[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Ftwo-fundamental-problems-of-particle-dynamics":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},250,"Two Fundamental Problems of Particle Dynamics","en","theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Ftwo-fundamental-problems-of-particle-dynamics","Direct and inverse dynamics problems: finding motion from forces and forces from a prescribed motion.","This topic explains the direct and inverse problems of particle dynamics and a systematic procedure for solving each type.","\u003Cp>Particle dynamics has two fundamental problem types. Both use the equation $m\\vec a=\\sum\\vec F$, but differ in which quantities are known and which must be determined.\u003C\u002Fp>\u003Ch2>First problem of dynamics\u003C\u002Fh2>\u003Cp>Given the particle's motion, its mass, and some of the forces, determine the unknown forces. Starting from $\\vec r(t)$, find velocity and acceleration and then use the equations of dynamics to determine the required forces or reactions.\u003C\u002Fp>\u003Ch2>Second problem of dynamics\u003C\u002Fh2>\u003Cp>Given the forces, mass, and initial conditions, determine the motion. Set up the differential equations, integrate them, and evaluate the integration constants from the initial conditions.\u003C\u002Fp>\u003Ch2>First problem: procedure\u003C\u002Fh2>\u003Col>\u003Cli>write the prescribed motion;\u003C\u002Fli>\u003Cli>differentiate the coordinates twice to obtain acceleration;\u003C\u002Fli>\u003Cli>draw the free-body diagram;\u003C\u002Fli>\u003Cli>write the components of $m\\vec a=\\sum\\vec F$;\u003C\u002Fli>\u003Cli>solve for the unknown forces.\u003C\u002Fli>\u003C\u002Fol>\u003Ch2>Second problem: procedure\u003C\u002Fh2>\u003Col>\u003Cli>isolate the particle and identify the forces;\u003C\u002Fli>\u003Cli>choose coordinates;\u003C\u002Fli>\u003Cli>form the differential equations of motion;\u003C\u002Fli>\u003Cli>integrate them;\u003C\u002Fli>\u003Cli>apply the initial conditions;\u003C\u002Fli>\u003Cli>check the resulting motion and units.\u003C\u002Fli>\u003C\u002Fol>\u003Ch2>Example of the first problem\u003C\u002Fh2>\u003Cp>If motion along an axis is prescribed by $x=2t^2$ m for a 3 kg particle, then $a_x=4$ m\u002Fs² and the required resultant force is $F_x=ma_x=12$ N.\u003C\u002Fp>\u003Ch2>Example of the second problem\u003C\u002Fh2>\u003Cp>If a constant force $F$ acts along an axis on a particle of mass $m$, then $\\ddot x=F\u002Fm$. With $x(0)=x_0$ and $\\dot x(0)=v_0$, integration gives $x=x_0+v_0t+Ft^2\u002F(2m)$.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>not distinguishing what is prescribed in the two problem types;\u003C\u002Fli>\u003Cli>omitting initial conditions in the second problem;\u003C\u002Fli>\u003Cli>trying to determine force from velocity instead of acceleration;\u003C\u002Fli>\u003Cli>omitting unknown constraint reactions from the equations of motion.\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},293,"Particle Dynamics","theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics",{"id":5,"name":6,"path":8},[],[],{"en":30,"uk":31},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Ftwo-fundamental-problems-of-particle-dynamics","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-materialnoi-tochky\u002Fdvi-osnovni-zadachi-dynamiky-materialnoi-tochky",1787712537019]