Equilateral Spring Triangle

Three point masses $m$ are attached together by identical springs. When placed at rest on a horizontal surface the masses form a triangle with side length $l$ . When the assembly is rotated about its center at angular velocity $\omega$ , the masses form a triangle with side length $2l$ . What is $\dfrac{k}{m\omega^2}$ for the springs, if $k$ is the spring constant?

Note : A version of this problem appeared on the 2012 F = ma exam.