{"id":4029,"date":"2021-08-08T08:32:32","date_gmt":"2021-08-08T03:02:32","guid":{"rendered":"https:\/\/acejee.com\/blog\/?p=4029"},"modified":"2024-09-30T08:10:41","modified_gmt":"2024-09-30T02:40:41","slug":"work-done-by-a-variable-force","status":"publish","type":"post","link":"https:\/\/acejee.com\/blog\/work-done-by-a-variable-force\/","title":{"rendered":"Work done by a variable force"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"4029\" class=\"elementor elementor-4029\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-daa15c1 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"daa15c1\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-e4a81dd\" data-id=\"e4a81dd\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f59bfd9 elementor-widget elementor-widget-text-editor\" data-id=\"f59bfd9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">Given a two-dimensional variable force $\\overrightarrow{F}(x,y)$ acting on a particle moving along a given curved path or moving along an unknown path, how would you determine the work done by this variable force as the particle moves from point $A$ to point $B$<\/span><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-a2bf6c3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"a2bf6c3\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-6bb2e96\" data-id=\"6bb2e96\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-a136e8d elementor-widget elementor-widget-image\" data-id=\"a136e8d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"768\" height=\"497\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-768x497.png\" class=\"attachment-medium_large size-medium_large wp-image-4076\" alt=\"Work done by a variable force - Sample Problem 1\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-768x497.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-300x194.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-1024x663.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-1536x994.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-2048x1326.png 2048w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-57ba5a8 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"57ba5a8\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-c533960\" data-id=\"c533960\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-755bddc elementor-widget elementor-widget-text-editor\" data-id=\"755bddc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<iframe src=\"https:\/\/anchor.fm\/ace-jee\/embed\/episodes\/04-Work-Energy-Power---QT6---Work-done-by-a-2D-variable-force-e15k00h\" height=\"150px\" width=\"400px\" frameborder=\"0\" scrolling=\"no\"><\/iframe>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-1faae4d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"1faae4d\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-35d6f47\" data-id=\"35d6f47\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-985e5b6 elementor-widget elementor-widget-text-editor\" data-id=\"985e5b6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><span style=\"font-size: 15px;\">$\\color{blue}{\\text{Solution Steps}}$<\/span><\/p><p>$W = \\int \\overrightarrow{F}.\\overrightarrow{ds}$ $=\\int (f_x(x,y) \\hat{i} +$ $f_y(x,y) \\hat{j}).$$(dx \\hat{i} +$ $dy \\hat{j})$ $=\\int f_x(x,y) dx +$ $f_y(x,y) dy$<\/p><p>Now depending on the given expressions for $f_x(x,y)$ and $f_y(x,y)$, there are 3 ways we can approach the problem to find the work done<\/p><p>a) If $f_x(x,y) = f(x)$ and $f_y(x,y) = g(y)$, then $W=\\int (f_x(x,y) dx +$ $f_y(x,y) dy)$ $=\\int_{x_i}^{x_f} f(x) dx +$ $\\int_{y_i}^{y_f} g(y) dy$<\/p><p>For example, let&#8217;s say that $\\overrightarrow{F} = $ $x \\hat{i} +$ $y \\hat{j}$, then $W=$ $=\\int_{x_i}^{x_f} x dx +$ $\\int_{y_i}^{y_f} y dy$, which you can easily solve.<\/p><p>Note that if you were given $\\overrightarrow{F} = $ $K(\\cfrac{x}{(x^2+y^2)^{1\/2}} \\hat{i} +$ $\\cfrac{y}{(x^2+y^2)^{1\/2}} \\hat{j})$ and it was also given that particle is moving along a circular path of radius &#8216;$a$&#8217;, centered at origin, then recall that $(x^2+y^2)^{1\/2} = a$ and so, we can write $\\overrightarrow{F} = $ $\\cfrac{K}{a}(x \\hat{i} +$ $y \\hat{j})$ and the rest of the steps for finding $W$ will be same as mentioned above.