Talk:PlanetPhysics/Kinetic Energy
Original TeX Content from PlanetPhysics Archive[edit source]
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: kinetic energy %%% Primary Category Code: 03.30.+p %%% Filename: KineticEnergy.tex %%% Version: 1 %%% Owner: pbruin %%% Author(s): pbruin %%% 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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\emph{Kinetic energy} is \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} associated to \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. The kinetic energy of a mechanical \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} is the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html} required to bring the system from its `rest' state to a `moving' state. When exactly a system is considered to be `at rest' depends on the context: a stone is usually considered to be at rest when its \htmladdnormallink{centre of mass}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} is fixed, but in situations where, for example, the stone undergoes a change in \htmladdnormallink{temperature}{http://planetphysics.us/encyclopedia/BoltzmannConstant.html} the movement of the individual \htmladdnormallink{particles}{http://planetphysics.us/encyclopedia/Particle.html} will play a role in the energetic description of the stone.
Kinetic energy is commonly denoted by various symbols, such as $E_{\mathrm{k}}$, $E_{\mathrm{kin}}$, $K$, or $T$ (the latter is the convention in \htmladdnormallink{Lagrangian}{http://planetphysics.us/encyclopedia/LagrangesEquations.html} \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html}). The SI unit of kinetic energy, like that of all sorts of energy, is the joule (J), which is the same as $\mathrm{kg\;m^2/s^2}$ in SI base units.
Energy associated to motion in a straight line is called \emph{translational kinetic energy}. For a particle or \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/RigidBody.html} with \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html} $m$ and \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $\mathbf{v}$, the translational kinetic energy is $$ E_{\mathrm{trans}}=\frac{1}{2}mv^2=\frac{1}{2}m\mathbf{v}\cdot\mathbf{v}. $$ Kinetic energy associated to rotation of a rigid body is called \emph{rotational kinetic energy}. It depends on the \htmladdnormallink{moment of inertia}{http://planetphysics.us/encyclopedia/MomentOfInertia.html} $I$ of the body with respect to the axis of rotation. When the body rotates around that axis at an angular velocity $\omega$, the rotational kinetic energy is $$ E_{\mathrm{rot}}=\frac{1}{2}I\omega^2. $$
In \htmladdnormallink{special relativity}{http://planetphysics.us/encyclopedia/SR.html}, the total energy of an \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of mass $m$ moving in a straight line with \htmladdnormallink{speed}{http://planetphysics.us/encyclopedia/Velocity.html} $v$ is $$ E=\gamma(v)mc^2, $$ where $c$ is the \htmladdnormallink{speed of light}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} and $\gamma(v)$ is the Lorentz factor: $$ \gamma(v)=\frac{1}{\sqrt{1-v^2/c^2}}. $$ In particular, the rest energy of this object (obtained by setting $v=0$) is equal to $mc^2$. The kinetic energy is therefore $$ E_{\mathrm{kin}}=\gamma(v)mc^2-mc^2=(\gamma(v)-1)mc^2. $$ For values of $v$ much smaller than $c$, this expression becomes approximately equal to $\frac{1}{2}mv^2$, the kinetic energy from \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}. This can be checked by expanding $\gamma(v)$ in a \htmladdnormallink{Taylor series}{http://planetphysics.us/encyclopedia/TaylorFormula.html} around $v=0$: $$ \gamma(v)=1+\frac{1}{2}\frac{v^2}{c^2}+\frac{3}{8}\frac{v^4}{c^4} +\frac{5}{16}\frac{v^6}{c^6}+\cdots $$ Substituting this into the expression for the kinetic energy gives the following expansion: $$ E_{\mathrm{kin}}=\frac{1}{2}mv^2+\frac{3}{8}mv^4/c^2 +\frac{5}{16}mv^6/c^4+\cdots $$ When $v$ approaches the speed of light, the factor $\gamma(v)$ goes to infinity. This is one way of seeing why objects with positive mass can never reach a speed $c$: an infinite amount of work would be required to accelerate the object to this speed.
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