Quark model - Revision history

en>Cuzkatzimhut: The actual paper itself, warts and all, instead of Just the flawed review building it up.

2015-01-02T13:37:10Z

The actual paper itself, warts and all, instead of Just the flawed review building it up.

← Older revision		Revision as of 15:37, 2 January 2015
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2001:630:D0:ED04:B4B6:3B88:CAD7:1944: /* Baryons */

2014-02-28T15:10:30Z

Baryons

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en>Suslindisambiguator: /* References and external links */

2013-09-06T19:30:30Z

References and external links

← Older revision		Revision as of 21:30, 6 September 2013
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	~~My name~~ is ~~Yong~~ and ~~I am studying Greek~~ and ~~Roman Culture~~ and ~~Playwriting at Hamburg Allermohe~~ / ~~Germany~~.<br><br>~~My web~~-~~site~~ :: [~~http~~://~~Farnborough~~-~~Hill~~.~~Org~~.uk/~~default~~/~~mailforms~~/~~dvdpreview~~/~~Libraries~~/~~Sixth_Form_DVD~~/~~Sixth_Form_DVD~~.~~aspx FIFA 15 coin hack~~]		[[File:Kleinwinkelnaeherungen.png\|thumb\|upright=1.5\|Approximately equal behavior of some (trigonometric) functions for x → 0]]

			The '''small-angle approximation''' is a useful simplification of the basic [[trigonometric functions]] which is approximately true in the [[limit (mathematics)\|limit]] where the angle approaches zero. They are truncations of the [[Taylor series]] for the basic trigonometric functions to a [[orders of approximation\|second-order approximation]]. This truncation gives:

			:<math>\sin \theta \approx \theta</math>

			:<math>\cos \theta \approx 1 - \frac{\theta^2}{2}</math>

			:<math>\tan \theta \approx \theta</math>,
			where θ is the angle in [[radian]]s.

			The small angle approximation is useful in many areas of physics, including [[mechanics]], [[electromagnetics]], [[optics]] (where it forms the basis of the [[paraxial approximation]]), [[cartography]], [[astronomy]], and so on.

			== Justifications ==

			=== Graphic ===
			The accuracy of the approximations can be seen below in Figure 1 and Figure 2. As the angle approaches zero, it is clear that the gap between the approximation and the original function quickly vanishes.
			<gallery widths="500px" heights="400px">
			File:Small_angle_compair_odd.svg\|'''Figure 1.''' A comparison of the basic odd trigonometric functions to θ. It is seen that as the angle approaches 0 the approximations become better.
			File:Small_angle_compare_even.svg\|'''Figure 2.''' A comparison of cos(θ) to 1 - θ<sup>2</sup>/2. It is seen that as the angle approaches 0 the approximation becomes better.
			</gallery>

			=== Geometric ===
			[[File:Small angle triangle.svg\|center]]
			The red section on the right, d, is the difference between the lengths of the hypotenuse, H, and the adjacent side, A. As is shown, H and A are almost the same length, meaning cos θ is close to 1 and <math>\theta^2/2</math> helps trim the red away.
			:<math> \cos{\theta} \approx 1 - \frac{\theta^2}{2}</math>

			The opposite leg, O, is approximately equal to the length of the blue arc, s. Gathering facts from geometry, s = A*θ, from trigonometry, sin θ = O/H and tan θ = O/A, and from the picture, <math> O \approx s </math> and <math> H \approx A </math> leads to:

			:<math>\sin \theta = {O\over H}\approx{O\over A} = \tan \theta = {O\over A} \approx {s\over A} = {{A*\theta}\over A} = \theta</math>.
			Simplifying leaves,

			:<math>\sin \theta \approx \tan \theta \approx \theta</math>.

			=== Algebraic ===
			The Maclaurin expansion (''viz.'', Taylor expansion about 0) of the relevant trigonometric function is<ref>{{cite book\|last=[[Mary L. Boas\|Boas]]\|first=Mary L.\|title=[[Mathematical Methods in the Physical Sciences]]\|year=2006\|publisher=Wiley\|location=p. 26\|isbn=978-0-471-19826-0\|pages=839}}</ref>

			:<math>\sin \theta = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots </math>

			where θ is the angle in radians. In plain-Jane terms,

			:<math>\sin \theta = \theta - \frac{\theta^3}{6} + \frac{\theta^5}{120} - \frac{\theta^7}{5040} + \cdots </math>

			It is readily seen that the second most significant (''viz.'', the third-order) term falls off as the cube of the first term; thus, even for a not-so-small argument such as 0.01, the value of the second most significant term is on the order of 0.000001, or one-ten-thousandth the first term. One can thus safely approximate:

			:<math>\sin \theta \approx \theta</math>

			By extension, since the cosine of a small angle is very, very nearly one, and the tangent is given by the sine divided by the cosine,

			:<math>\tan \theta \approx \sin \theta \approx \theta</math>.

			==Error of the approximations==
			[[File:Small angle compare error.svg\|thumb\|upright=2\|'''Figure 3.''' A graph of the [[relative error]]s for the small angle approximations.]]
			Figure 3 shows the relative errors of the small angle approximations. The angles at which the relative error exceeds 1% are as follows:
			* tan θ ≈ θ at about 0.176 radians(10°).
			* sin θ ≈ θ at about 0.244 radians (14°).
			* cos θ ≈ 1 - θ<sup>2</sup>/2 at about 0.664 radians (38°).

			==Specific uses==

			=== Astronomy ===
			In astronomy, the angle subtended by the image of a distant object is often only a few [[arcsecond]]s, so it is well suited to the small angle approximation. The linear size (''D'') is related to the angular size (''X'') and the distance from the observer (''d'') by the simple formula

			:''D'' = ''X'' · ''d'' / 206,265

			where ''X'' is measured in arcseconds.

			The number 206,265 is approximately equal to the number of arcseconds in a [[circle]] (1,296,000), divided by 2π.

			The exact formula is

			:''D'' = ''d'' tan(''X''·2π/1,296,000)

			and the above approximation follows when tan(''X'') is replaced by ''X''.

			=== Motion of a pendulum ===
			The second order Cos approximation is especially useful in calculating the potential energy of a pendulum, which can then be applied with a Lagrangian to find the indirect (energy) equation of motion.

			When calculating the period of a simple pendulum, the small-angle approximation for sine is used to allow the resulting differential equation to be solved easily by comparison with the differential equation describing [[simple harmonic motion]].

			=== Structural mechanics ===
			The small angle approximation also appears in structural mechanics especially in stability and bifurcation analyses (mainly of axially-loaded columns ready to undergo buckling) leading to significant simplifications, though at a cost in accuracy and insight into the true behaviour.

			=== Piloting ===
			The [[1 in 60 rule]] used in [[air navigation]] has its basis in the small-angle approximation, plus the fact that one radian is approximately 60 degrees.

			== See also ==
			* [[Skinny triangle]]
			* [[Pendulum_(mathematics)#Infinitesimal_oscillations_of_a_pendulum\|Infinitesimal oscillations of a pendulum]]

			== References ==
			{{reflist}}

			{{DEFAULTSORT:Small-Angle Approximation}}
			[[Category:Trigonometry]]

en>Headbomb: Disambiguated: orbital angular momentum → angular momentum operator; Help needed: Adjoint representation

2012-08-07T23:48:50Z

Disambiguated: orbital angular momentum → angular momentum operator; Help needed: Adjoint representation

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