Approximation of string extension length

[gboff21]

Homework Statement

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=$\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}$ and therefore a length extension ΔL= $\sqrt{a^{2}+\frac{y^{2}}{a^{2}}}-a$ which simplifies to ${a(\sqrt{1+\frac{y}{a}^{2}}-1}[\itex]. <br /> <br /> Apparently when the displacement y is small, it can be approximated to y^2/2a<br /> <br /> How does that even work??$

 

[clamtrox]

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

 

[gboff21]

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! :)