[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Fangular-momentum-theorem-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},262,"Angular-Momentum Theorem for a Mechanical System","en","theoretical-mechanics\u002Fdynamics\u002Fdynamics-of-mechanical-systems\u002Fangular-momentum-theorem-mechanical-system","Angular momentum of a mechanical system about a point and axis, external moments, and conservation of angular momentum.","This topic develops the angular-momentum theorem for a mechanical system and explains conservation when the resultant external moment vanishes.","\u003Cp>The \u003Cstrong>angular momentum of a mechanical system\u003C\u002Fstrong> characterizes the rotational aspect of system motion relative to a selected point or axis.\u003C\u002Fp>\u003Ch2>Angular momentum about a point\u003C\u002Fh2>\u003Cp>About a fixed point $O$:\u003C\u002Fp>\u003Cp>$$\\vec K_O=\\sum_{i=1}^{n}\\vec r_i\\times m_i\\vec v_i.$$\u003C\u002Fp>\u003Cp>Angular momentum about an axis is the projection of this vector onto that axis.\u003C\u002Fp>\u003Ch2>Angular-momentum theorem\u003C\u002Fh2>\u003Cp>For a fixed point $O$ in an inertial frame:\u003C\u002Fp>\u003Cp>$$\\frac{d\\vec K_O}{dt}=\\sum\\vec M_O^{e}.$$\u003C\u002Fp>\u003Cp>The time derivative of system angular momentum equals the resultant moment of external forces about the same point.\u003C\u002Fp>\u003Ch2>Role of internal forces\u003C\u002Fh2>\u003Cp>For central pairwise internal forces, the moments of each interaction pair cancel. Thus only the total moment of external forces remains in the equation for the complete system.\u003C\u002Fp>\u003Ch2>Theorem about an axis\u003C\u002Fh2>\u003Cp>Projecting onto a fixed $z$ axis gives:\u003C\u002Fp>\u003Cp>$$\\frac{dK_z}{dt}=\\sum M_z^{e}.$$\u003C\u002Fp>\u003Ch2>Conservation\u003C\u002Fh2>\u003Cp>If $\\sum\\vec M_O^{e}=0$, then $\\vec K_O=const$. If only the resultant external moment about a particular axis is zero, only the corresponding angular-momentum component is conserved.\u003C\u002Fp>\u003Ch2>Rigid-body rotation\u003C\u002Fh2>\u003Cp>For a rigid body rotating about a fixed principal axis $z$, $K_z=I_z\\omega$. With constant $I_z$, the equation becomes $I_z\\dot\\omega=\\sum M_z^{e}$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>If a rigid body has $I_z=2$ kg·m² and rotates at $\\omega=5$ rad\u002Fs, then $K_z=10$ kg·m²\u002Fs. With zero external moment about the axis, this value remains constant.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing angular momentum with moment of force;\u003C\u002Fli>\u003Cli>taking force moments about one point and angular momentum about another;\u003C\u002Fli>\u003Cli>using $K_z=I_z\\omega$ without checking the motion and axis conditions;\u003C\u002Fli>\u003Cli>claiming conservation of the full vector when only one external-moment component is zero.\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\u002Fangular-momentum-theorem-mechanical-system","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-mekhanichnoi-systemy\u002Fteorema-pro-zminu-kinetychnoho-momentu-systemy",1787712537670]