{"page":"\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/resources-c03aa079.css\" /\u003e\n\u003clink rel=\"stylesheet\" href=\"https://lessonplanet.com/assets/packs/css/lp_boclips_stylesheets-517835be.css\" media=\"all\" /\u003e\n\u003cdiv data-title='Conservation of Mechanical Energy Proof (2DoF)' data-url='/boclips/videos/628c91db487b257f8b9d81a1' data-video-url='/boclips/videos/628c91db487b257f8b9d81a1' id='bo_player_modal'\u003e\n\u003cdiv class='boclips-resource-page modal-dialog panel-container'\u003e\n\u003cdiv class='react-notifications-root'\u003e\u003c/div\u003e\n\u003cdiv class='rp-header'\u003e\n\u003cdiv class='rp-type'\u003e\n\u003ci aria-hidden='true' class='fai fa-regular fa-circle-play'\u003e\u003c/i\u003e\nVideo\n\u003c/div\u003e\n\u003ch1 class='rp-title' id='video-title'\u003e\nConservation of Mechanical Energy Proof (2DoF)\n\u003c/h1\u003e\n\u003cdiv class='rp-actions'\u003e\n\u003cdiv class='mr-1'\u003e\n\u003ca class=\"btn btn-success\" data-posthog-event=\"Signup: LP Signup Activity\" data-posthog-location=\"body_link_boclips\" data-remote=\"true\" href=\"/subscription/new\"\u003e\u003cspan\u003e\u003cspan\u003eGet Free Access\u003c/span\u003e\u003cspan class=\"\"\u003e for 10 Days\u003c/span\u003e\u003cspan\u003e!\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-body'\u003e\n\u003cdiv class='rp-info'\u003e\n\u003cdiv aria-label='Hide resource details' class='rp-hide-info' role='button' tabindex='0'\u003e\u0026times;\u003c/div\u003e\n\u003ci aria-label='Expand resource details' class='rp-expand-info fai fa-solid fa-up-right-and-down-left-from-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003ci aria-label='Compress resource details' class='rp-compress-info fai fa-solid fa-down-left-and-up-right-to-center' role='button' tabindex='0'\u003e\u003c/i\u003e\n\u003cdiv class='rp-rating'\u003e\n\u003cspan class='resource-pool'\u003e\n\u003cspan class='pool-label'\u003ePublisher:\u003c/span\u003e\n\u003cspan class='pool-name'\u003e\n\u003cspan class='text'\u003e\u003ca data-publisher-id=\"30360345\" href=\"/search?publisher_ids%5B%5D=30360345\"\u003eVirtually Passed\u003c/a\u003e\u003c/span\u003e\n\u003c/span\u003e\n\u003c/span\u003e\n\u003c/div\u003e\n\u003cdiv class='rp-description'\u003e\n\u003cspan class='short-description'\u003eHere is a quick math heavy proof of the conservation of mechanical energy in a 2 dimensional system. The proof in 3 dimensions is the exact same. Notice that the definition of the potential in the one dimensional system dV/dx = - F is...\u003c/span\u003e\n\u003cspan class='full-description hide'\u003eHere is a quick math heavy proof of the conservation of mechanical energy in a 2 dimensional system. The proof in 3 dimensions is the exact same. Notice that the definition of the potential in the one dimensional system dV/dx = - F is still consistent with the defubutuib for potential in the 2 dimensional system since Grad(V) in 1 dimension simply becomes dV/dx. \u003cbr/\u003e\u003cbr/\u003eHow does this all connect together with Work? Well, Work is defined by the integral of the force times infinitesimal displacement in the direction of the force. The total work is the work done by non conservative forces F(x,y) and non conservative forces. W = W_c + W_nc\u003cbr/\u003e\u003cbr/\u003eW = integral F(x,y) . dr   +  W_nc \u003cbr/\u003e\u003cbr/\u003eW = - integral grad(V) . dr  + W_nc\u003cbr/\u003e\u003cbr/\u003eW = - delta(V) + W_nc\u003cbr/\u003e\u003cbr/\u003eTherefore work done by conservative forces is the negative change in potential energy.\u003cbr/\u003e\u003cbr/\u003eBut we're not done, we can prove W = W_c + W_nc = delta (T)     where T = kinetic energy.\u003cbr/\u003e\u003cbr/\u003eTherefore\u003cbr/\u003e\u003cbr/\u003edelta(T) = - delta(V) + W_nc\u003cbr/\u003e\u003cbr/\u003edelta(T) + delta(V) = W_nc\u003cbr/\u003e\u003cbr/\u003eSo the work done by non conservative forces is equal to the change in kinetic energy + change in potential energy.\u003c/span\u003e\n\u003c/div\u003e\n\u003cdiv class='action-container flex justify-between'\u003e\n\u003cbutton aria-expanded='false' aria-label='Read more description' class='rp-full-description' type='button'\u003e\n\u003ci class='fai fa-solid 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