{"id":475,"date":"2025-02-13T19:45:18","date_gmt":"2025-02-13T19:45:18","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-apply-it\/"},"modified":"2025-02-13T19:45:18","modified_gmt":"2025-02-13T19:45:18","slug":"moments-and-centers-of-mass-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-apply-it\/","title":{"raw":"Moments and Centers of Mass: Apply It","rendered":"Moments and Centers of Mass: Apply It"},"content":{"raw":"\n<section class=\"textbox learningGoals\">\n<ul>\n\t<li>Find the balance point (center of mass) of straight objects and flat surfaces<\/li>\n\t<li>Utilize a shape\u2019s symmetry to find the centroid, or geometric center, of flat objects<\/li>\n\t<li>Use Pappus\u2019s theorem to calculate the volume of an object<\/li>\n<\/ul>\n<\/section>\n<h2>Engineering The Grand Canyon Skywalk<\/h2>\n<p id=\"fs-id1167793524940\">The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended [latex]4000[\/latex] ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure).<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"700\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213307\/CNX_Calc_Figure_06_06_011.jpg\" alt=\"This figure is a picture of the Grand Canyon skywalk. It is a building at the edge of the canyon with a walkway extending out over the canyon\" width=\"700\" height=\"526\"> Figure 11. The Grand Canyon Skywalk offers magnificent views of the canyon. (credit: 10da_ralta, Wikimedia Commons)[\/caption]\n\n<p>&nbsp;<\/p>\n<p>The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend [latex]46[\/latex] ft down into bedrock. The structure was built to withstand [latex]100[\/latex]-mph winds and an [latex]8.0[\/latex]-magnitude earthquake within [latex]50[\/latex] mi, and is capable of supporting more than [latex]70,000,000[\/latex] lb.<\/p>\n<p>One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform.<\/p>\n<p id=\"fs-id1167793502542\">The observation platform is U-shaped. The legs of the U are [latex]10[\/latex] ft wide and begin on land, under the visitors\u2019 center, [latex]48[\/latex] ft from the edge of the canyon. The platform extends [latex]70[\/latex] ft over the edge of the canyon.<\/p>\n<p id=\"fs-id1167793547133\">To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the [latex]xy[\/latex]-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted [latex]{R}_{1},[\/latex] consists of the curved part of the U. We model [latex]{R}_{1}[\/latex] as a semicircular annulus, with inner radius [latex]25[\/latex] ft and outer radius [latex]35[\/latex] ft, centered at the origin (see the following figure).<\/p>\n\n[caption id=\"\" align=\"aligncenter\" width=\"700\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213311\/CNX_Calc_Figure_06_06_012.jpg\" alt=\"This figure is a sketch of the Grand Canyon walkway. It is on the xy coordinate system. The walkway is upside-down \u201cu\u201d shaped. It has been divided into three regions. The first region at the top is labeled \u201cRsub1\u201d. It is a semi-circle with outer radius of 35 feet and inner radius of 25 feet. The second region is labeled \u201cRsub2\u201d. It has two rectangles with width of 10 feet each and height of 35 feet. The third region is labeled \u201cRsub3\u201d and is two rectangles. They have a width of 10 feet and height of 48 feet. These represent the part of the walkway inside of the visitor center.\" width=\"700\" height=\"571\"> Figure 12. We model the Skywalk with three sub-regions.[\/caption]\n\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793630335\">The legs of the platform, extending [latex]35[\/latex] ft between [latex]{R}_{1}[\/latex] and the canyon wall, comprise the second sub-region, [latex]{R}_{2}.[\/latex] Last, the ends of the legs, which extend [latex]48[\/latex] ft under the visitor center, comprise the third sub-region, [latex]{R}_{3}.[\/latex] Assume the density of the lamina is constant and assume the total weight of the platform is [latex]1,200,000[\/latex] lb (not including the weight of the visitor center; we will consider that later). Use [latex]g=32{\\text{ft\/sec}}^{2}.[\/latex]<\/p>\n<ol id=\"fs-id1167793506232\">\n\t<li>Compute the area of each of the three sub-regions. Note that the areas of regions [latex]{R}_{2}[\/latex] and [latex]{R}_{3}[\/latex] should include the areas of the legs only, not the open space between them. Round answers to the nearest square foot.<br>\n<br>\n<br>\n[reveal-answer q=\"806511\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"806511\"]<br>\nThe area of the three masses are [latex]R_1 = 942[\/latex] ft<sup>2<\/sup>, [latex]R_2 = 700[\/latex] ft<sup>2<\/sup> , and [latex]R_3=960[\/latex]ft<sup>2<br>\n<\/sup>[\/hidden-answer]<\/li>\n\t<li>Determine the mass associated with each of the three sub-regions.