{"id":195485,"date":"2025-02-28T10:00:38","date_gmt":"2025-02-28T10:00:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195485"},"modified":"2025-02-28T10:00:42","modified_gmt":"2025-02-28T10:00:42","slug":"consider-the-wedge-and-the-brace","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/28\/consider-the-wedge-and-the-brace\/","title":{"rendered":"Consider the wedge and the brace"},"content":{"rendered":"\n<p>Consider the wedge and the brace shown in (Figure 1).<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-1651.png\" alt=\"\" class=\"wp-image-195486\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>To determine the force ( P ) required to lift the brace supporting the load ( F ) using the wedge, we must analyze the system&#8217;s static equilibrium, considering frictional forces at the contact surfaces.<\/p>\n\n\n\n<p><strong>Free-Body Diagrams (FBDs):<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Brace (Block):<\/strong> The brace experiences:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Its weight ( F ) acting downward.<\/li>\n\n\n\n<li>A normal force ( N_1 ) from the wedge acting perpendicular to the contact surface.<\/li>\n\n\n\n<li>A frictional force ( f_1 = \\mu_s N_1 ) opposing the relative motion between the brace and the wedge.<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Wedge:<\/strong> The wedge is subjected to:<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The applied force ( P ) horizontally to the right.<\/li>\n\n\n\n<li>The normal force ( N_1 ) from the brace, acting at an angle corresponding to the wedge&#8217;s inclination.<\/li>\n\n\n\n<li>A frictional force ( f_1 ) opposing the motion relative to the brace.<\/li>\n\n\n\n<li>A normal force ( N_2 ) from the ground, acting perpendicular to the base of the wedge.<\/li>\n\n\n\n<li>A frictional force ( f_2 = \\mu_s N_2 ) opposing the motion relative to the ground.<\/li>\n<\/ul>\n\n\n\n<p><strong>Equilibrium Equations:<\/strong><\/p>\n\n\n\n<p>For the brace:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertical forces:<\/strong> [ N_1 \\cos(\\theta) &#8211; f_1 \\sin(\\theta) = F ]<\/li>\n<\/ul>\n\n\n\n<p>For the wedge:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Horizontal forces:<\/strong> [ P = f_1 \\cos(\\theta) + N_1 \\sin(\\theta) + f_2 ]<\/li>\n\n\n\n<li><strong>Vertical forces:<\/strong> [ N_2 = N_1 \\cos(\\theta) &#8211; f_1 \\sin(\\theta) ]<\/li>\n<\/ul>\n\n\n\n<p><strong>Solving for ( P ):<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Express ( f_1 ) and ( f_2 ) in terms of ( N_1 ) and ( N_2 ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>( f_1 = \\mu_s N_1 )<\/li>\n\n\n\n<li>( f_2 = \\mu_s N_2 )<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Substitute ( f_1 ) into the brace&#8217;s vertical equilibrium equation to solve for ( N_1 ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[ N_1 \\cos(\\theta) &#8211; \\mu_s N_1 \\sin(\\theta) = F ]<\/li>\n\n\n\n<li>[ N_1 (\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)) = F ]<\/li>\n\n\n\n<li>[ N_1 = \\frac{F}{\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)} ]<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Substitute ( N_1 ) into the wedge&#8217;s vertical equilibrium equation to find ( N_2 ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[ N_2 = N_1 (\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)) ]<\/li>\n\n\n\n<li>[ N_2 = F ]<\/li>\n<\/ul>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Substitute ( N_1 ), ( f_1 ), and ( f_2 ) into the wedge&#8217;s horizontal equilibrium equation to solve for ( P ):<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>[ P = \\mu_s N_1 \\cos(\\theta) + N_1 \\sin(\\theta) + \\mu_s N_2 ]<\/li>\n\n\n\n<li>[ P = N_1 (\\mu_s \\cos(\\theta) + \\sin(\\theta)) + \\mu_s F ]<\/li>\n\n\n\n<li>[ P = \\frac{F (\\mu_s \\cos(\\theta) + \\sin(\\theta))}{\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)} + \\mu_s F ]<\/li>\n\n\n\n<li>[ P = F \\left( \\frac{\\mu_s \\cos(\\theta) + \\sin(\\theta)}{\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)} + \\mu_s \\right) ]<\/li>\n<\/ul>\n\n\n\n<p><strong>Conclusion:<\/strong><\/p>\n\n\n\n<p>The force ( P ) required to lift the brace is:<\/p>\n\n\n\n<p>[ P = F \\left( \\frac{\\mu_s \\cos(\\theta) + \\sin(\\theta)}{\\cos(\\theta) &#8211; \\mu_s \\sin(\\theta)} + \\mu_s \\right) ]<\/p>\n\n\n\n<p>This equation shows that ( P ) depends on the load ( F ), the wedge angle ( \\theta ), and the coefficient of static friction ( \\mu_s ). A smaller wedge angle or higher friction coefficient reduces the required force ( P ).<\/p>\n\n\n\n<p>For a visual explanation and step-by-step solution of a similar wedge problem, you may find the following video helpful:<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the wedge and the brace shown in (Figure 1). The Correct Answer and Explanation is : To determine the force ( P ) required to lift the brace supporting the load ( F ) using the wedge, we must analyze the system&#8217;s static equilibrium, considering frictional forces at the contact surfaces. Free-Body Diagrams (FBDs): [&hellip;]<\/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-195485","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195485","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=195485"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195485\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195485"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195485"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195485"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}