{"id":238117,"date":"2025-06-17T20:34:49","date_gmt":"2025-06-17T20:34:49","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238117"},"modified":"2025-06-17T20:34:51","modified_gmt":"2025-06-17T20:34:51","slug":"derive-for-both-the-single-particle-and-system-of-particles-the-work-energy-principle-or-work-energy-theorem","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/derive-for-both-the-single-particle-and-system-of-particles-the-work-energy-principle-or-work-energy-theorem\/","title":{"rendered":"Derive, for both the single particle and system of particles, the Work-Energy Principle or Work-Energy Theorem"},"content":{"rendered":"\n<p> Derive, for both the single particle and system of particles, the Work-Energy Principle or Work-Energy Theorem. 2. Draw a system of particles and write all the basic equations that describe the mechanics of a System of Particles. 3. Compare and contrast between Hamilton&#8217;s H and Lagrange&#8217;s L.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>1. Derivation of the Work-Energy Theorem<\/strong><\/p>\n\n\n\n<p><strong>Single Particle:<\/strong><\/p>\n\n\n\n<p>Start with Newton\u2019s Second Law:F\u20d7=ma\u20d7\\vec{F} = m \\vec{a}F=ma<\/p>\n\n\n\n<p>Multiply both sides by the displacement dr\u20d7\\vec{dr}dr:F\u20d7\u22c5dr\u20d7=ma\u20d7\u22c5dr\u20d7\\vec{F} \\cdot \\vec{dr} = m \\vec{a} \\cdot \\vec{dr}F\u22c5dr=ma\u22c5dr<\/p>\n\n\n\n<p>Since a\u20d7=dv\u20d7dt\\vec{a} = \\frac{d\\vec{v}}{dt}a=dtdv\u200b, and v\u20d7=dr\u20d7dt\\vec{v} = \\frac{d\\vec{r}}{dt}v=dtdr\u200b, then:a\u20d7\u22c5dr\u20d7=dv\u20d7dt\u22c5v\u20d7dt=v\u20d7\u22c5dv\u20d7\\vec{a} \\cdot \\vec{dr} = \\frac{d\\vec{v}}{dt} \\cdot \\vec{v} dt = \\vec{v} \\cdot d\\vec{v}a\u22c5dr=dtdv\u200b\u22c5vdt=v\u22c5dv<\/p>\n\n\n\n<p>Thus:F\u20d7\u22c5dr\u20d7=mv\u20d7\u22c5dv\u20d7\\vec{F} \\cdot \\vec{dr} = m \\vec{v} \\cdot d\\vec{v}F\u22c5dr=mv\u22c5dv<\/p>\n\n\n\n<p>Integrating both sides:\u222bF\u20d7\u22c5dr\u20d7=\u222bmv\u20d7\u22c5dv\u20d7=12mv2\u221212mv02\\int \\vec{F} \\cdot \\vec{dr} = \\int m \\vec{v} \\cdot d\\vec{v} = \\frac{1}{2} m v^2 &#8211; \\frac{1}{2} m v_0^2\u222bF\u22c5dr=\u222bmv\u22c5dv=21\u200bmv2\u221221\u200bmv02\u200b<\/p>\n\n\n\n<p>So,W=\u0394KW = \\Delta KW=\u0394K<\/p>\n\n\n\n<p><strong>System of Particles:<\/strong><\/p>\n\n\n\n<p>Sum of external work on all particles equals change in total kinetic energy:\u2211Wext=\u0394Ktotal\\sum W_{\\text{ext}} = \\Delta K_{\\text{total}}\u2211Wext\u200b=\u0394Ktotal\u200b<\/p>\n\n\n\n<p>Here, internal forces cancel due to Newton&#8217;s Third Law. So:\u2211i=1nF\u20d7ext,i\u22c5dr\u20d7i=\u0394(\u2211i=1n12mivi2)\\sum_{i=1}^{n} \\vec{F}_{\\text{ext},i} \\cdot \\vec{dr}_i = \\Delta \\left( \\sum_{i=1}^{n} \\frac{1}{2} m_i v_i^2 \\right)i=1\u2211n\u200bFext,i\u200b\u22c5dri\u200b=\u0394(i=1\u2211n\u200b21\u200bmi\u200bvi2\u200b)<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>2. System of Particles and Basic Equations<\/strong><\/p>\n\n\n\n<p><strong>Diagram<\/strong> (You can draw this by sketching several particles labeled m1,m2,\u2026m_1, m_2, \\ldotsm1\u200b,m2\u200b,\u2026 with external forces F\u20d7ext\\vec{F}_{\\text{ext}}Fext\u200b and internal forces between them)<\/p>\n\n\n\n<p><strong>Basic Equations:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Total external force:<\/li>\n<\/ul>\n\n\n\n<p>F\u20d7ext=Ma\u20d7CM\\vec{F}_{\\text{ext}} = M \\vec{a}_{\\text{CM}}Fext\u200b=MaCM\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Momentum conservation:<\/li>\n<\/ul>\n\n\n\n<p>P\u20d7=\u2211miv\u20d7i\\vec{P} = \\sum