[{"data":1,"prerenderedAt":33},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fkinetic-energy-of-a-particle-work-energy-theorem":3},{"topic":4,"trail":14,"children":28,"tasks":29,"alternates":30},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},255,"Kinetic Energy of a Particle. Work-Energy Theorem","en","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fkinetic-energy-of-a-particle-work-energy-theorem","Kinetic Energy of a Particle and Work-Energy Theorem","Particle kinetic energy and the work-energy theorem relating its change to the work of forces.","This topic explains particle kinetic energy and the work-energy theorem and shows how it is used to solve dynamics problems.","\u003Cp>\u003Cstrong>Kinetic energy\u003C\u002Fstrong> is a scalar measure of the mechanical motion of a particle. Unlike linear momentum, it does not depend on the direction of velocity.\u003C\u002Fp>\u003Ch2>Kinetic energy of a particle\u003C\u002Fh2>\u003Cp>For a particle of mass $m$ moving at speed $v$:\u003C\u002Fp>\u003Cp>$$T=\\frac{mv^2}{2}.$$\u003C\u002Fp>\u003Cp>Kinetic energy is nonnegative and is measured in joules in SI.\u003C\u002Fp>\u003Ch2>Differential form of the theorem\u003C\u002Fh2>\u003Cp>From $m\\vec a=\\sum\\vec F$ and $d\\vec r=\\vec vdt$:\u003C\u002Fp>\u003Cp>$$dT=\\sum\\vec F\\cdot d\\vec r=\\sum dA.$$\u003C\u002Fp>\u003Cp>Thus, the elementary change in kinetic energy equals the sum of the elementary works of the forces.\u003C\u002Fp>\u003Ch2>Integral form\u003C\u002Fh2>\u003Cp>Between positions 1 and 2:\u003C\u002Fp>\u003Cp>$$T_2-T_1=\\sum A_{1\\to2}.$$\u003C\u002Fp>\u003Cp>This is the work-energy theorem for a particle.\u003C\u002Fp>\u003Ch2>Advantages of the energy method\u003C\u002Fh2>\u003Cp>The method directly relates speed and position and often avoids solving for time. Reactions of ideal fixed constraints that are perpendicular to displacement do no work and therefore need not appear in the energy equation.\u003C\u002Fp>\u003Ch2>Sign of work and change of speed\u003C\u002Fh2>\u003Cp>Positive net work increases kinetic energy, while negative net work decreases it. However, because kinetic energy depends on $v^2$, the theorem determines speed rather than the direction of the velocity vector.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A 2 kg body starts from rest and the total work of all forces over a displacement is 36 J. Then $mv^2\u002F2=36$, giving $v=6$ m\u002Fs.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>assigning kinetic energy a sign based on velocity direction;\u003C\u002Fli>\u003Cli>including the work of only one force instead of the total work;\u003C\u002Fli>\u003Cli>confusing the work-energy theorem with conservation of mechanical energy;\u003C\u002Fli>\u003Cli>using the energy equation to determine the velocity direction directly.\u003C\u002Fli>\u003C\u002Ful>",[],[15,19,23,27],{"id":16,"name":17,"path":18},81,"Theoretical Mechanics","theoretical-mechanics",{"id":20,"name":21,"path":22},138,"Dynamics","theoretical-mechanics\u002Fdynamics",{"id":24,"name":25,"path":26},294,"General Theorems of Particle Dynamics","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics",{"id":5,"name":6,"path":8},[],[],{"en":31,"uk":32},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fkinetic-energy-of-a-particle-work-energy-theorem","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fzahalni-teoremy-dynamiky-materialnoi-tochky\u002Fkinetychna-enerhiia-tochky-teorema-pro-zminu",1787712537247]