{"id":29197,"date":"2021-03-25T13:00:39","date_gmt":"2021-03-25T13:00:39","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=29197"},"modified":"2024-09-22T16:12:21","modified_gmt":"2024-09-22T16:12:21","slug":"pascals-principle","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/pascals-principle\/","title":{"rendered":"Pascal&#8217;s Principle"},"content":{"rendered":"<p><center><\/p>\n<h1>Pascal&#8217;s Principle<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nIn this class, Pascal&#8217;s Principle will be explained, which states that pressure applied to an incompressible fluid is distributed uniformly. We will see its consequences, such as the constant level in containers and the relationship between displaced volumes. Additionally, we will show applications in hydraulic machines and solve practical examples like leveling land, differences in liquid levels of various densities, and the operation of a hydraulic lift.<br \/>\n<\/em><br \/>\n<strong>Learning Objectives:<\/strong><\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> Pascal&#8217;s Principle and its application in incompressible fluids.<\/li>\n<li><strong>Analyze<\/strong> the consequences of the principle, such as uniform pressure distribution and the behavior of liquid levels.<\/li>\n<li><strong>Apply<\/strong> Pascal&#8217;s Principle in practical situations, such as the operation of hydraulic machines.<\/li>\n<li><strong>Solve<\/strong> practical problems regarding land leveling and differences in liquid levels with different densities.<\/li>\n<li><strong>Explain<\/strong> the use of Torricelli&#8217;s Barometer to measure atmospheric pressure and the functioning of hydraulic lifts.<\/li>\n<\/ol>\n<p><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\">Transmission of forces in liquids<\/a><br \/>\n<a href=\"#2\">Consequences of Pascal&#8217;s Principle<\/a><br \/>\n<a href=\"#3\">Pascal&#8217;s Principle Applied in Hydraulic Machines<\/a><br \/>\n<a href=\"#4\">Application Examples<\/a>\n<\/p>\n<p><\/center><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/9IAFRXx0Cns\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Transmission of forces in liquids<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=84s\" target=\"_blank\" rel=\"noopener\"><strong>Pascal&#8217;s Principle applies when dealing with incompressible liquids<\/strong><\/a>. The particles in such substances exert forces on each other that do not preserve position but rather the average relative distance between them. If this did not occur, the particles would change their average relative distance in some regions, altering the total volume, indicating that we would be dealing with a compressible liquid. Most liquids are, in fact, highly incompressible, and therefore, when a force is applied in a given region, it is transmitted entirely throughout the fluid.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Consequences of Pascal&#8217;s Principle<\/h2>\n<h3>The level is the same everywhere<\/h3>\n<p style=\"text-align: justify;\">A clear example of this is shown in the following figure. If this is filled with a liquid, the level will rise equally in all available spaces, regardless of the shape of the conduits.<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-SD1TKOVRBrw\/YF0upu1TqEI\/AAAAAAAAEtc\/3r34Qw4kXgwJpbXqylLBaKJX45tFH1S9gCLcBGAsYHQ\/s763\/principio%2Bde%2Bpascal%2B-%2Bnivel%2Bde%2Bliquido.png\" alt=\"Pascal's Principle\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-SD1TKOVRBrw\/YF0upu1TqEI\/AAAAAAAAEtc\/3r34Qw4kXgwJpbXqylLBaKJX45tFH1S9gCLcBGAsYHQ\/s763\/principio%2Bde%2Bpascal%2B-%2Bnivel%2Bde%2Bliquido.png\" alt=\"Pascal's Principle\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">This is because, as we have already seen, pressure is a function of height.<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P = \\rho g h<\/span>\n<p style=\"text-align: justify;\">If there were a difference in height between different parts of the liquid, then there would necessarily be a difference in pressure between those parts.