A particle moving in a horizontal circle, suspended from the ceiling by an elastic string of natural length
and modulus of elasticity
must move with an angular velocity less than some minimum
Applying
horizontally gives
(1)
so (1) can be written
hence
(2)
Write
so that
where
is the extension of the string. Substitute this into (2) to obtain
Rearranging this gives
Since
we must have