Finding Tension in an Elastic String?

Six years later:

We are given that the natural length of the string is $2l$. When the mass is suspended at the midpoint, $O$, of the string, the extension perpendicular to $AB$ is $x$. Let $M$ be the position of the mass at perpendicular extension $x$. Then the extended length of the string from $A$ to $M$ (the hypotenuse of a triangle with sides $a$ and $x$) is: $$AM=sqrt(a^{2}+x^{2})$$ Since the string is extended at its mid-point, this is the same as the extension from M to B. Hence, the total length, $AMB$, of the extended string is $$L=2sqrt(a^{2}+x^{2})$$ The extension of the string from its natural length is $$delta {l}=2sqrt(a^{2}+x^{2})-2l=2left[{sqrt(a^{2}+x^{2})-l} right]$$

Using the formula for Tension in a string and substituting for $delta {l}$: $$begin{align} T &=lambda frac{delta {l}}{l} &=lambda frac{2left[{sqrt(a^{2}+x^{2})-l} right]}{2l} &=lambda frac{left[{sqrt(a^{2}+x^{2})-l} right]}{l}end{align} $$

Finally, if $x^{2} ll 1$, this gives: $$begin{align} T &=lambda frac{left[{sqrt(a^{2})-l} right]}{l} &=lambda frac{(a-l)}{l} end{align}$$