<\/p>\n<h3>Displaced Volume = Ceded Volume<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=323s\" target=\"_blank\" rel=\"noopener\"><strong>If pressure is applied at some point in the liquid,<\/strong><\/a> it will displace while maintaining the same level in all other parts. The ceded volume <span class=\"katex-eq\" data-katex-display=\"false\">v_1<\/span> would equal the displaced volume <span class=\"katex-eq\" data-katex-display=\"false\">v_2+v_3.<\/span>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-nDStNasYWZQ\/YF0upq_20xI\/AAAAAAAAEtg\/bPPlgi4vpug4i70kHoMpkPsvtRtfbQrVACLcBGAsYHQ\/s767\/principio%2Bde%2Bpascalnivel%2Bde%2Bliquido2.png\" alt=\"Pascal's principle, ceded volume equals displaced volume\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-nDStNasYWZQ\/YF0upq_20xI\/AAAAAAAAEtg\/bPPlgi4vpug4i70kHoMpkPsvtRtfbQrVACLcBGAsYHQ\/s767\/principio%2Bde%2Bpascalnivel%2Bde%2Bliquido2.png\" alt=\"Pascal's principle, ceded volume equals displaced volume\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><br \/>\n<a name=\"3\"><\/a><\/p>\n<h2>Pascal&#8217;s Principle Applied in Hydraulic Machines<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=426s\" target=\"_blank\" rel=\"noopener\"><strong>Pascal&#8217;s Principle is used in the construction of hydraulic presses.<\/strong><\/a> As pressure is distributed uniformly throughout all parts of a liquid, the following expression holds:<\/p>\n<p><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-hpJeg_iRsgw\/YF0uplM2GnI\/AAAAAAAAEtY\/tBztNE2eUJQe2uHlrzgZlh_y0QPLGyCnwCLcBGAsYHQ\/s805\/prensahidr%25C3%25A1ulica.PNG\" alt=\"Pascal's Principle, Hydraulic Lift\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"200\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-hpJeg_iRsgw\/YF0uplM2GnI\/AAAAAAAAEtY\/tBztNE2eUJQe2uHlrzgZlh_y0QPLGyCnwCLcBGAsYHQ\/s805\/prensahidr%25C3%25A1ulica.PNG\" alt=\"Pascal's Principle, Hydraulic Lift\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"200\" \/><\/noscript><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P_2 = P_1<\/span>\n<p style=\"text-align: justify;\">But since pressure can be expressed as force per unit area, <span class=\"katex-eq\" data-katex-display=\"false\">P=F\/A<\/span>, we have:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{F_2}{A_2} = \\frac{F_1}{A_1}<\/span>\n<p style=\"text-align: justify;\">This means that if we apply a force <span class=\"katex-eq\" data-katex-display=\"false\">F_1<\/span> on piston 1, a force <span class=\"katex-eq\" data-katex-display=\"false\">F_2<\/span> will result on piston 2 as:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F_2 = \\frac{A_2}{A_1} \\cdot F_1<\/span>\n<p style=\"text-align: justify;\">The factor <span class=\"katex-eq\" data-katex-display=\"false\">A_2\/A_1<\/span> is what we will call the \u00abAmplification or reduction coefficient of the force.\u00bb It will amplify if it is greater than 1, and reduce if it is between 0 and 1.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Application Examples<\/h2>\n<ol>\n<li style=\"text-align: justify;\">Two people want to level a sloping terrain. To do this, they take a hose and fill it with water, then place its ends at a horizontal distance of 275[cm]. The water level at one point is 110[cm] and at the other is 175[cm].<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-qmCq1t6szPU\/YF0uopFj2HI\/AAAAAAAAEtM\/Bo__H3vPJ9YZCms__pgwk-26VxjBbiYmgCLcBGAsYHQ\/s625\/nivel%2Bde%2Bagua.PNG\" alt=\"Pascal's Principle, Water Level\" class=\"aligncenter size-full lazyload\" width=\"625\" height=\"313\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-qmCq1t6szPU\/YF0uopFj2HI\/AAAAAAAAEtM\/Bo__H3vPJ9YZCms__pgwk-26VxjBbiYmgCLcBGAsYHQ\/s625\/nivel%2Bde%2Bagua.PNG\" alt=\"Pascal's Principle, Water Level\" class=\"aligncenter size-full lazyload\" width=\"625\" height=\"313\" \/><\/noscript>a) What is the difference in terrain level?\n<p>b) What is the angle of inclination of the terrain?