m_i \\vec{v}_iP=\u2211mi\u200bvi\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Angular momentum about origin:<\/li>\n<\/ul>\n\n\n\n<p>L\u20d7=\u2211r\u20d7i\u00d7miv\u20d7i\\vec{L} = \\sum \\vec{r}_i \\times m_i \\vec{v}_iL=\u2211ri\u200b\u00d7mi\u200bvi\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Torque:<\/li>\n<\/ul>\n\n\n\n<p>\u03c4\u20d7ext=dL\u20d7dt\\vec{\\tau}_{\\text{ext}} = \\frac{d\\vec{L}}{dt}\u03c4ext\u200b=dtdL\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Work-energy for system:<\/li>\n<\/ul>\n\n\n\n<p>\u2211F\u20d7ext\u22c5dr\u20d7=\u0394Ktotal\\sum \\vec{F}_{\\text{ext}} \\cdot \\vec{dr} = \\Delta K_{\\text{total}}\u2211Fext\u200b\u22c5dr=\u0394Ktotal\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>3. Comparison: Hamilton&#8217;s H vs Lagrange&#8217;s L<\/strong><\/p>\n\n\n\n<p><strong>Lagrangian L=T\u2212VL = T &#8211; VL=T\u2212V<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>It is the difference between kinetic and potential energy.<\/li>\n\n\n\n<li>Leads to Lagrange\u2019s equations:<\/li>\n<\/ul>\n\n\n\n<p>ddt(\u2202L\u2202q\u02d9i)\u2212\u2202L\u2202qi=0\\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}_i} \\right) &#8211; \\frac{\\partial L}{\\partial q_i} = 0dtd\u200b(\u2202q\u02d9\u200bi\u200b\u2202L\u200b)\u2212\u2202qi\u200b\u2202L\u200b=0<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Best for systems with constraints and generalized coordinates.<\/li>\n<\/ul>\n\n\n\n<p><strong>Hamiltonian H=T+VH = T + VH=T+V<\/strong> (in many cases)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Represents total energy if potential energy does not depend on velocity.<\/li>\n\n\n\n<li>Used in Hamilton&#8217;s equations:<\/li>\n<\/ul>\n\n\n\n<p>q\u02d9i=\u2202H\u2202pi,p\u02d9i=\u2212\u2202H\u2202qi\\dot{q}_i = \\frac{\\partial H}{\\partial p_i}, \\quad \\dot{p}_i = -\\frac{\\partial H}{\\partial q_i}q\u02d9\u200bi\u200b=\u2202pi\u200b\u2202H\u200b,p\u02d9\u200bi\u200b=\u2212\u2202qi\u200b\u2202H\u200b<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>More powerful for analytical mechanics and quantum mechanics.<\/li>\n<\/ul>\n\n\n\n<p><strong>Contrast:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Lagrangian depends on positions and velocities; Hamiltonian uses positions and momenta.<\/li>\n\n\n\n<li>Lagrangian is better suited to derive equations of motion; Hamiltonian is more symmetric and useful in phase space analysis.<\/li>\n<\/ul>\n\n\n\n<p>Both are reformulations of Newtonian mechanics and lead to the same physical predictions.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-990.jpeg\" alt=\"\" class=\"wp-image-238118\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Derive, for both the single particle and system of particles, the Work-Energy Principle or Work-Energy Theorem. 2. Draw a system of particles and write all the basic equations that describe the mechanics of a System of Particles. 3. Compare and contrast between Hamilton&#8217;s H and Lagrange&#8217;s L. The Correct Answer and Explanation is: 1. Derivation [&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-238117","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238117","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=238117"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238117\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238117"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238117"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238117"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}