{"id":181006,"date":"2025-01-09T12:22:42","date_gmt":"2025-01-09T12:22:42","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=181006"},"modified":"2025-01-09T12:22:45","modified_gmt":"2025-01-09T12:22:45","slug":"determine-the-magnitude-and-direction","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/09\/determine-the-magnitude-and-direction\/","title":{"rendered":"Determine the magnitude and direction"},"content":{"rendered":"\n<p>Determine the magnitude and direction ? of FASo that the resultant force is directed along the positive x-axis and has a magnitude of 1250 N. I. 2. Determine the magnitude and direction, measured counterclockwise from the axis, of the resultant force acting on the ring at O,if FA- 750 N and -4s positive x- FA 30\u00b0 FB=800 N<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/01\/image-70.png\" alt=\"\" class=\"wp-image-181007\"\/><\/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>\ue203To solve these problems, we&#8217;ll decompose each force into its horizontal (x-axis) and vertical (y-axis) components, sum these components to find the resultant force, and then determine the required magnitudes and directions.\ue204<\/p>\n\n\n\n<p><strong>Problem 1: Determining the Magnitude and Direction of FAF_A<\/strong><\/p>\n\n\n\n<p>\ue203Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Force FB=800\u2009NF_B = 800 \\, \\text{N} at an angle of 30\u221830^\\circ from the positive x-axis.\ue204<\/li>\n\n\n\n<li>The resultant force FRF_R is directed along the positive x-axis with a magnitude of 1250\u2009N1250 \\, \\text{N}.\ue204<\/li>\n<\/ul>\n\n\n\n<p>\ue203We need to find the magnitude and direction \u03b8\\theta of FAF_A such that the resultant force meets these conditions.\ue204<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Resolve FBF_B into components:<\/strong>\n<ul class=\"wp-block-list\">\n<li>FBx=FBcos\u2061(30\u2218)=800\u00d7cos\u2061(30\u2218)=800\u00d732=800\u00d70.866=692.8\u2009NF_{Bx} = F_B \\cos(30^\\circ) = 800 \\times \\cos(30^\\circ) = 800 \\times \\frac{\\sqrt{3}}{2} = 800 \\times 0.866 = 692.8 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>FBy=FBsin\u2061(30\u2218)=800\u00d7sin\u2061(30\u2218)=800\u00d70.5=400\u2009NF_{By} = F_B \\sin(30^\\circ) = 800 \\times \\sin(30^\\circ) = 800 \\times 0.5 = 400 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Express the resultant force components:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Since FRF_R is along the positive x-axis:\ue206\n<ul class=\"wp-block-list\">\n<li>FRx=1250\u2009NF_{Rx} = 1250 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>FRy=0\u2009NF_{Ry} = 0 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Set up equations for the components:<\/strong>\n<ul class=\"wp-block-list\">\n<li>In the x-direction:\ue206\n<ul class=\"wp-block-list\">\n<li>FAx+FBx=FRxF_{Ax} + F_{Bx} = F_{Rx}\ue206<\/li>\n\n\n\n<li>FAsin\u2061(\u03b8)+692.8=1250F_A \\sin(\\theta) + 692.8 = 1250\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>In the y-direction:\ue206\n<ul class=\"wp-block-list\">\n<li>FAy+FBy=FRyF_{Ay} + F_{By} = F_{Ry}\ue206<\/li>\n\n\n\n<li>FAcos\u2061(\u03b8)\u2212400=0F_A \\cos(\\theta) &#8211; 400 = 0\ue206<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Solve for FAF_A and \u03b8\\theta:<\/strong>\n<ul class=\"wp-block-list\">\n<li>From the y-component equation:\ue206\n<ul class=\"wp-block-list\">\n<li>FAcos\u2061(\u03b8)=400F_A \\cos(\\theta) = 400\ue206<\/li>\n\n\n\n<li>FA=400cos\u2061(\u03b8)F_A = \\frac{400}{\\cos(\\theta)}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Substitute FAF_A into the x-component equation:\ue206\n<ul class=\"wp-block-list\">\n<li>400sin\u2061(\u03b8)cos\u2061(\u03b8)+692.8=1250\\frac{400 \\sin(\\theta)}{\\cos(\\theta)} + 692.8 = 1250\ue206<\/li>\n\n\n\n<li>400tan\u2061(\u03b8)+692.8=1250400 \\tan(\\theta) + 692.8 = 1250\ue206<\/li>\n\n\n\n<li>400tan\u2061(\u03b8)=557.2400 \\tan(\\theta) = 557.2\ue206<\/li>\n\n\n\n<li>tan\u2061(\u03b8)=557.2400=1.393\\tan(\\theta) = \\frac{557.2}{400} = 1.393\ue206<\/li>\n\n\n\n<li>\u03b8=tan\u2061\u22121(1.393)\u224854.3\u2218\\theta = \\tan^{-1}(1.393) \\approx 54.3^\\circ\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li>Now, calculate