Editing Linear approximation (section)

==Applications==

===Optics===
{{main|Gaussian optics}}
''Gaussian optics'' is a technique in [[geometrical optics]] that describes the behaviour of light rays in optical systems by using the [[paraxial approximation]], in which only rays which make small angles with the [[optical axis]] of the system are considered.<ref>{{cite book |first1=A. |last1=Lipson |first2=S. G. |last2=Lipson |first3=H. |last3=Lipson |url=https://books.google.com/books?id=aow3o0dhyjYC&pg=PA51 |title=Optical Physics |edition=4th |year=2010 |publisher=Cambridge University Press |location=Cambridge, UK |page=51 |isbn=978-0-521-49345-1 }}</ref> In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a [[sphere]]. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.

===Period of oscillation===
{{main|Pendulum}}
The period of swing of a [[Pendulum (mathematics)#Simple gravity pendulum|simple gravity pendulum]] depends on its [[length]], the local [[Gravitational acceleration|strength of gravity]], and to a small extent on the maximum [[angle]] that the pendulum swings away from vertical, {{math|''θ''<sub>0</sub>}}, called the [[amplitude]].<ref name="Milham1945">{{cite book |last=Milham |first=Willis I. |title=Time and Timekeepers |year=1945 |publisher=MacMillan |pages=188–194 |oclc=1744137 }}</ref> It is independent of the [[mass]] of the bob. The true period ''T'' of a simple pendulum, the time taken for a complete cycle of an ideal simple gravity pendulum, can be written in several different forms (see [[Pendulum (mathematics)|pendulum]]), one example being the [[infinite series]]:<ref name="Nelson">{{cite journal | last = Nelson | first = Robert |author2=M. G. Olsson | title = The pendulum – Rich physics from a simple system | journal = American Journal of Physics | volume = 54 | issue = 2 | pages = 112–121 | date = February 1987 | url = http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf | doi = 10.1119/1.14703 | access-date = 2008-10-29 | bibcode = 1986AmJPh..54..112N | s2cid = 121907349 }}</ref><ref>{{Cite EB1911|wstitle= Clock |volume= 06 |last1= Beckett |first1= Edmund |last2= and three more| pages = 534&ndash;553; see page 538, second para |quote= Pendulum.—}} includes a derivation</ref>
<math display="block">
T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right)
</math>

where '''''L''''' is the length of the pendulum and '''''g''''' is the local [[Gravitational acceleration|acceleration of gravity]].

However, if one takes the linear approximation (i.e. if the amplitude is limited to small swings,<ref group = Note>A "small" swing is one in which the angle θ is small enough that sin(θ) can be approximated by θ when θ is measured in radians</ref> ) the [[Frequency|period]] is:<ref>{{cite book |last = Halliday
 |first      = David
 |author2    = Robert Resnick
 |author3    = Jearl Walker
 |title      = Fundamentals of Physics, 5th Ed.
 |publisher  = John Wiley & Sons.
 |year       = 1997
 |location   = New York
 |page       = [https://archive.org/details/fundamentalsofp000davi/page/381 381]
 |url        = https://archive.org/details/fundamentalsofp000davi/page/381
 |url-access = registration
 |isbn       = 0-471-14854-7
}}</ref>
{{NumBlk||<math display="block">T \approx 2\pi \sqrt\frac{L}{g} \qquad \qquad \qquad \theta_0 \ll 1 </math>|{{EquationRef|1}}}}

In the linear approximation, the period of swing is approximately the same for different size swings: that is, ''the period is independent of amplitude''. This property, called [[isochronism]], is the reason pendulums are so useful for timekeeping.<ref>{{cite book
  | last = Cooper
  | first = Herbert J.
  | title = Scientific Instruments
  | publisher = Hutchinson's
  | year = 2007
  | location = New York
  | page = 162
  | url = https://books.google.com/books?id=t7OoPLzXwiQC&pg=PA162
  | isbn = 978-1-4067-6879-4}}</ref> Successive swings of the pendulum, even if changing in amplitude, take the same amount of time.

===Electrical resistivity===
{{main|Electrical resistivity}}
The electrical resistivity of most materials changes with temperature. If the temperature ''T'' does not vary too much, a linear approximation is typically used:
<math display="block">\rho(T) = \rho_0[1+\alpha (T - T_0)]</math>
where <math>\alpha</math> is called the ''temperature coefficient of resistivity'', <math>T_0</math> is a fixed reference temperature (usually room temperature), and <math>\rho_0</math> is the resistivity at temperature <math>T_0</math>. The parameter <math>\alpha</math> is an empirical parameter fitted from measurement data. Because the linear approximation is only an approximation, <math>\alpha</math> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math>\alpha</math> was measured at with a suffix, such as <math>\alpha_{15}</math>, and the relationship only holds in a range of temperatures around the reference.<ref>{{cite book |first=M. R. |last=Ward |year=1971 |title=Electrical Engineering Science |pages=36–40 |publisher=McGraw-Hill |isbn=0-07-094255-2 }}</ref> When the temperature varies over a large temperature range, the linear approximation is inadequate and a more detailed analysis and understanding should be used.