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| [[File:Free body frictionless.svg|right|300px|thumb|Key:<br>N = [[normal force]] that is perpendicular to the plane<br>m = [[mass]] of object<br>g = acceleration due to [[gravity]]<br>θ ([[theta]]) = angle of elevation of the plane, measured from the horizontal]]
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| The '''frictionless plane''' is a concept from the writings of [[Galileo Galilei]]. In his 1608 ''The Two New Sciences'', Galileo presented a formula that predicted the motion of an object moving down an [[inclined plane]]. His formula was based upon his past experimentation with free-falling bodies.<ref>Drake, Stillman, ''Galileo’s Experimental Confirmation of Horizontal Inertia: Unpublished Manuscripts.'' ''Isis'': Vol. 64, No. 3, p.296.</ref> However, his model was not based upon experimentation with objects moving down an inclined plane, but from his conceptual modeling of the forces acting upon the object. Galileo understood the mechanics of the inclined plane as the combination of horizontal and vertical [[Vector (geometric)|vectors]]; the result of [[gravity]] acting upon the object, diverted by the [[slope]] of the plane.<ref>Settle, T.B."An Experiment in the History of Science,” ''Science'', 1061 ''133'' 19-23.</ref>
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| However, Galileo's equations do not contemplate [[friction]], and therefore do not perfectly predict the results of an actual [[experiment]]. This is because some energy is always lost when one mass applies a non-zero [[normal force]] to another. Therefore, the observed [[speed]], [[acceleration]] and distance traveled should be less than Galileo predicts. <ref> Jenkin, Fleeming. ''On Friction Between Surfaces at Low Speeds.'' Proceedings of the Royal Society of London, Vol. 26 p. 93-94 </ref> This energy is lost in forms like sound and heat. However, from Galileo’s predictions of an object moving down an inclined plane in a frictionless environment, he created the theoretical foundation for extremely fruitful real-world experimental prediction.<ref>Drake, at p. 297-99</ref>
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| Frictionless planes do not exist in the real world. However, if they did, one can be all but certain that objects on them would behave exactly as Galileo predicts. Despite their nonexistence, they have considerable value in the design of engines, motors, roadways, and even tow-truck beds, to name but a few examples. <ref> Koyré, Alexandre ''Metaphysics and Measurement'', pp. 83-84 (1992). </ref>
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| The effect of friction on an object moving down an inclined plane can be calculated as
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| :<math> F_\mathrm{f} = \mu_\mathrm{k} F_\mathrm{N}, </math> | |
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| where <math> F_\mathrm{f} </math> is the force of friction exerted by the object and the inclined plane on each other, parallel to the surface of the plane, <math>F_\mathrm{N}</math> is the [[normal force]] exerted by the object and the plane on each other, directed perpendicular to the plane, and <math>\mu_\mathrm{k}</math> is the coefficient of [[Friction#Kinetic_friction|kinetic friction]].<ref> Koyré, pp. 84-86. </ref>
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| Unless the inclined plane is in a [[vacuum]], a (usually) small amount of [[potential energy]] is also lost to [[air drag]].
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| ==See also== | |
| *[[Atwood machine]]
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| *[[Spherical cow]]
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| == References ==
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| <references/>
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| [[Category:Abstraction]]
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| [[Category:Physics education]]
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