Talk:PlanetPhysics/Work in Classic Mechanics
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: work in classic mechanics %%% Primary Category Code: 40. %%% Filename: WorkInClassicMechanics.tex %%% Version: 37 %%% Owner: bloftin %%% Author(s): bloftin, shmulik %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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\htmladdnormallink{Work}{http://planetphysics.us/encyclopedia/Work.html} is defined as the change of \htmladdnormallink{kinetic energy}{http://planetphysics.us/encyclopedia/KineticEnergy.html} of an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} caused by a force along a distance.
Work is commonly denoted by the latter $W$
The SI unit for work is joule [J] which is the same as $\frac{kg\cdot m^2}{s^2}$ in SI base units.
When focusing on an object moving along a straight line under the effect of constant forces $\Sigma F$. Let's define a $x$ axis along the line of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}.
According to Newton: $\Sigma F_x$ = $ma_x$
The \htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html} $a$ is constant (the sum of forces is constant, and so the following \htmladdnormallink{kinematic}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} \htmladdnormallink{formula}{http://planetphysics.us/encyclopedia/Formula.html} is relevant: $ v_f{}^2 = v_i{}^2 + 2a \Delta x $ ($v_f$-The final \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html}, $v_i$-The initial velocity) \[ a = \frac{v_f{}^2 - v_i{}^2}{2\Delta x}\]
When inserting the previous equation into Newton's second law: \[ \Sigma F_x = \frac{{m(v_f{}^2 - v_i{}^2)}}{2\Delta x}\] And after a few algebric actions we get: \[ \Sigma F_x \Delta x = 0.5mv_f{}^2 - 0.5mv_i{}^2 = \Delta E_k \] From that we can conclude that $ W = F\Delta x $ ; $F_x\Delta x$ is the work done by the force $F_x$ along the route $\Delta x$
Work like \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} is a \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} but is defined as the product of two vectorial \htmladdnormallink{parameters}{http://planetphysics.us/encyclopedia/Parameter.html}: $\nabla F\cdot \nabla \Delta S$ and so in a two dimentional space work is defined as the \htmladdnormallink{scalar product}{http://planetphysics.us/encyclopedia/DotProduct.html} of force $\nabla F$ and change in place $\nabla \Delta S$. \[ W = |\nabla F|\cdot |\nabla \Delta S|\cos \theta \]
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