[{"data":1,"prerenderedAt":33},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fpotential-force-field-potential-energy":3},{"topic":4,"trail":14,"children":28,"tasks":29,"alternates":30},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},256,"Potential Force Field. Potential Energy","en","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fpotential-force-field-potential-energy","Potential Force Field and Potential Energy","Conservative forces, potential force fields, potential energy, and the relation between work and potential-energy change.","This topic introduces conservative forces, potential force fields, and potential energy and relates force work to changes in potential energy.","\u003Cp>A \u003Cstrong>potential force field\u003C\u002Fstrong> is one in which the work done between two positions is independent of the path and depends only on the initial and final positions. Such forces are called conservative.\u003C\u002Fp>\u003Ch2>Potential energy\u003C\u002Fh2>\u003Cp>For a conservative force, potential energy $\\Pi$ is defined so that:\u003C\u002Fp>\u003Cp>$$A_{1\\to2}=\\Pi_1-\\Pi_2=-\\Delta\\Pi.$$\u003C\u002Fp>\u003Cp>The zero level of potential energy is arbitrary; only differences in potential energy have physical significance.\u003C\u002Fp>\u003Ch2>Force and potential energy\u003C\u002Fh2>\u003Cp>In three dimensions, a conservative force is related to potential energy by $\\vec F=-\\nabla\\Pi$. In one-dimensional motion this becomes $F_x=-d\\Pi\u002Fdx$.\u003C\u002Fp>\u003Ch2>Gravity near Earth's surface\u003C\u002Fh2>\u003Cp>With height $h$ measured upward, gravitational potential energy may be written $\\Pi_g=mgh+C$. Choosing zero potential at $h=0$ gives $\\Pi_g=mgh$.\u003C\u002Fp>\u003Ch2>Elastic force\u003C\u002Fh2>\u003Cp>For a linear spring with $F_x=-kx$, the elastic potential energy is:\u003C\u002Fp>\u003Cp>$$\\Pi_s=\\frac{kx^2}{2}+C.$$\u003C\u002Fp>\u003Cp>It is common to choose $\\Pi_s=0$ at $x=0$.\u003C\u002Fp>\u003Ch2>Properties of conservative forces\u003C\u002Fh2>\u003Cp>The work of a conservative force around any closed path is zero. Dry sliding friction and most resistance models are nonconservative because their work depends on the path traveled.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A 3 kg body descends by 2 m. Its change in gravitational potential energy is $\\Delta\\Pi=-3g\\cdot2$, while the work of gravity is $A_g=6g\\approx58.9$ J for $g=9.81$ m\u002Fs².\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>confusing work of a conservative force with change in potential energy; their signs are opposite;\u003C\u002Fli>\u003Cli>treating the absolute value of potential energy as unique without choosing a reference level;\u003C\u002Fli>\u003Cli>assigning potential energy to dry friction;\u003C\u002Fli>\u003Cli>omitting the minus sign in $\\vec F=-\\nabla\\Pi$.\u003C\u002Fli>\u003C\u002Ful>",[],[15,19,23,27],{"id":16,"name":17,"path":18},81,"Theoretical Mechanics","theoretical-mechanics",{"id":20,"name":21,"path":22},138,"Dynamics","theoretical-mechanics\u002Fdynamics",{"id":24,"name":25,"path":26},294,"General Theorems of Particle Dynamics","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics",{"id":5,"name":6,"path":8},[],[],{"en":31,"uk":32},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Ftheoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fpotential-force-field-potential-energy","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fzahalni-teoremy-dynamiky-materialnoi-tochky\u002Fpotentsialne-sylove-pole-potentsialna-enerhiia",1787712537263]