{"id":196151,"date":"2025-03-03T05:13:05","date_gmt":"2025-03-03T05:13:05","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=196151"},"modified":"2025-03-03T05:13:14","modified_gmt":"2025-03-03T05:13:14","slug":"locate-the-centroid-y-of-the-area","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/03\/locate-the-centroid-y-of-the-area\/","title":{"rendered":"Locate the centroid y of the area"},"content":{"rendered":"\n

Locate the centroid y of the area.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

To determine the centroid (y\u0304) of a given area, we need to calculate the first moment of area about the x-axis (Q_x) and divide it by the total area (A). The formula is:<\/p>\n\n\n\n

[ y\u0304 = \\frac{Q_x}{A} ]\ue206<\/p>\n\n\n\n

Steps to Calculate the Centroid y\u0304:<\/strong><\/p>\n\n\n\n

    \n
  1. Divide the Area into Simple Shapes:<\/strong> Break down the complex shape into simpler, known shapes (rectangles, triangles, circles, etc.).\ue206<\/li>\n\n\n\n
  2. Calculate the Area (A) of Each Shape:<\/strong> Use standard geometric formulas to find the area of each simple shape.\ue206<\/li>\n\n\n\n
  3. Determine the Centroid of Each Shape:<\/strong> Find the centroid (center of gravity) of each simple shape. For standard shapes, these are located at specific points:\ue206<\/li>\n<\/ol>\n\n\n\n
      \n
    • Rectangle:<\/strong> Centroid at the intersection of the diagonals.\ue206<\/li>\n\n\n\n
    • Triangle:<\/strong> Centroid at the intersection of the medians.\ue206<\/li>\n\n\n\n
    • Circle:<\/strong> Centroid at the center.\ue206<\/li>\n<\/ul>\n\n\n\n
        \n
      1. Calculate the First Moment of Area (Q_x):<\/strong> For each shape, compute Q_x using:\ue206
        [ Q_x = A \\times y_{\\text{centroid}} ]
        where ( y_{\\text{centroid}} ) is the distance from the x-axis to the centroid of the shape.<\/li>\n\n\n\n
      2. Compute the Total Area (A):<\/strong> Sum the areas of all simple shapes.\ue206<\/li>\n\n\n\n
      3. Calculate the Overall Centroid (y\u0304):<\/strong> Use the formula:\ue206
        [ y\u0304 = \\frac{\\sum Q_{x}}{A} ]
        where the summation is over all simple shapes.<\/li>\n<\/ol>\n\n\n\n

        Example:<\/strong><\/p>\n\n\n\n

        Consider an L-shaped area composed of two rectangles:<\/p>\n\n\n\n

          \n
        • Rectangle 1:<\/strong> Width = 4 units, Height = 10 units.\ue206<\/li>\n\n\n\n
        • Rectangle 2:<\/strong> Width = 6 units, Height = 4 units.\ue206<\/li>\n<\/ul>\n\n\n\n

          Calculations:<\/strong><\/p>\n\n\n\n

            \n
          1. Areas:<\/strong><\/li>\n<\/ol>\n\n\n\n
              \n
            • A\u2081 = 4 \u00d7 10 = 40 units\u00b2.\ue206<\/li>\n\n\n\n
            • A\u2082 = 6 \u00d7 4 = 24 units\u00b2.\ue206<\/li>\n\n\n\n
            • Total Area A = A\u2081 + A\u2082 = 40 + 24 = 64 units\u00b2.\ue206<\/li>\n<\/ul>\n\n\n\n
                \n
              1. Centroids:<\/strong><\/li>\n<\/ol>\n\n\n\n
                  \n
                • For Rectangle 1: y\u2081 = (10\/2) = 5 units.\ue206<\/li>\n\n\n\n
                • For Rectangle 2: y\u2082 = (4\/2) = 2 units.\ue206<\/li>\n<\/ul>\n\n\n\n
                    \n
                  1. First Moments of Area:<\/strong><\/li>\n<\/ol>\n\n\n\n
                      \n
                    • Q\u2081 = A\u2081 \u00d7 y\u2081 = 40 \u00d7 5 = 200 units\u00b3.\ue206<\/li>\n\n\n\n
                    • Q\u2082 = A\u2082 \u00d7 y\u2082 = 24 \u00d7 2 = 48 units\u00b3.\ue206<\/li>\n<\/ul>\n\n\n\n
                        \n
                      1. Total First Moment of Area:<\/strong><\/li>\n<\/ol>\n\n\n\n
                          \n
                        • \u03a3Q_x = Q\u2081 + Q\u2082 = 200 + 48 = 248 units\u00b3.\ue206<\/li>\n<\/ul>\n\n\n\n
                            \n
                          1. Centroid y\u0304:<\/strong><\/li>\n<\/ol>\n\n\n\n
                              \n
                            • y\u0304 = \u03a3Q_x \/ A = 248 \/ 64 \u2248 3.875 units.\ue206<\/li>\n<\/ul>\n\n\n\n

                              Therefore, the centroid y\u0304 of the L-shaped area is approximately 3.875 units from the x-axis.<\/p>\n","protected":false},"excerpt":{"rendered":"

                              Locate the centroid y of the area. The Correct Answer and Explanation is : To determine the centroid (y\u0304) of a given area, we need to calculate the first moment of area about the x-axis (Q_x) and divide it by the total area (A). The formula is: [ y\u0304 = \\frac{Q_x}{A} ]\ue206 Steps to Calculate […]<\/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-196151","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196151","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=196151"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/196151\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=196151"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=196151"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=196151"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}