<br>\n<br>\n<br>\n[reveal-answer q=\"203970\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"203970\"]<br>\nThe masses associated are [latex]m_1 = 434,431.6[\/latex] lb, [latex]m_2 = 322,826[\/latex] lb, and [latex]m_3=442,732.8[\/latex] lb<br>\n[\/hidden-answer]<br>\n<br>\n<\/li>\n\t<li>Calculate the center of mass of each of the three sub-regions.<br>\n<br>\n<br>\n[reveal-answer q=\"888681\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"888681\"] <br>\n<br>\nThe center of masses are [latex]y_1 = 19.28 [\/latex] ft, [latex]y_2=-17.5[\/latex] ft, and [latex]y_3=-59[\/latex] ft<br>\n[\/hidden-answer]<br>\n<br>\n<\/li>\n\t<li>Now, treat each of the three sub-regions as a point mass located at the center of mass of the corresponding sub-region. Using this representation, calculate the center of mass of the entire platform.<br>\n<br>\n<br>\n[reveal-answer q=\"97102\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"97102\"]<br>\n<br>\nThe COM of the platform lies at [latex](0,-19.5)[\/latex] ft<br>\n<br>\n[\/hidden-answer]<br>\n<br>\n<\/li>\n\t<li>Assume the visitor center weighs [latex]2,200,000[\/latex] lb, with a center of mass corresponding to the center of mass of [latex]{R}_{3}.[\/latex] Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change?<br>\n<br>\n<br>\n[reveal-answer q=\"561310\"]Show Answer[\/reveal-answer]<br>\n[hidden-answer a=\"561310\"]<br>\n<br>\nThe COM of the platform including the center of mass lies at [latex](0,-45)[\/latex] ft<br>\n[\/hidden-answer]<br>\n<br>\n<\/li>\n<\/ol>\n","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Find the balance point (center of mass) of straight objects and flat surfaces<\/li>\n<li>Utilize a shape\u2019s symmetry to find the centroid, or geometric center, of flat objects<\/li>\n<li>Use Pappus\u2019s theorem to calculate the volume of an object<\/li>\n<\/ul>\n<\/section>\n<h2>Engineering The Grand Canyon Skywalk<\/h2>\n<p id=\"fs-id1167793524940\">The Grand Canyon Skywalk opened to the public on March 28, 2007. This engineering marvel is a horseshoe-shaped observation platform suspended [latex]4000[\/latex] ft above the Colorado River on the West Rim of the Grand Canyon. Its crystal-clear glass floor allows stunning views of the canyon below (see the following figure).<\/p>\n<figure style=\"width: 700px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213307\/CNX_Calc_Figure_06_06_011.jpg\" alt=\"This figure is a picture of the Grand Canyon skywalk. It is a building at the edge of the canyon with a walkway extending out over the canyon\" width=\"700\" height=\"526\" \/><figcaption class=\"wp-caption-text\">Figure 11. The Grand Canyon Skywalk offers magnificent views of the canyon. (credit: 10da_ralta, Wikimedia Commons)<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The Skywalk is a cantilever design, meaning that the observation platform extends over the rim of the canyon, with no visible means of support below it. Despite the lack of visible support posts or struts, cantilever structures are engineered to be very stable and the Skywalk is no exception. The observation platform is attached firmly to support posts that extend [latex]46[\/latex] ft down into bedrock. The structure was built to withstand [latex]100[\/latex]-mph winds and an [latex]8.0[\/latex]-magnitude earthquake within [latex]50[\/latex] mi, and is capable of supporting more than [latex]70,000,000[\/latex] lb.<\/p>\n<p>One factor affecting the stability of the Skywalk is the center of gravity of the structure. We are going to calculate the center of gravity of the Skywalk, and examine how the center of gravity changes when tourists walk out onto the observation platform.<\/p>\n<p id=\"fs-id1167793502542\">The observation platform is U-shaped. The legs of the U are [latex]10[\/latex] ft wide and begin on land, under the visitors\u2019 center, [latex]48[\/latex] ft from the edge of the canyon. The platform extends [latex]70[\/latex] ft over the edge of the canyon.<\/p>\n<p id=\"fs-id1167793547133\">To calculate the center of mass of the structure, we treat it as a lamina and use a two-dimensional region in the [latex]xy[\/latex]-plane to represent the platform. We begin by dividing the region into three subregions so we can consider each subregion separately. The first region, denoted [latex]{R}_{1},[\/latex] consists of the curved part of the U. We model [latex]{R}_{1}[\/latex] as a semicircular annulus, with inner radius [latex]25[\/latex] ft and outer radius [latex]35[\/latex] ft, centered at the origin (see the following figure).