[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fconservation-of-mechanical-energy":3},{"topic":4,"trail":13,"children":27,"tasks":28,"alternates":29},{"id":5,"name":6,"locale":7,"path":8,"seo_title":6,"seo_description":9,"seo_text":10,"content_html":11,"content_chunks":12},257,"Conservation of Mechanical Energy","en","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fconservation-of-mechanical-energy","Conservation of the sum of kinetic and potential energies in conservative mechanical systems.","This topic explains the conditions for conservation of mechanical energy and the use of energy balance in dynamics problems.","\u003Cp>The \u003Cstrong>mechanical energy\u003C\u002Fstrong> of a system is the sum of its kinetic and potential energies: $E=T+\\Pi$. For a system acted on only by conservative forces, this sum remains constant.\u003C\u002Fp>\u003Ch2>Conservation law\u003C\u002Fh2>\u003Cp>If the work of all nonconservative forces is zero, then:\u003C\u002Fp>\u003Cp>$$T_1+\\Pi_1=T_2+\\Pi_2=const.$$\u003C\u002Fp>\u003Cp>Kinetic and potential energy may transform into each other while their sum remains unchanged.\u003C\u002Fp>\u003Ch2>Energy balance\u003C\u002Fh2>\u003Cp>More generally, the change in mechanical energy equals the work of nonconservative forces:\u003C\u002Fp>\u003Cp>$$E_2-E_1=A_{nc}.$$\u003C\u002Fp>\u003Cp>For example, negative work of dry friction reduces mechanical energy.\u003C\u002Fp>\u003Ch2>Gravitational system\u003C\u002Fh2>\u003Cp>For a particle moving in a uniform gravitational field without resistance, $mv^2\u002F2+mgh=const$. A decrease in height is accompanied by an increase in kinetic energy.\u003C\u002Fp>\u003Ch2>Elastic system\u003C\u002Fh2>\u003Cp>For a mass attached to an ideal spring with no losses, $mv^2\u002F2+kx^2\u002F2=const$. At extreme positions the speed may be zero while elastic potential energy is maximum.\u003C\u002Fp>\u003Ch2>Choice of potential-energy reference\u003C\u002Fh2>\u003Cp>The conservation law is independent of the chosen zero level of $\\Pi$, provided the same reference is used consistently in all states.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A body falls from rest through 5 m without resistance. Taking $\\Pi=0$ at the lower level gives $mgh=mv^2\u002F2$, so $v=\\sqrt{2gh}\\approx9.90$ m\u002Fs for $g=9.81$ m\u002Fs².\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>applying mechanical-energy conservation while omitting friction work;\u003C\u002Fli>\u003Cli>mixing different zero levels of potential energy;\u003C\u002Fli>\u003Cli>assuming kinetic and potential energy are separately constant;\u003C\u002Fli>\u003Cli>confusing conservation of mechanical energy with conservation of total energy of the physical system.\u003C\u002Fli>\u003C\u002Ful>",[],[14,18,22,26],{"id":15,"name":16,"path":17},81,"Theoretical Mechanics","theoretical-mechanics",{"id":19,"name":20,"path":21},138,"Dynamics","theoretical-mechanics\u002Fdynamics",{"id":23,"name":24,"path":25},294,"General Theorems of Particle Dynamics","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics",{"id":5,"name":6,"path":8},[],[],{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fconservation-of-mechanical-energy","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fzahalni-teoremy-dynamiky-materialnoi-tochky\u002Fzakon-zberezhennia-mekhanichnoi-enerhii",1787712537284]