<\/p><p>b) If $f_x(x,y)$ and $f_y(x,y)$ are such that $f_x(x,y) dx + f_y(x,y)dy$ can be expressed as exact differential, i.e. $f_x(x,y) dx + f_y(x,y)dy$ $=d(g(x,y))$ then $W=\\int f_x(x,y) dx +$ $f_y(x,y) dy$ $=\\int_{x_i, \\ y_i}^{x_f, \\ y_f} d(g(x,y))$ $=g(x_f,y_f) &#8211; g(x_i,y_i)$<\/p><p>Recall that $\\cfrac{d(AB)}{dt} =A \\cfrac{d(B)}{dt} + B\\cfrac{d(A)}{dt}$<br \/>Or, $d(AB) = Ad(B) + Bd(A)$<\/p><p>For example, let&#8217;s say that $\\overrightarrow{F} = $ $xy^2 \\hat{i} +$ $x^2y \\hat{j}$, then $W=\\int (xy^2 dx +$ $x^2y dy)$ $=\\int_{x_i, \\ y_i}^{x_f, \\ y_f} d(\\cfrac{1}{2}x^2y^2)$ $=\\cfrac{1}{2}(x_f^2y_f^2 &#8211; x_i^2y_i^2)$<\/p><p>c) If $f_x(x,y)$ and $f_y(x,y)$ are such that we CANNOT express $f_x(x,y) dx + f_y(x,y)dy$ as exact differential and if the trajectory of the path is specified, say for example $(x-a)^2+(y-b)^2 = c^2$<\/p><p>Then, to solve $W=\\int f_x(x,y) dx +$ $f_y(x,y) dy$, we will write $y$ in terms of $x$ using the equation of trajectory for $f_x$ and write $x$ in terms of $y$ for $f_y$, then we should be able to integrate the two terms. Here you will need to be careful when writing $x$ in terms of $y$ or vice versa for the given trajectory as the expression might be different for different parts of the trajectory, so break the integral limits as appropriate and then integrate.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-73e2f46 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"73e2f46\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-33eb922\" data-id=\"33eb922\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-11de0ea elementor-widget elementor-widget-heading\" data-id=\"11de0ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Work done by a variable force | Sample Problems<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-5b0681a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"5b0681a\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-72881f7\" data-id=\"72881f7\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-dfa36de elementor-widget elementor-widget-text-editor\" data-id=\"dfa36de\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Q1 Given $\\overrightarrow{F}=$ $xy^2 \\hat {i} + x^2y \\hat{j}$ newtons, find the work done by $\\overrightarrow{F}$ when a particle is taken along the semicircular path $AB$ where the coordinates of $B$ are ($2,0$)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-894a491 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"894a491\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d453768\" data-id=\"d453768\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4b04122 elementor-widget elementor-widget-image\" data-id=\"4b04122\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" width=\"768\" height=\"497\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-768x497.png\" class=\"attachment-medium_large size-medium_large wp-image-4076\" alt=\"Work done by a variable force - Sample Problem 1\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-768x497.png 768w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-300x194.png 300w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-1024x663.png 1024w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-1536x994.png 1536w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem1-2048x1326.png 2048w\" sizes=\"(max-width: 768px) 100vw, 768px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-7e2243c elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"7e2243c\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-1ee7305\" data-id=\"1ee7305\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-648cca0 elementor-widget elementor-widget-toggle\" data-id=\"648cca0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1051\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1051\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Answer<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1051\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1051\"><p>Zero.<\/p><p>$W = \\int F_x dx + F_y dy$\u00a0<span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$=\\int (xy^2 dx +$ $x^2y dy)$ $=\\int_{x_i, \\ y_i}^{x_f, \\ y_f} d(\\cfrac{1}{2}x^2y^2)$ $=<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">\\cfrac{1}{2}(<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">x_f^2y_f^2 &#8211; x_i^2y_i^2) =\u00a0<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">\\cfrac{1}{2}(<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">4.0-0.0) = 0$<\/span><\/p><p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-c2ddd95 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"c2ddd95\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-d7c295f\" data-id=\"d7c295f\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-701c2d1 elementor-widget elementor-widget-text-editor\" data-id=\"701c2d1\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Q2. Particle moves from point A to point B along the line shown in figure under the action of force $\\overrightarrow{F}$ $=\u2212x \\hat{i} + y \\hat{j}$. Determine the work done on the particle by $\\overrightarrow{F}$ in moving the particle from point $A$ to point $B$ (all quantities are in SI units).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4e59286 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"4e59286\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-59462fa\" data-id=\"59462fa\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-f422c5c elementor-widget elementor-widget-image\" data-id=\"f422c5c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"image.