\[m\]
is attached to an elastic string of natural length \[l\]
, and modulus of elasticity \[\lambda\]
and is moving in a horizontal circle of radius \[\]
.
\[T=\frac{\lambda (r-l)}{l}\]
.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.