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Radius of Rotation of a Particle Attached to Elastic String Moving in Horizontal Circle

Suppose a particle of mass  
\[m\]
  is attached to an elastic string of natural length  
\[l\]
, and modulus of elasticity  
\[\lambda\]
  and is moving in a horizontal circle of radius  
\[\]
.

The tension in the string is  
\[T=\frac{\lambda (r-l)}{l}\]
.
Applying  
\[F=ma=mv^2/r\]
  to the particle gives  
\[\frac{mv^2}{r}=\frac{\lambda (r-l)}{l}\]

Rearranging gives  
\[\lambda r^2 - \lambda l r -mv^2l=0\]
.
Solving for  
\[r\]
  gives  
\[r=\frac{\lambda l \pm \sqrt{\lambda^2 l^2 +4 \lambda mv^2 l}}{2 \lambda} = \frac{l \pm \sqrt{l^2+4mv^2l/ \lambda}}{2}\]

Only the positive option is viable - the negative option gives a valur for  
\[r\]
  less than zero.