Maximum Angular Speed for Horizontal Circular Motion of a Particle on an Elastic String

A particle moving in a horizontal circle, suspended from the ceiling by an elastic string of natural lengthand modulus of elasticitymust move with an angular velocity less than some minimum

Applyinghorizontally gives (1)

so (1) can be writtenhence (2)

Writeso thatwhereis the extension of the string. Substitute this into (2) to obtain

Rearranging this gives

Sincewe must have