PlanetPhysics/Work in Classic Mechanics

[[../Work/|Work]] is defined as the change of [[../KineticEnergy/|kinetic energy]] of an [[../TrivialGroupoid/|object]] 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 $\mathrm{\Sigma }F$. Let's define a $x$ axis along the line of [[../CosmologicalConstant/|motion]].

According to Newton: $\mathrm{\Sigma }{F}_x$ = $m{a}_x$

The [[../Acceleration/|acceleration]] $a$ is constant (the sum of forces is constant, and so the following [[../MathematicalFoundationsOfQuantumTheories/|kinematic]] [[../Formula/|formula]] is relevant: </math> v_f{}^2 = v_i{}^2 + 2a \Delta x $($v_f$-Thefinal[[../Velocity/|velocity]],$v_i$-Theinitialvelocity)<math>a=\frac{v_f{}^2-v_i{}^2}{2\mathrm{\Delta }x}$

When inserting the previous equation into Newton's second law: $\mathrm{\Sigma }{F}_x=\frac{m(v_f{}^2-v_i{}^2)}{2\mathrm{\Delta }x}$And after a few algebric actions we get: $\mathrm{\Sigma }{F}_x\mathrm{\Delta }x=0.5m{v}_f{}^2-0.5m{v}_i{}^2=\mathrm{\Delta }{E}_k$From that we can conclude that $W=F\mathrm{\Delta }x$ ; $F_x\mathrm{\Delta }x$ is the work done by the force $F_x$ along the route $\mathrm{\Delta }x$

Work like [[../CosmologicalConstant/|energy]] is a [[../Vectors/|scalar]] but is defined as the product of two vectorial [[../Parameter/|parameters]]: $\mathrm{\nabla }F\cdot \mathrm{\nabla }\mathrm{\Delta }S$ and so in a two dimentional space work is defined as the [[../DotProduct/|scalar product]] of force $\mathrm{\nabla }F$ and change in place $\mathrm{\nabla }\mathrm{\Delta }S$. $W=|\mathrm{\nabla }F|\cdot |\mathrm{\nabla }\mathrm{\Delta }S|\cos \theta$ Template:CourseCat