## 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.