A particle suspended from an elastic string will extend the string slightly and if the particle is made to move in a horizontal circle, the will be a component of the centripetal force extending the string a little more. To find the extension and the tension in the string, we can resolve vertically and horizontally.
The tension in the string is
Resolving vertically gives (1)
Applying F=ma horizontally gives (2)
(2) divided by (1) gives