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Approximation of string extension length

  1. Apr 5, 2012 #1
    1. The problem statement, all variables and given/known data

    A string of length a is stretched to a height of y when it is attached to the origin so making a triangle with length L=[itex]\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}[/itex] and therefore a length extension ΔL= [itex]\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a[/itex] which simplifies to [itex]{a(\sqrt{1+\frac{y}{a}^{2}}-1}[\itex].

    Apparently when the displacement y is small, it can be approximated to y^2/2a

    How does that even work??
     
  2. jcsd
  3. Apr 5, 2012 #2
    Are you sure you got the result right? Seems to me like there should be a³, not a. This works by a method called Taylor's series. You can read about it in wikipedia: http://en.wikipedia.org/wiki/Taylor_series
     
  4. Apr 5, 2012 #3
    Ah of course. I'm familiar with Taylor but I just didn't apply it. However, I've just done it and it is over 2a. Thanks for pointing me down the right track! :)
     
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