<\/p>\n<figure style=\"width: 700px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11213311\/CNX_Calc_Figure_06_06_012.jpg\" alt=\"This figure is a sketch of the Grand Canyon walkway. It is on the xy coordinate system. The walkway is upside-down \u201cu\u201d shaped. It has been divided into three regions. The first region at the top is labeled \u201cRsub1\u201d. It is a semi-circle with outer radius of 35 feet and inner radius of 25 feet. The second region is labeled \u201cRsub2\u201d. It has two rectangles with width of 10 feet each and height of 35 feet. The third region is labeled \u201cRsub3\u201d and is two rectangles. They have a width of 10 feet and height of 48 feet. These represent the part of the walkway inside of the visitor center.\" width=\"700\" height=\"571\" \/><figcaption class=\"wp-caption-text\">Figure 12. We model the Skywalk with three sub-regions.<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1167793630335\">The legs of the platform, extending [latex]35[\/latex] ft between [latex]{R}_{1}[\/latex] and the canyon wall, comprise the second sub-region, [latex]{R}_{2}.[\/latex] Last, the ends of the legs, which extend [latex]48[\/latex] ft under the visitor center, comprise the third sub-region, [latex]{R}_{3}.[\/latex] Assume the density of the lamina is constant and assume the total weight of the platform is [latex]1,200,000[\/latex] lb (not including the weight of the visitor center; we will consider that later). Use [latex]g=32{\\text{ft\/sec}}^{2}.[\/latex]<\/p>\n<ol id=\"fs-id1167793506232\">\n<li>Compute the area of each of the three sub-regions. Note that the areas of regions [latex]{R}_{2}[\/latex] and [latex]{R}_{3}[\/latex] should include the areas of the legs only, not the open space between them. Round answers to the nearest square foot.\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q806511\">Show Answer<\/button><\/p>\n<div id=\"q806511\" class=\"hidden-answer\" style=\"display: none\">\nThe area of the three masses are [latex]R_1 = 942[\/latex] ft<sup>2<\/sup>, [latex]R_2 = 700[\/latex] ft<sup>2<\/sup> , and [latex]R_3=960[\/latex]ft<sup>2<br \/>\n<\/sup><\/div>\n<\/div>\n<\/li>\n<li>Determine the mass associated with each of the three sub-regions.\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q203970\">Show Answer<\/button><\/p>\n<div id=\"q203970\" class=\"hidden-answer\" style=\"display: none\">\nThe masses associated are [latex]m_1 = 434,431.6[\/latex] lb, [latex]m_2 = 322,826[\/latex] lb, and [latex]m_3=442,732.8[\/latex] lb\n<\/div>\n<\/div>\n<\/li>\n<li>Calculate the center of mass of each of the three sub-regions.\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q888681\">Show Answer<\/button><\/p>\n<div id=\"q888681\" class=\"hidden-answer\" style=\"display: none\">\n<p>The center of masses are [latex]y_1 = 19.28[\/latex] ft, [latex]y_2=-17.5[\/latex] ft, and [latex]y_3=-59[\/latex] ft\n<\/div>\n<\/div>\n<\/li>\n<li>Now, treat each of the three sub-regions as a point mass located at the center of mass of the corresponding sub-region. Using this representation, calculate the center of mass of the entire platform.\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q97102\">Show Answer<\/button><\/p>\n<div id=\"q97102\" class=\"hidden-answer\" style=\"display: none\">\n<p>The COM of the platform lies at [latex](0,-19.5)[\/latex] ft<\/p>\n<\/div>\n<\/div>\n<\/li>\n<li>Assume the visitor center weighs [latex]2,200,000[\/latex] lb, with a center of mass corresponding to the center of mass of [latex]{R}_{3}.[\/latex] Treating the visitor center as a point mass, recalculate the center of mass of the system. How does the center of mass change?\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q561310\">Show Answer<\/button><\/p>\n<div id=\"q561310\" class=\"hidden-answer\" style=\"display: none\">\n<p>The COM of the platform including the center of mass lies at [latex](0,-45)[\/latex] ft\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n","protected":false},"author":6,"menu_order":16,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":450,"module-header":"","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/475"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/475\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/450"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/475\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=475"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=475"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=475"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=475"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}