{"id":798234,"date":"2023-11-28T13:31:21","date_gmt":"2023-11-28T08:01:21","guid":{"rendered":"https:\/\/leverageedu.com\/discover\/?p=798234"},"modified":"2023-11-28T13:31:21","modified_gmt":"2023-11-28T08:01:21","slug":"derivation-of-work-energy-theorem","status":"publish","type":"post","link":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/","title":{"rendered":"Derivation of Work-Energy Theorem"},"content":{"rendered":"\n<p>We do various activities in our daily life, sitting and reading a newspaper is also doing some work. According to <strong>Physics<\/strong>, any work done that does not involve the displacement of the body is not considered work. Thus, to describe the phenomenon of work energy we will derive the work energy theorem. The <a href=\"https:\/\/leverageedu.com\/blog\/work-energy-and-power-class-11\/#:~:text=T%20%3D%20Time%20taken-,Definition%20of%20Work%20Energy%20Theorem,defined%20as%20Work%2DEnergy%20Theorem.\"><strong>work-energy<\/strong><\/a> theorem explains the reasons behind the physics of no work. Stay tuned and continue reading this article to get the derivation of the work-energy theorem!<\/p>\n\n\n\n<p class=\"has-background\" style=\"background:linear-gradient(135deg,rgb(254,205,165) 51%,rgb(254,45,45) 100%,rgb(107,0,62) 100%)\"><strong>Also Read: <\/strong><strong>Class 11 Physics Chapter 3 \u2013 Motion in a Straight Line<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-what-is-work\">What is Work?<\/h2>\n\n\n\n<p>Work is defined as the product of force and displacement. It means when the force acting on an object displaces the object from its original position, then it is said to be work done.\u00a0<\/p>\n\n\n\n<p>You must also know that moving objects possess <strong>kinetic energy<\/strong>. Thus, the work-energy theorem gives the relationship between work and kinetic energy.<\/p>\n\n\n\n<p>Work is the change in kinetic energy.<\/p>\n\n\n\n<p class=\"has-black-color has-text-color has-background has-medium-font-size\" style=\"background-color:#8dd2fc38\"><strong>W= \u0394K<\/strong><\/p>\n\n\n\n<p class=\"has-background\" style=\"background:linear-gradient(135deg,rgb(254,205,165) 32%,rgb(254,45,45) 100%,rgb(107,0,62) 100%)\"><strong>Also Read: <\/strong><strong>Class 9 Motion\u00a0<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-work-energy-theorem\">Work-Energy Theorem<\/h2>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Work Energy Theorem: Block on a Slope\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/dlkiGqQ2Neg?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe>\n<\/div><figcaption class=\"wp-element-caption\">Source: Physics Ninja<\/figcaption><\/figure>\n\n\n\n<p>The work-energy theorem is defined as the work done by the net force applied on an object or body is directly proportional to the change in kinetic energy.<\/p>\n\n\n\n<p>The formula of the Work-Energy Theorem is\u00a0<\/p>\n\n\n\n<p class=\"has-black-color has-text-color has-background has-medium-font-size\" style=\"background-color:#8dd2fc38\"><strong>Kf \u2013 Ki = W<\/strong><\/p>\n\n\n\n<p>Here,\u00a0<\/p>\n\n\n\n<p>Kf= Final kinetic energy<\/p>\n\n\n\n<p>Ki= Initial Kinetic energy<\/p>\n\n\n\n<p>Kf \u2013 Ki is a change in Kinetic energy<\/p>\n\n\n\n<p>W= Work done on an object<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-derivation-of-work-energy-theorem\">Derivation of Work-Energy Theorem<\/h2>\n\n\n\n<p>Equations of motion,<\/p>\n\n\n\n<p>v<sup>2<\/sup> = u<sup>2<\/sup> + 2as<\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<p>v = final velocity of an object<\/p>\n\n\n\n<p>u = initial velocity of an object<\/p>\n\n\n\n<p>a = constant acceleration<\/p>\n\n\n\n<p>s = displacement of the object<\/p>\n\n\n\n<p>The above equation can also be written as:<\/p>\n\n\n\n<p>v<sup>2<\/sup> \u2013 u<sup>2 <\/sup>= 2as<\/p>\n\n\n\n<p>Substituting the values of the vector quantities, the equation would be:<\/p>\n\n\n\n<p>v<sup>2<\/sup> \u2013 u<sup>2 <\/sup>= 2*a*d<\/p>\n\n\n\n<p>Now, multiply both sides of the equation by m\/2, we get:<\/p>\n\n\n\n<p>\u00bd mv<sup>2 <\/sup>\u2013 \u00bd mu<sup>2<\/sup> = ma*d<\/p>\n\n\n\n<p>According to Newton\u2019s second <strong>law of motion<\/strong>, F= ma,\u00a0<\/p>\n\n\n\n<p>Now, the above equation can be written as:<\/p>\n\n\n\n<p>\u00bd mv<sup>2 <\/sup>\u2013 \u00bd mu<sup>2<\/sup> = F*d<\/p>\n\n\n\n<p>We also know that W= F.d and, Kinetic energy = (mv\u00b2)\/2,<\/p>\n\n\n\n<p>Thus the equation becomes:<\/p>\n\n\n\n<p class=\"has-background has-medium-font-size\" style=\"background-color:#8dd2fc5e\"><strong>K<sub>f <\/sub>\u2013 K<sub>i <\/sub>= W<\/strong><\/p>\n\n\n\n<p>Hence, we have:<\/p>\n\n\n\n<p>\u0394K = W<\/p>\n\n\n\n<p>Here \u0394K = K<sub>f <\/sub>\u2013 K<sub>i<\/sub>\u00a0<\/p>\n\n\n\n<p>\u0394K= Change in kinetic energy<\/p>\n\n\n\n<p class=\"has-background\" style=\"background:linear-gradient(135deg,rgb(254,205,165) 42%,rgb(254,45,45) 100%,rgb(107,0,62) 100%)\"><strong>Also Read: <\/strong><a href=\"https:\/\/leverageedu.com\/blog\/basic-physics-formulas\/\"><strong>Basic Physics Formulas and Notes for Competitive Exams\u00a0<\/strong><\/a><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-derivation-of-work-energy-theorem-for-variable-force\">Derivation of Work-Energy Theorem for Variable Force<\/h2>\n\n\n\n<p>Here is the proof of the work-energy theorem for variable force:<\/p>\n\n\n\n<p>F= Variable Force<\/p>\n\n\n\n<p>t= Time<\/p>\n\n\n\n<p>Work Done by Force is given by:<\/p>\n\n\n\n<p>W= \u222b<sup>X1<\/sup><sub>X0\u00a0 <\/sub>F* dx \u2014equation 1<\/p>\n\n\n\n<p>Here,\u00a0<\/p>\n\n\n\n<p>x<sub>0<\/sub>= initial position<\/p>\n\n\n\n<p>x<sub>1 <\/sub>= Final Position<\/p>\n\n\n\n<p>We know that,\u00a0<\/p>\n\n\n\n<p>Kinetic energy KE= \u00bd *mv<sup>2.<\/sup><\/p>\n\n\n\n<p>This means that when x<sub>0 <\/sub>then K<sub>0 <\/sub>and when x<sub>1 <\/sub>then K<sub>1<\/sub>.<\/p>\n\n\n\n<p>Now, differentiate K w.r.t time (t):<\/p>\n\n\n\n<p>dK\/dt= m*v* dv\/dt<\/p>\n\n\n\n<p>dK\/dt=m*a*dx\/dt<\/p>\n\n\n\n<p>dK\/dt= F*dx\/dt<\/p>\n\n\n\n<p>dK= F*dx<\/p>\n\n\n\n<p class=\"has-background has-medium-font-size\" style=\"background-color:#8dd2fc4d\"><strong>\u222b<sub>K0<\/sub><sup>K1<\/sup> dK = \u222b<sub>x0<\/sub><sup>x1<\/sup> F* dx<\/strong><\/p>\n\n\n\n<p>\u0394K = W<\/p>\n\n\n\n<p>Hence Proved.<\/p>\n\n\n\n<p class=\"has-background\" style=\"background:linear-gradient(135deg,rgb(254,205,165) 34%,rgb(254,45,45) 100%,rgb(107,0,62) 100%)\"><strong>Also Read: <\/strong><strong>Laws of Motion Class 11<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-derivation-of-work-energy-theorem-for-constant-force\">Derivation of Work-Energy Theorem for Constant Force<\/h2>\n\n\n\n<p>From Newton\u2019s Second Law of Motion: F = ma<\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<p>a = acceleration of the object<\/p>\n\n\n\n<p>The velocity of the object increases i.e. v<sub>1<\/sub> to v<sub>2<\/sub> by applying acceleration, and the object gets displaced by a distance d.