FAF_A:\ue206\n<ul class=\"wp-block-list\">\n<li>FA=400cos\u2061(54.3\u2218)=4000.584\u2248685.8\u2009NF_A = \\frac{400}{\\cos(54.3^\\circ)} = \\frac{400}{0.584} \\approx 685.8 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p><strong>Answer:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Magnitude of FAF_A: 686\u2009N686 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>Direction of FAF_A: 54.3\u221854.3^\\circ from the positive x-axis\ue206<\/li>\n<\/ul>\n\n\n\n<p><strong>Problem 2: Determining the Resultant Force with Given FAF_A<\/strong><\/p>\n\n\n\n<p>\ue203Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>FA=750\u2009NF_A = 750 \\, \\text{N} at 45\u221845^\\circ from the positive x-axis.\ue204<\/li>\n\n\n\n<li>FB=800\u2009NF_B = 800 \\, \\text{N} at 30\u221830^\\circ from the positive x-axis.\ue204<\/li>\n<\/ul>\n\n\n\n<p>\ue203We need to find the magnitude and direction of the resultant force acting on the ring at O, measured counterclockwise from the positive x-axis.\ue204<\/p>\n\n\n\n<p><strong>Solution:<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Resolve FAF_A into components:<\/strong>\n<ul class=\"wp-block-list\">\n<li>FAx=FAcos\u2061(45\u2218)=750\u00d7cos\u2061(45\u2218)=750\u00d70.707=530.3\u2009NF_{Ax} = F_A \\cos(45^\\circ) = 750 \\times \\cos(45^\\circ) = 750 \\times 0.707 = 530.3 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>FAy=FAsin\u2061(45\u2218)=750\u00d7sin\u2061(45\u2218)=750\u00d70.707=530.3\u2009NF_{Ay} = F_A \\sin(45^\\circ) = 750 \\times \\sin(45^\\circ) = 750 \\times 0.707 = 530.3 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Resolve FBF_B into components:<\/strong>\n<ul class=\"wp-block-list\">\n<li>FBx=FBcos\u2061(30\u2218)=800\u00d7cos\u2061(30\u2218)=800\u00d70.866=692.8\u2009NF_{Bx} = F_B \\cos(30^\\circ) = 800 \\times \\cos(30^\\circ) = 800 \\times 0.866 = 692.8 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>FBy=FBsin\u2061(30\u2218)=800\u00d7sin\u2061(30\u2218)=800\u00d70.5=400\u2009NF_{By} = F_B \\sin(30^\\circ) = 800 \\times \\sin(30^\\circ) = 800 \\times 0.5 = 400 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Sum the components to find the resultant force:<\/strong>\n<ul class=\"wp-block-list\">\n<li>FRx=FAx+FBx=530.3+692.8=1223.1\u2009NF_{Rx} = F_{Ax} + F_{Bx} = 530.3 + 692.8 = 1223.1 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>FRy=FAy\u2212FBy=530.3\u2212400=130.3\u2009NF_{Ry} = F_{Ay} &#8211; F_{By} = 530.3 &#8211; 400 = 130.3 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Calculate the magnitude of the resultant force:<\/strong>\n<ul class=\"wp-block-list\">\n<li>FR=FRx2+FRy2=1223.12+130.32\u22481230\u2009NF_R = \\sqrt{F_{Rx}^2 + F_{Ry}^2} = \\sqrt{1223.1^2 + 130.3^2} \\approx 1230 \\, \\text{N}\ue206<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Determine the direction of the resultant force:<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u03b8=tan\u2061\u22121(FRyFRx)=tan\u2061\u22121(130.31223.1)\u22486.08\u2218\\theta = \\tan^{-1}\\left(\\frac{F_{Ry}}{F_{Rx}}\\right) = \\tan^{-1}\\left(\\frac{130.3}{1223.1}\\right) \\approx 6.08^\\circ\ue206<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p><strong>Answer:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Magnitude of the resultant force: 1230\u2009N1230 \\, \\text{N}\ue206<\/li>\n\n\n\n<li>Direction: ( 6.08^\\circ<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Determine the magnitude and direction ? of FASo that the resultant force is directed along the positive x-axis and has a magnitude of 1250 N. I. 2. Determine the magnitude and direction, measured counterclockwise from the axis, of the resultant force acting on the ring at O,if FA- 750 N and -4s positive x- FA [&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-181006","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181006","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=181006"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/181006\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=181006"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=181006"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=181006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}