[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Fdifferential-equations-of-motion-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},249,"Differential Equations of Motion of a Particle","en","theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Fdifferential-equations-of-motion-of-a-particle","Particle equations of motion in vector, Cartesian, and natural-coordinate forms and their use in dynamics.","This topic develops the fundamental equation of particle dynamics and its vector, Cartesian, and natural-coordinate representations.","\u003Cp>Differential equations of motion connect particle kinematics with the forces acting on the particle. For a particle of constant mass, they follow directly from Newton's second law.\u003C\u002Fp>\u003Ch2>Vector equation\u003C\u002Fh2>\u003Cp>The fundamental equation of motion is:\u003C\u002Fp>\u003Cp>$$m\\frac{d^2\\vec r}{dt^2}=\\sum_i\\vec F_i.$$\u003C\u002Fp>\u003Cp>If forces depend on position, velocity, or time, the right-hand side may be written as $\\vec F(\\vec r,\\vec v,t)$.\u003C\u002Fp>\u003Ch2>Cartesian equations\u003C\u002Fh2>\u003Cp>Projection onto fixed coordinate axes gives:\u003C\u002Fp>\u003Cp>$$m\\ddot x=\\sum F_x,\\qquad m\\ddot y=\\sum F_y,\\qquad m\\ddot z=\\sum F_z.$$\u003C\u002Fp>\u003Cp>This is a system of second-order differential equations for the particle coordinates.\u003C\u002Fp>\u003Ch2>Natural coordinates\u003C\u002Fh2>\u003Cp>For motion along a known path, projection onto the tangent and principal normal is often convenient:\u003C\u002Fp>\u003Cp>$$m\\frac{dv}{dt}=\\sum F_\\tau,\\qquad m\\frac{v^2}{\\rho}=\\sum F_n,$$\u003C\u002Fp>\u003Cp>where $\\rho$ is the radius of curvature of the path.\u003C\u002Fp>\u003Ch2>Initial conditions\u003C\u002Fh2>\u003Cp>To determine the motion uniquely, initial position and velocity are normally specified, for example $x(t_0)=x_0$ and $\\dot x(t_0)=v_{0x}$. The integration constants are found from these conditions.\u003C\u002Fp>\u003Ch2>Choosing coordinates\u003C\u002Fh2>\u003Cp>The coordinate system should match the geometry of motion and force directions. One axis is enough for rectilinear motion; natural coordinates are often efficient for motion along a curved path.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>For vertical free fall without air resistance, with the $y$ axis directed upward, $m\\ddot y=-mg$, hence $\\ddot y=-g$. Integrating twice and applying the initial conditions gives the equation of motion.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing the sign of a force component with the force magnitude;\u003C\u002Fli>\u003Cli>omitting constraint reactions;\u003C\u002Fli>\u003Cli>integrating without applying initial conditions;\u003C\u002Fli>\u003Cli>treating $v^2\u002F\\rho$ as the total acceleration rather than its normal component.\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\u002Fdifferential-equations-of-motion-of-a-particle","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-materialnoi-tochky\u002Fdyferentsialni-rivniannia-rukhu-materialnoi-tochky",1787778342617]