<\/p>\n\n\n\n<p>Thus the equation becomes:<\/p>\n\n\n\n<p>v<sub>2<\/sub><sup>2<\/sup>\u2212v<sub>1<\/sub><sup>2<\/sup>= 2*a*d<\/p>\n\n\n\n<p>We can also write it as:<\/p>\n\n\n\n<p>a = (v<sub>2<\/sub><sup>2<\/sup>\u2212v<sub>1<\/sub><sup>2<\/sup>)\/2d, or<\/p>\n\n\n\n<p>Now substituting the value of a in F= ma<\/p>\n\n\n\n<p>F = m (v<sub>2<\/sub><sup>2<\/sup>\u2212v<sub>1<\/sub><sup>2<\/sup>)\/2d, or<\/p>\n\n\n\n<p>Fd = m (v<sub>2<\/sub><sup>2<\/sup>\u2212v<sub>1<\/sub><sup>2<\/sup>)\/2d, or<\/p>\n\n\n\n<p>Fd = \u00bd m*v<sub>2<\/sub><sup>2<\/sup> \u2013 \u00bd m*v<sub>1<\/sub><sup>2<\/sup> \u2014Equation 1<\/p>\n\n\n\n<p>Fd is the work done by the force F to move the object through a distance d.<\/p>\n\n\n\n<p>In equation 1, the quantity<\/p>\n\n\n\n<p>K<sub>2 <\/sub>= m*v<sub>2<\/sub><sup>2<\/sup>\/2 is the final Kinetic energy of the object.<\/p>\n\n\n\n<p>K<sub>1<\/sub>=m*v<sub>1<\/sub><sup>2<\/sup>\/2 is the initial Kinetic of the object.<\/p>\n\n\n\n<p>Thus equation (1) becomes<\/p>\n\n\n\n<p class=\"has-background has-medium-font-size\" style=\"background-color:#8dd2fc61\"><strong>W=K<sub>2<\/sub>-K<sub>1<\/sub>=\u0394K- Equation 2<\/strong><\/p>\n\n\n\n<p>Here,<\/p>\n\n\n\n<p>\u0394K = change in Kinetic energy of the object.<\/p>\n\n\n\n<p>From equation 2, it is clear that the work done by a force is equal to the change in kinetic energy.<\/p>\n\n\n\n<p class=\"has-text-align-center has-vivid-red-color has-text-color has-medium-font-size\"><strong>Relevant Blogs<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/who-is-the-father-of-modern-politics\/\"><strong>Who is Known as the Father of Modern Politics?<\/strong><\/a><\/td><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/gk-quiz-for-class-5\/\"><strong>GK Quiz for Class 5<\/strong><\/a><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/what-is-the-total-weight-of-chandrayaan-3\/\"><strong>What is the Total Weight of Chandrayaan-3?<\/strong><\/a><\/td><td class=\"has-text-align-center\" data-align=\"center\"><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/world-science-day-for-peace-and-development\/\"><strong>World Science Day for Peace and Development\u00a0<\/strong><\/a><\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\"><strong><\/strong><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/national-science-day\/\"><strong>National Science Day<\/strong><\/a><\/td><td class=\"has-text-align-center\" data-align=\"center\"><strong><\/strong><a href=\"https:\/\/leverageedu.com\/discover\/trending-events\/national-techies-day\/\"><strong>National Techies Day<\/strong><\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"h-faqs\">FAQs<\/h2>\n\n\n\n<div class=\"schema-faq wp-block-yoast-faq-block\"><div class=\"schema-faq-section\" id=\"faq-question-1701158164159\"><strong class=\"schema-faq-question\">What is the formula of the Work-Energy Theorem?<\/strong> <p class=\"schema-faq-answer\">The Formula for the Work-Energy Theorem is W= Kf \u2013 Ki\u00a0<br\/>Here,\u00a0Kf= Final kinetic energy, Ki= Initial Kinetic energy<br\/>Kf \u2013 Ki is a change in Kinetic energy<br\/>W= Work done on an object<\/p> <\/div> <div class=\"schema-faq-section\" id=\"faq-question-1701158210532\"><strong class=\"schema-faq-question\">What is the statement of the Work-Energy Theorem?<\/strong> <p class=\"schema-faq-answer\">The work-energy theorem states that the total work done on an object by the application of force is equal to the change in its kinetic energy.<\/p> <\/div> <div class=\"schema-faq-section\" id=\"faq-question-1701158235691\"><strong class=\"schema-faq-question\">What is the derivation of the work equation?<\/strong> <p class=\"schema-faq-answer\">The derivation of the work equation is W= F*d. Here\u00a0 W= work and it is the product of Force (F) and Displacement (d).