{"id":1052,"date":"2020-12-21T15:01:44","date_gmt":"2020-12-21T15:01:44","guid":{"rendered":"https:\/\/pickedshares.com\/?p=1052"},"modified":"2021-05-08T12:28:43","modified_gmt":"2021-05-08T12:28:43","slug":"engineering-mechanics-1-exercise-36-center-of-gravity-of-a-hemisphere","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-36-center-of-gravity-of-a-hemisphere\/","title":{"rendered":"Center of gravity of a hemisphere"},"content":{"rendered":"\n<p>This exercise is about the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>How to calculate the center of gravity of a hemisphere in cartesian coordinates?<\/li><li>How is the formula for the center of gravity of a hemisphere derived?<\/li><\/ul>\n\n\n\n<p>In the solution below, the formula for the area of \u200b\u200ba circular area is assumed to be known.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>For a hemisphere with the radius R, the center of gravity coordinate in the x direction has to be calculated.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"593\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-1024x593.png\" alt=\"Center of gravity of a hemisphere\" class=\"wp-image-1038\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-1024x593.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-300x174.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-768x445.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-400x232.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36-800x463.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36.png 1100w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Center of gravity of a hemisphere<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<p>In the solution shown here the result is calculated with cartesian coordinates.<\/p>\n\n\n\n<p>The following sketch shows the geometrical relationships for calculating the face A(x) or the radius r(x). The face A(x) lays in the y-z-plane. The product A(x)dx represents the volume element or elements that are added up.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"650\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-1024x650.png\" alt=\"Infinitesimal volume element of the hemisphere\" class=\"wp-image-1042\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-1024x650.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-300x190.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-768x487.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-400x254.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution-800x507.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2020\/12\/tm1-36solution.png 1165w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Infinitesimal volume element of the hemisphere<\/figcaption><\/figure>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\require{cancel} \\]\n\n<p>To calculate the center of gravity, the volume of the hemisphere is calculated first. This is done by integrating over the cross-sectional area along the x coordinate. The area of \u200b\u200bthe circle is known to be:<\/p>\n<p>\\[ \\tag{1} A = \\pi \\cdot r^2 \\]<\/p>\n<p>The radius r of the cross-sectional area depends on x and can be expressed as follows:<\/p>\n<p>\\[ \\tag{2} r(x) = \\pi \\cdot \\sqrt{R^2-x^2} \\]<\/p>\n<p>So that is the cross section<\/p>\n<p>\\[ \\tag{3} A(x) = \\pi \\cdot \\left( R^2-x^2 \\right) \\]<\/p>\n<p>The volume of the hemisphere is<\/p>\n<p>\\[ \\tag{4} V = \\int\\limits_0^R{\\pi \\cdot \\left( R^2-x^2 \\right) dx} \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4.1} V = \\left[ \\pi \\cdot \\left( R^2 \\cdot x - \\frac{1}{3} \\cdot x^3 \\right) \\right]_0^R  \\]<\/p>\n<p>\\[ \\tag{4.2} V = \\left[ \\pi \\cdot \\left( R^2 \\cdot R - \\frac{1}{3} \\cdot R^3 \\right) \\right] - \\bcancel{\\left[ \\pi \\cdot \\left( R^2 \\cdot 0 - \\frac{1}{3} \\cdot 0^3 \\right) \\right]}  \\]<\/p>\n<\/div>\n<p>\\[ \\tag{5} V = \\frac{2}{3} \\cdot \\pi \\cdot R^3 \\]<\/p>\n<p>The center of gravity coordinate x<sub>S<\/sub> is calculated as follows<\/p>\n\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{6} x_S = \\frac{\\int\\limits_0^R{\\pi \\cdot \\left( R^2-x^2 \\right)  \\cdot x \\, dx}}{V} \\]<\/p>\n\n<p>\\[ \\tag{6.1} x_S = \\frac{\\left[ \\pi \\cdot \\left( \\frac{1}{2} \\cdot R^2 \\cdot x^2 - \\frac{1}{4} \\cdot x^4 \\right) \\right]_0^R}{\\frac{2}{3} \\cdot \\pi \\cdot R^3} \\]<\/p>\n\n<p>\\[ \\tag{6.2} x_S = \\frac{\\left[ \\pi \\cdot \\left( \\frac{1}{2} \\cdot R^2 \\cdot R^2 - \\frac{1}{4} \\cdot R^4 \\right) \\right] - \\bcancel{ \\left[ \\pi \\cdot \\left( \\frac{1}{2} \\cdot R^2 \\cdot 0^2 - \\frac{1}{4} \\cdot 0^4 \\right) \\right]}}{\\frac{2}{3} \\cdot \\pi \\cdot R^3} \\]<\/p>\n<\/div>\n<p>\\[ \\tag{6.3} x_S = \\frac{ \\frac{3}{4} \\cdot \\pi \\cdot   R^4  }{2 \\cdot \\pi \\cdot R^3} \\]<\/p>\n\n<p>from which follows<\/p>\n<p>\\[ \\tag{7} x_S = \\frac{3}{8} \\cdot R \\]<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Center of gravity of a hemisphere\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-1-exercise-36-center-of-gravity-of-a-hemisphere\/\" aria-label=\"Read more about Center of gravity of a hemisphere\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":1048,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,60],"tags":[50],"class_list":["post-1052","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-i","category-exercises","tag-centre-points","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Center of gravity of a hemisphere &#8226; pickedshares<\/title>\n<meta name=\"description\" content=\"Calculate the x-coordinate of the center of gravity for a hemisphere with the radius R. 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