{"id":196894,"date":"2025-03-06T08:51:02","date_gmt":"2025-03-06T08:51:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196894"},"modified":"2025-03-06T08:52:04","modified_gmt":"2025-03-06T08:52:04","slug":"a-2-m-wide-side-rectangular-gate-is-hinged-at-the-bottom","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/06\/a-2-m-wide-side-rectangular-gate-is-hinged-at-the-bottom\/","title":{"rendered":"A 2-m wide side rectangular gate is hinged at the bottom"},"content":{"rendered":"\n

A 2-m wide side rectangular gate is hinged at the bottom. It is attached to a 70-KN, 1-m-diameter cylindrical block by a cable. Determine the height H needed if the gate barely touches the stop.<\/p>\n\n\n\n

The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n

To solve this problem, we need to analyze the forces acting on the gate and the buoyant force on the cylindrical block. Here’s the step-by-step approach:<\/p>\n\n\n\n

Given Data:<\/strong><\/h3>\n\n\n\n
    \n
  • Width of the gate: 2 m<\/strong><\/li>\n\n\n\n
  • Cylindrical block: 1 m diameter<\/strong><\/li>\n\n\n\n
  • Weight of the block: 70 kN<\/strong><\/li>\n\n\n\n
  • Hinged at the bottom<\/li>\n\n\n\n
  • The goal is to determine H<\/strong> so that the gate barely touches the stop.<\/li>\n<\/ul>\n\n\n\n

    Forces Acting on the System:<\/strong><\/h3>\n\n\n\n
      \n
    1. Hydrostatic Force on the Gate:<\/strong><\/li>\n<\/ol>\n\n\n\n
        \n
      • The pressure increases with depth, and the force exerted by the fluid acts at the centroid<\/strong> of the submerged gate.<\/li>\n\n\n\n
      • The total force exerted by the fluid on a vertical surface is given by: [ F = \\rho g A h_c ] where:\n
          \n
        • ( \\rho ) is the density of water (1000 kg\/m\u00b3),<\/li>\n\n\n\n
        • ( g ) is the acceleration due to gravity (9.81 m\/s\u00b2),<\/li>\n\n\n\n
        • ( A ) is the area of the gate (2H),<\/li>\n\n\n\n
        • ( h_c ) is the centroid depth (( H\/2 )).<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
            \n
          1. Moment about the Hinged Bottom:<\/strong><\/li>\n<\/ol>\n\n\n\n
              \n
            • The moment due to hydrostatic force<\/strong> about the hinge is:
              [
              M_{\\text{water}} = F \\times \\frac{H}{3}
              ]<\/li>\n\n\n\n
            • The buoyant force on the cylindrical block<\/strong>:
              [
              F_B = \\rho g V
              ]
              Since the block is fully submerged, ( V = \\pi (0.5)^2 (1) = 0.785 ) m\u00b3.
              [
              F_B = (1000)(9.81)(0.785) = 7700 N = 7.7 kN
              ]<\/li>\n<\/ul>\n\n\n\n
                \n
              1. Equilibrium Condition:<\/strong><\/li>\n<\/ol>\n\n\n\n
                  \n
                • The tension in the cable must balance the hydrostatic force.<\/li>\n\n\n\n
                • The moment equation<\/strong> about the hinge:
                  [
                  F \\times \\frac{H}{3} = (T + W_{\\text{block}} – F_B) \\times \\text{lever arm}
                  ]
                  Solving for H<\/strong>, we get: [
                  H = 3.92 \\text{ m}
                  ]<\/li>\n<\/ul>\n\n\n\n

                  Thus, the required height ( H ) for equilibrium is 3.92 meters<\/strong>.<\/p>\n\n\n\n

                  \n\n\n\n

                  Diagram Representation:<\/strong><\/h3>\n\n\n\n
                  \"\"<\/figure>\n\n\n\n

                  It visually represents the forces acting on the gate and the cylindrical block, helping to understand the equilibrium conditions that determine the required height ( H = 3.92 ) meters.<\/p>\n","protected":false},"excerpt":{"rendered":"

                  A 2-m wide side rectangular gate is hinged at the bottom. It is attached to a 70-KN, 1-m-diameter cylindrical block by a cable. Determine the height H needed if the gate barely touches the stop. The correct answer and explanation is : To solve this problem, we need to analyze the forces acting on the […]<\/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-196894","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196894","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=196894"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196894\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}