<\/p> <\/div> <\/div>\n\n\n\n<p>For more information about such informative articles, make sure to check the <a href=\"https:\/\/leverageedu.com\/discover\/category\/trending-events\/\"><strong>trending events<\/strong> <\/a>page of <a href=\"https:\/\/leverageedu.com\/\"><strong>Leverage Edu<\/strong><\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"We do various activities in our daily life, sitting and reading a newspaper is also doing some work.&hellip;\n","protected":false},"author":97,"featured_media":798236,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"editor_notices":[],"footnotes":""},"categories":[400],"tags":[],"class_list":{"0":"post-798234","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-trending-events"},"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v27.5 (Yoast SEO v27.5) - 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I have 2-years of writing experience in Ed-tech (digital marketing) company. I am passionate towards writing blogs and am on the path of discovering true potential professionally in the field of content marketing. I am engaged in writing creative content for students which is simple yet creative and engaging and leaves an impact on the reader's mind.","sameAs":["https:\/\/www.linkedin.com\/in\/kajal-thareja-6b600a191"],"honorificPrefix":"Ms","birthDate":"1998-11-08","gender":"Female","url":"https:\/\/leverageedu.com\/discover\/author\/kajal\/"},{"@type":"Question","@id":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158164159","position":1,"url":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158164159","name":"What is the formula of the Work-Energy Theorem?","answerCount":1,"acceptedAnswer":{"@type":"Answer","text":"The Formula for the Work-Energy Theorem is W= Kf \u2013 Ki\u00a0<br\/>Here,\u00a0Kf= Final kinetic energy, Ki= Initial Kinetic energy<br\/>Kf \u2013 Ki is a change in Kinetic energy<br\/>W= Work done on an object","inLanguage":"en-US"},"inLanguage":"en-US"},{"@type":"Question","@id":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158210532","position":2,"url":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158210532","name":"What is the statement of the Work-Energy Theorem?","answerCount":1,"acceptedAnswer":{"@type":"Answer","text":"The work-energy theorem states that the total work done on an object by the application of force is equal to the change in its kinetic energy.","inLanguage":"en-US"},"inLanguage":"en-US"},{"@type":"Question","@id":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158235691","position":3,"url":"https:\/\/leverageedu.com\/discover\/trending-events\/derivation-of-work-energy-theorem\/#faq-question-1701158235691","name":"What is the derivation of the work equation?","answerCount":1,"acceptedAnswer":{"@type":"Answer","text":"The derivation of the work equation is W= F*d. Here\u00a0 W= work and it is the product of Force (F) and Displacement (d).","inLanguage":"en-US"},"inLanguage":"en-US"}]}},"acf":[],"_links":{"self":[{"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/posts\/798234","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/users\/97"}],"replies":[{"embeddable":true,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/comments?post=798234"}],"version-history":[{"count":0,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/posts\/798234\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/media\/798236"}],"wp:attachment":[{"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/media?parent=798234"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/categories?post=798234"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/leverageedu.com\/discover\/wp-json\/wp\/v2\/tags?post=798234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}