[{"data":1,"prerenderedAt":32},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Fparticle-motion-under-typical-forces":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},251,"Particle Motion Under Typical Forces","en","theoretical-mechanics\u002Fdynamics\u002Fparticle-dynamics\u002Fparticle-motion-under-typical-forces","Particle dynamics under gravity, elastic force, dry friction, and resistance of a medium.","This topic examines particle-motion models under common forces: gravity, elasticity, dry friction, and resistance of a medium.","\u003Cp>Dynamics problems repeatedly use several standard force models. Writing these forces correctly and choosing consistent directions makes the differential equation of motion much easier to formulate.\u003C\u002Fp>\u003Ch2>Gravity\u003C\u002Fh2>\u003Cp>Near Earth's surface, gravity is commonly treated as constant: $\\vec F_g=m\\vec g$. If air resistance is neglected, vertical motion has constant acceleration $g$ directed downward.\u003C\u002Fp>\u003Ch2>Elastic force\u003C\u002Fh2>\u003Cp>For a linear spring within Hooke's-law behavior, the restoring force is proportional to deformation and opposite to it:\u003C\u002Fp>\u003Cp>$$F_s=-kx.$$\u003C\u002Fp>\u003Cp>With no other variable forces, $m\\ddot x+kx=0$ describes free harmonic oscillation.\u003C\u002Fp>\u003Ch2>Dry sliding friction\u003C\u002Fh2>\u003Cp>In the simple Coulomb model, the sliding-friction magnitude is $F_f=\\mu N$, and its direction opposes relative sliding velocity. The sign of its component must be chosen from the actual direction of motion.\u003C\u002Fp>\u003Ch2>Resistance of a medium\u003C\u002Fh2>\u003Cp>At relatively low speeds, a linear model $\\vec R=-c\\vec v$ is often used. In other regimes, a quadratic model with resistance magnitude proportional to $v^2$ may be appropriate. The problem statement must specify or justify the model.\u003C\u002Fp>\u003Ch2>Motion on an inclined plane\u003C\u002Fh2>\u003Cp>With an axis along a plane inclined by $\\alpha$ to the horizontal, the gravity component along the plane has magnitude $mg\\sin\\alpha$, while the normal component is $mg\\cos\\alpha$. If no other normal forces act, $N=mg\\cos\\alpha$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A body slides down an incline with friction coefficient $\\mu$. Taking positive direction down the plane gives $ma=mg\\sin\\alpha-\\mu mg\\cos\\alpha$, hence $a=g(\\sin\\alpha-\\mu\\cos\\alpha)$.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>automatically directing friction opposite to the coordinate axis rather than opposite to sliding;\u003C\u002Fli>\u003Cli>writing spring force without its restoring direction;\u003C\u002Fli>\u003Cli>assuming $N=mg$ for every geometry;\u003C\u002Fli>\u003Cli>mixing linear and quadratic resistance models.\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\u002Fparticle-motion-under-typical-forces","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fdynamika-materialnoi-tochky\u002Frukh-materialnoi-tochky-pid-diieiu-typovykh-syl",1787712537044]