[{"data":1,"prerenderedAt":33},["ShallowReactive",2],{"topic-en-theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fwork-of-a-force-power":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},254,"Work of a Force. Power","en","theoretical-mechanics\u002Fdynamics\u002Fgeneral-theorems-of-particle-dynamics\u002Fwork-of-a-force-power","Work of a Force and Power in Dynamics","Elementary and finite work of a force, work of common forces, and mechanical power.","This topic covers work done by a force along a displacement, mechanical power, the sign of work, and work of common forces.","\u003Cp>\u003Cstrong>Work of a force\u003C\u002Fstrong> measures the action of a force through a displacement, while \u003Cstrong>power\u003C\u002Fstrong> measures the rate at which work is done.\u003C\u002Fp>\u003Ch2>Elementary work\u003C\u002Fh2>\u003Cp>For an infinitesimal displacement $d\\vec r$:\u003C\u002Fp>\u003Cp>$$dA=\\vec F\\cdot d\\vec r=F\\,ds\\cos\\alpha,$$\u003C\u002Fp>\u003Cp>where $\\alpha$ is the angle between the force and displacement direction.\u003C\u002Fp>\u003Ch2>Work over a finite displacement\u003C\u002Fh2>\u003Cp>From point 1 to point 2:\u003C\u002Fp>\u003Cp>$$A_{1\\to2}=\\int_1^2\\vec F\\cdot d\\vec r.$$\u003C\u002Fp>\u003Cp>For a constant force over a straight displacement $s$, $A=Fs\\cos\\alpha$.\u003C\u002Fp>\u003Ch2>Sign of work\u003C\u002Fh2>\u003Cp>Work is positive for an acute angle between force and displacement, negative for an obtuse angle, and zero when the force is perpendicular to the instantaneous displacement.\u003C\u002Fp>\u003Ch2>Work of gravity\u003C\u002Fh2>\u003Cp>Near Earth's surface, the work of gravity depends only on the change in height: $A_g=mg(h_1-h_2)$. It is positive for downward motion and negative for upward motion.\u003C\u002Fp>\u003Ch2>Work of a spring force\u003C\u002Fh2>\u003Cp>For a spring with $F_x=-kx$:\u003C\u002Fp>\u003Cp>$$A_s=\\frac{kx_1^2}{2}-\\frac{kx_2^2}{2}.$$\u003C\u002Fp>\u003Ch2>Power\u003C\u002Fh2>\u003Cp>Instantaneous power is:\u003C\u002Fp>\u003Cp>$$P=\\frac{dA}{dt}=\\vec F\\cdot\\vec v.$$\u003C\u002Fp>\u003Cp>The SI unit is the watt: $1\\,\\text{W}=1\\,\\text{J}\u002F\\text{s}$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>A constant 50 N force moves a point 3 m in the force direction. The work is 150 J. If this occurs over 5 s at a uniform average rate of doing work, the average power is 30 W.\u003C\u002Fp>\u003Ch2>Common mistakes\u003C\u002Fh2>\u003Cul>\u003Cli>using $Fs$ without accounting for the angle;\u003C\u002Fli>\u003Cli>confusing work and power;\u003C\u002Fli>\u003Cli>assuming work is always positive;\u003C\u002Fli>\u003Cli>using average power where instantaneous $\\vec F\\cdot\\vec v$ is required.\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\u002Fwork-of-a-force-power","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fteoretychna-mekhanika\u002Fdynamika\u002Fzahalni-teoremy-dynamiky-materialnoi-tochky\u002Frobota-syly-potuzhnist",1787712537228]