For the mass m, attached to a spring, to move in a circle centripetal force and restoring force of the spring must be equal. Centripetal force is given with this formula:
[tex]F_c=\frac{mv^2}{r}=\frac{mw^2r^2}{r}=mw^2r[/tex]
Restoring force only occurs if the spring is stretched. If L is the length of unstretched spring we have the following formula for restoring force:
[tex]F=k(r-L)[/tex]
r is the length of a circle that mass m is traveling along.
As said above, these two forces have to be equal:
[tex]F=k(r-L)=mw^2r[/tex]
We solve for r:
[tex]F=k(r-L)=mw^2r\\
kr-kL=mw^2r\\
mw^2r-kr=-kL\\
r(mw^2-k)=-kL\\
r=\frac{-kL}{mw^2-k}\\
r(w)=\frac{kL}{k-mw^2}
[/tex]
r(w) will go to infinity when denominator is equal to zero:
[tex]k-mw^2=0\\
k=mw^2\\
w^2=\frac{k}{m}\\
w_{crit}=\sqrt{\frac{k}{m}}[/tex]
Please keep in mind that Hooke's law has it's limitations, and before we reach our critial value of angular velocity spring will be strecthed to a point where Hooke's law does not aply anymore.
