[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Fkinetic-energy-of-a-mechanical-system":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},263,"Kinetic Energy of a Mechanical System","en","theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Fkinetic-energy-of-a-mechanical-system","Kinetic energy of particle systems and rigid bodies in translation, fixed-axis rotation, and plane motion.","This topic covers kinetic energy of a mechanical system and the standard expressions for the principal types of rigid-body motion.","\u003Cp>The \u003Cstrong>kinetic energy of a mechanical system\u003C\u002Fstrong> is the sum of the kinetic energies of all its particles. It is a scalar quantity, so the energies of individual parts are added algebraically.\u003C\u002Fp>\u003Ch2>General definition\u003C\u002Fh2>\u003Cp>For a system of $n$ particles:\u003C\u002Fp>\u003Cp>$$T=\\sum_{i=1}^{n}\\frac{m_iv_i^2}{2}.$$\u003C\u002Fp>\u003Cp>System kinetic energy is nonnegative and is measured in joules.\u003C\u002Fp>\u003Ch2>König's theorem\u003C\u002Fh2>\u003Cp>The kinetic energy of a system can be decomposed into the kinetic energy of translational motion of its center of mass and the kinetic energy of motion relative to the center of mass:\u003C\u002Fp>\u003Cp>$$T=\\frac{Mv_C^2}{2}+T_C.$$\u003C\u002Fp>\u003Cp>This decomposition is especially useful for rigid bodies.\u003C\u002Fp>\u003Ch2>Rigid-body translation\u003C\u002Fh2>\u003Cp>In pure translation, all points have the same velocity, so:\u003C\u002Fp>\u003Cp>$$T=\\frac{Mv_C^2}{2}.$$\u003C\u002Fp>\u003Ch2>Rotation about a fixed axis\u003C\u002Fh2>\u003Cp>If a rigid body rotates with angular velocity $\\omega$ about a fixed $z$ axis:\u003C\u002Fp>\u003Cp>$$T=\\frac{I_z\\omega^2}{2},$$\u003C\u002Fp>\u003Cp>where $I_z$ is the mass moment of inertia about the rotation axis.\u003C\u002Fp>\u003Ch2>Plane motion\u003C\u002Fh2>\u003Cp>For plane motion of a rigid body:\u003C\u002Fp>\u003Cp>$$T=\\frac{Mv_C^2}{2}+\\frac{I_C\\omega^2}{2},$$\u003C\u002Fp>\u003Cp>where $I_C$ is the mass moment of inertia about the axis through the center of mass perpendicular to the plane of motion.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A 4 kg disk has center-of-mass speed 3 m\u002Fs, central mass moment of inertia 0.5 kg·m², and angular speed 4 rad\u002Fs. Then $T=4\\cdot3^2\u002F2+0.5\\cdot4^2\u002F2=18+4=22$ J.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>using only translational energy for a body that is also rotating;\u003C\u002Fli>\u003Cli>using a mass moment of inertia about the wrong axis;\u003C\u002Fli>\u003Cli>adding velocities instead of kinetic energies;\u003C\u002Fli>\u003Cli>forgetting that relative kinetic energy is zero in pure translation.\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},295,"Dynamics of Mechanical Systems","theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems",{"id":5,"name":6,"path":8},[],[],{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Fkinetic-energy-of-a-mechanical-system","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-mekhanichnoi-systemy\u002Fkinetychna-enerhiia-mekhanichnoi-systemy",1787712537702]