Resonance in a Mechanical System

[amk1995]

Homework Statement

A mass m is attached to one end of a massless spring with a force constant k and an unstretched length l₀. The other end of the spring is free to turn about a nail driven into a frictionless, horizontal surface. The mass is made to revolve in a circle with an angular frequency of revolution ω.
Question:
Calculate the length l of the spring as a function of ω.

Homework Equations

F = ω²*l*m
F_Spring = k*l

The Attempt at a Solution

Fspring = Fcirc
-k*l = ω²*l*m

The issue I am having is how to get L by itself, or what I am supposed to do with l₀. Can someone guide me on where I should head in order to solve this problem? The issue is l cancels when I divide, unless I am supposed to use l₀ in place of l
Thanks!

 

[nasu]

You are using "l" to designate two different things.
If "l" in the first equation is the total length of the spring (radius of the circle), then the elastic force is not correct. The force does not depend on the total length.
What is the force when the length is lo?

 

[amk1995]

The force when length lo is just omega^2 *m*lo correct?

 

[nasu]

No. What is the elastic force ("produced" by the spring) when the spring's length is lo?
Hint: at lo the spring is unstretched.