[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Ftheorem-on-motion-of-center-of-mass":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},259,"Theorem on the Motion of the Center of Mass","en","theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Ftheorem-on-motion-of-center-of-mass","Equation of motion of a mechanical system's center of mass and consequences for systems with zero resultant external force.","This topic relates the acceleration of the center of mass to the resultant external force and develops important conservation consequences.","\u003Cp>The \u003Cstrong>theorem on the motion of the center of mass\u003C\u002Fstrong> describes the translational motion of a mechanical system by a single equation in which internal forces do not appear explicitly.\u003C\u002Fp>\u003Ch2>Derivation\u003C\u002Fh2>\u003Cp>For each particle, $m_i\\vec a_i=\\vec F_i^{e}+\\vec F_i^{i}$. Summing over the system cancels the internal forces, while $\\sum m_i\\vec a_i=M\\vec a_C$.\u003C\u002Fp>\u003Ch2>Fundamental equation\u003C\u002Fh2>\u003Cp>Therefore:\u003C\u002Fp>\u003Cp>$$M\\vec a_C=\\sum\\vec F^{e}=\\vec R^{e},$$\u003C\u002Fp>\u003Cp>where $\\vec R^{e}$ is the resultant of the external forces.\u003C\u002Fp>\u003Ch2>Coordinate components\u003C\u002Fh2>\u003Cp>In Cartesian coordinates:\u003C\u002Fp>\u003Cp>$$M\\ddot x_C=\\sum F_x^{e},\\qquad M\\ddot y_C=\\sum F_y^{e},\\qquad M\\ddot z_C=\\sum F_z^{e}.$$\u003C\u002Fp>\u003Cp>These equations have the same form as the equations of motion of a particle of mass $M$.\u003C\u002Fp>\u003Ch2>Zero external resultant\u003C\u002Fh2>\u003Cp>If $\\sum\\vec F^{e}=0$, then $\\vec a_C=0$ and $\\vec v_C=const$. The center of mass is either at rest or moves uniformly in a straight line.\u003C\u002Fp>\u003Ch2>Conservation along one coordinate\u003C\u002Fh2>\u003Cp>If the sum of external-force components along one axis, say $x$, is zero, then $v_{Cx}=const$. If additionally $v_{Cx}(0)=0$, the coordinate $x_C$ remains constant.\u003C\u002Fp>\u003Ch2>Internal motions\u003C\u002Fh2>\u003Cp>Internal forces can strongly change the relative positions of system parts, but by themselves cannot change the center-of-mass motion of an isolated system. Motion of one part is therefore accompanied by compensating motion of other parts.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A horizontal external resultant of 30 N acts on a system of mass 10 kg. Regardless of internal interactions, $a_C=30\u002F10=3$ m\u002Fs² in the force direction.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>including internal forces in the center-of-mass equation after summing the complete system;\u003C\u002Fli>\u003Cli>assuming $\\sum\\vec F^e=0$ means the center of mass must be stationary rather than have constant velocity;\u003C\u002Fli>\u003Cli>applying the theorem to only part of a system without reclassifying forces;\u003C\u002Fli>\u003Cli>identifying center-of-mass motion with the motion of every system particle.\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\u002Ftheorem-on-motion-of-center-of-mass","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-mekhanichnoi-systemy\u002Fteorema-pro-rukh-tsentra-mas-systemy",1787712537343]