[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fangular-momentum-of-a-particle":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},253,"Angular Momentum of a Particle","en","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fangular-momentum-of-a-particle","Angular momentum of a particle about a point and an axis and the angular-momentum theorem.","This topic covers angular momentum of a particle and relates its rate of change to the moment of the applied force.","\u003Cp>\u003Cstrong>Angular momentum of a particle\u003C\u002Fstrong> characterizes the rotational aspect of particle motion relative to a selected point or axis.\u003C\u002Fp>\u003Ch2>Angular momentum about a point\u003C\u002Fh2>\u003Cp>For particle $M$ relative to point $O$:\u003C\u002Fp>\u003Cp>$$\\vec L_O=\\vec r\\times m\\vec v,$$\u003C\u002Fp>\u003Cp>where $\\vec r=\\overrightarrow{OM}$. The direction of $\\vec L_O$ follows from the right-hand rule.\u003C\u002Fp>\u003Ch2>Magnitude\u003C\u002Fh2>\u003Cp>The magnitude is $L_O=mvr\\sin\\theta=mv h$, where $h$ is the perpendicular distance from point $O$ to the velocity line.\u003C\u002Fp>\u003Ch2>Angular momentum about an axis\u003C\u002Fh2>\u003Cp>Angular momentum about an axis is the projection of $\\vec L_O$ onto that axis. For the $z$ axis, $L_z=(\\vec r\\times m\\vec v)\\cdot\\vec e_z$.\u003C\u002Fp>\u003Ch2>Angular-momentum theorem\u003C\u002Fh2>\u003Cp>For a fixed point $O$ in an inertial frame:\u003C\u002Fp>\u003Cp>$$\\frac{d\\vec L_O}{dt}=\\vec M_O,$$\u003C\u002Fp>\u003Cp>where $\\vec M_O=\\vec r\\times\\sum\\vec F$ is the moment of the resultant force about $O$.\u003C\u002Fp>\u003Ch2>Conservation\u003C\u002Fh2>\u003Cp>If the resultant external moment about the point is zero, then $\\vec L_O=const$. Likewise, if the sum of moments about a fixed axis is zero, the corresponding component of angular momentum is conserved.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A 2 kg particle moves at 4 m\u002Fs perpendicular to a 0.5 m position vector. Its angular momentum magnitude is $L_O=mvr=2\\cdot4\\cdot0.5=4$ kg·m²\u002Fs.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing angular momentum with moment of force;\u003C\u002Fli>\u003Cli>reversing the cross-product order;\u003C\u002Fli>\u003Cli>using the full distance $r$ instead of the perpendicular lever arm $h$ when the vectors are not perpendicular;\u003C\u002Fli>\u003Cli>applying conservation without checking the external moment.\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\u002Fangular-momentum-of-a-particle","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fzahalni-teoremy-dynamiky-materialnoi-tochky\u002Fmoment-kilkosti-rukhu-tochky",1787712537180]