pipe diameter, we can use <strong>continuity equation<\/strong> for fluid flow. According to the continuity equation, for an incompressible fluid (like the liquid in the pipe), the flow rate QQQ is constant throughout the pipe. This equation is expressed as:Q=A1v1=A2v2Q = A_1 v_1 = A_2 v_2Q=A1\u200bv1\u200b=A2\u200bv2\u200b<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A1A_1A1\u200b and A2A_2A2\u200b are the cross-sectional areas at Locations 1 and 2, respectively,<\/li>\n\n\n\n<li>v1v_1v1\u200b and v2v_2v2\u200b are the flow speeds at Locations 1 and 2, respectively.<\/li>\n<\/ul>\n\n\n\n<p>The cross-sectional area of a pipe is related to its diameter by the formula:A=\u03c0d24A = \\frac{\\pi d^2}{4}A=4\u03c0d2\u200b<\/p>\n\n\n\n<p>From this, the flow speeds at the two locations are related to the pipe diameters as follows:v1v2=A2A1=d12d22\\frac{v_1}{v_2} = \\frac{A_2}{A_1} = \\frac{d_1^2}{d_2^2}v2\u200bv1\u200b\u200b=A1\u200bA2\u200b\u200b=d22\u200bd12\u200b\u200b<\/p>\n\n\n\n<p>Given that the diameter at Location 2 (d2=2.17d_2 = 2.17d2\u200b=2.17 cm) is smaller than at Location 1 (d1=5.75d_1 = 5.75d1\u200b=5.75 cm), the area at Location 2 is smaller, and the flow speed at Location 2 must be higher in order to maintain the same flow rate.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Pressure Difference Calculation<\/h3>\n\n\n\n<p>Now, to calculate the pressure difference \u0394P\\Delta P\u0394P, we apply <strong>Bernoulli\u2019s equation<\/strong>, which relates the pressure and velocity of a fluid in steady, incompressible flow. Bernoulli\u2019s equation is given by:P1+12\u03c1v12=P2+12\u03c1v22P_1 + \\frac{1}{2} \\rho v_1^2 = P_2 + \\frac{1}{2} \\rho v_2^2P1\u200b+21\u200b\u03c1v12\u200b=P2\u200b+21\u200b\u03c1v22\u200b<\/p>\n\n\n\n<p>where:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>P1P_1P1\u200b and P2P_2P2\u200b are the pressures at Locations 1 and 2,<\/li>\n\n\n\n<li>\u03c1\\rho\u03c1 is the density of the liquid (1290 kg\/m\u00b3),<\/li>\n\n\n\n<li>v1v_1v1\u200b and v2v_2v2\u200b are the velocities at Locations 1 and 2.<\/li>\n<\/ul>\n\n\n\n<p>Rearranging the equation to find the pressure difference:\u0394P=P2\u2212P1=12\u03c1(v12\u2212v22)\\Delta P = P_2 &#8211; P_1 = \\frac{1}{2} \\rho \\left( v_1^2 &#8211; v_2^2 \\right)\u0394P=P2\u200b\u2212P1\u200b=21\u200b\u03c1(v12\u200b\u2212v22\u200b)<\/p>\n\n\n\n<p>Since v2v_2v2\u200b is higher than v1v_1v1\u200b, we can see that v12\u2212v22v_1^2 &#8211; v_2^2v12\u200b\u2212v22\u200b is negative, which means:\u0394P&lt;0\\Delta P &lt; 0\u0394P&lt;0<\/p>\n\n\n\n<p>Thus, <strong>\u0394P\\Delta P\u0394P is negative<\/strong>, meaning the <strong>pressure at Location 2 is lower<\/strong> than at Location 1.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Final Answer:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The flow speed at Location 2 is <strong>higher<\/strong>.<\/li>\n\n\n\n<li>The pressure difference \u0394P\\Delta P\u0394P is <strong>negative<\/strong>, meaning the pressure at Location 2 is <strong>lower<\/strong> than at Location 1.<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner8-36.jpeg\" alt=\"\" class=\"wp-image-240783\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>A horizontal pipe carries a smoothly flowing liquid of density 1290 kg\/m^3. At Locations 1 and 2 along the pipe, the diameters are d1 = 5.75 cm and d2 = 2.17 cm, respectively. Side view of pipe with two diameters. At location 1, the diameter of the pipe is d subscript 1. At location 2, [&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-240778","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240778","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=240778"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/240778\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=240778"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=240778"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=240778"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}