<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=691s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/li>\n<li style=\"text-align: justify;\">A U-shaped hose has water inside, with a density of <span class=\"katex-eq\" data-katex-display=\"false\">1000 [kg\/m^3].<\/span> If oil is poured into one end of the hose, filling it to a height of <span class=\"katex-eq\" data-katex-display=\"false\">20[cm]<\/span>, with a density of <span class=\"katex-eq\" data-katex-display=\"false\">800[kg\/m^3]<\/span>, what level difference will be produced between the two ends of the hose?<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-EYBgWw2b6jY\/YF0uoq4c5AI\/AAAAAAAAEtU\/LCAin5GVeSoRDCIgn3widW9kaTwfJwFRACLcBGAsYHQ\/s596\/nivel-agua-aceite.PNG\" alt=\"Pascal's Principle - Mixtures\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-EYBgWw2b6jY\/YF0uoq4c5AI\/AAAAAAAAEtU\/LCAin5GVeSoRDCIgn3widW9kaTwfJwFRACLcBGAsYHQ\/s596\/nivel-agua-aceite.PNG\" alt=\"Pascal's Principle - Mixtures\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300\" \/><\/noscript><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=867s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/li>\n<li style=\"text-align: justify;\"><strong>Torricelli&#8217;s Barometer:<\/strong> Torricelli designed the following instrument to measure atmospheric pressure:<img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-hB2h_82Yg6A\/YF0uol0V-OI\/AAAAAAAAEtQ\/1w25pez7rj4LE3-tanK4NMo5NBy_0xJQgCLcBGAsYHQ\/s572\/Principio%2Bde%2Bpascal%2B-%2Btorricelli.PNG\" principle=\"\" de=\"\" pascal=\"\" bar=\"\" metro=\"\" torricelli=\"\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300 alt=\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-hB2h_82Yg6A\/YF0uol0V-OI\/AAAAAAAAEtQ\/1w25pez7rj4LE3-tanK4NMo5NBy_0xJQgCLcBGAsYHQ\/s572\/Principio%2Bde%2Bpascal%2B-%2Btorricelli.PNG\" principle=\"\" de=\"\" pascal=\"\" bar=\"\" metro=\"\" torricelli=\"\" class=\"aligncenter size-full lazyload\" width=\"400\" height=\"300 alt=\" \/><\/noscript>a) If the density of mercury is <span class=\"katex-eq\" data-katex-display=\"false\">13.534,0[kg\/m3]<\/span> what will be the height of the mercury column due to atmospheric pressure? Consider <span class=\"katex-eq\" data-katex-display=\"false\">P_{atm}=1,0[atm]=101.325,0[Pa]<\/span>?\n<p>b) If instead of mercury, water were used, how tall would the water column need to be to measure the same pressure?<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=1049s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/li>\n<li style=\"text-align: justify;\">A hydraulic lift has two pistons, one with a 2[foot] and the other with a 30[foot] radius. a) What weight must be placed on the smaller piston if a one-ton block is to be lifted on the larger piston? b) If the larger piston is to lift a concrete block by a height of <span class=\"katex-eq\" data-katex-display=\"false\">50[cm]<\/span>, how far must the smaller piston move?\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=9IAFRXx0Cns&amp;t=1403s\" target=\"_blank\" rel=\"noopener\"><strong>SOLUTION<\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Pascal&#8217;s Principle Summary: In this class, Pascal&#8217;s Principle will be explained, which states that pressure applied to an incompressible fluid is distributed uniformly. We will see its consequences, such as the constant level in containers and the relationship between displaced volumes. Additionally, we will show applications in hydraulic machines and solve practical examples like leveling [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":29196,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":21,"footnotes":""},"categories":[889,635],"tags":[],"class_list":["post-29197","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fluid-mechanics","category-physics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Pascal&#039;s Principle - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Pascal&#039;s Principle states that when pressure is applied to an incompressible fluid, it is transmitted uniformly in all directions. 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