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<img decoding=\"async\" width=\"684\" height=\"452\" src=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem2.jpg\" class=\"attachment-medium_large size-medium_large wp-image-4080\" alt=\"Work done by a variable force - Sample Problem 2\" srcset=\"https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem2.jpg 684w, https:\/\/acejee.com\/blog\/wp-content\/uploads\/2021\/08\/Sample-Problem2-300x198.jpg 300w\" sizes=\"(max-width: 684px) 100vw, 684px\" \/>\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-f3fbc41 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"f3fbc41\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-7c94fc0\" data-id=\"7c94fc0\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-705a8e9 elementor-widget elementor-widget-toggle\" data-id=\"705a8e9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"toggle.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle\">\n\t\t\t\t\t\t\t<div class=\"elementor-toggle-item\">\n\t\t\t\t\t<div id=\"elementor-tab-title-1171\" class=\"elementor-tab-title\" data-tab=\"1\" role=\"button\" aria-controls=\"elementor-tab-content-1171\" aria-expanded=\"false\">\n\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon elementor-toggle-icon-left\" aria-hidden=\"true\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-closed\"><i class=\"fas fa-caret-right\"><\/i><\/span>\n\t\t\t\t\t\t\t\t<span class=\"elementor-toggle-icon-opened\"><i class=\"elementor-toggle-icon-opened fas fa-caret-up\"><\/i><\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-toggle-title\" tabindex=\"0\">Answer<\/a>\n\t\t\t\t\t<\/div>\n\n\t\t\t\t\t<div id=\"elementor-tab-content-1171\" class=\"elementor-tab-content elementor-clearfix\" data-tab=\"1\" role=\"region\" aria-labelledby=\"elementor-tab-title-1171\"><p>1J<\/p><p>$W = \\int F_x dx + F_y dy$\u00a0<span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$=\\int (-x dx +$ $y dy)$ $=-\\int_{1}^{0} xdx$\u00a0<\/span><span style=\"font-family: 'PT Sans'; font-size: 15px; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: 400;\">$+\\int_{0}^{1} ydy$<\/span><\/p><p>\u00a0<\/p><\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-41fc777 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"41fc777\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-33bb34e\" data-id=\"33bb34e\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-7de705a elementor-align-center elementor-widget elementor-widget-button\" data-id=\"7de705a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"button.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<div class=\"elementor-button-wrapper\">\n\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm\" href=\"https:\/\/acejee.com\/sign-up\" target=\"_blank\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">Additional Practice Problems<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Given a two or three dimensional variable force vector, F, acting on the particle as it moves along a two or three dimensional path, the work done by F is&#8230;<\/p>\n","protected":false},"author":1,"featured_media":4031,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"0","ocean_second_sidebar":"0","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"0","ocean_custom_header_template":"0","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"0","ocean_menu_typo_font_family":"0","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"on","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"off","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"0","cybocfi_hide_featured_image":"yes","ocean_post_oembed":"","ocean_post_self_hosted_media":"","ocean_post_video_embed":"","ocean_link_format":"","ocean_link_format_target":"self","ocean_quote_format":"","ocean_quote_format_link":"post","ocean_gallery_link_images":"off","ocean_gallery_id":[],"footnotes":""},"categories":[16],"tags":[],"class_list":["post-4029","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-iit-jee-physics-chapter-wise-approach","entry","has-media"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/4029","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/comments?post=4029"}],"version-history":[{"count":1,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/4029\/revisions"}],"predecessor-version":[{"id":9440,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/posts\/4029\/revisions\/9440"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media\/4031"}],"wp:attachment":[{"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/media?parent=4029"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/categories?post=4029"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/acejee.com\/blog\/wp-json\/wp\/v2\/tags?post=4029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}