# Mass-spring-mass system orbiting a third mass [closed]

I'm having a small problem finishing the solution of the following problem: Assume we got a mass $$m$$ of radius $$r_m$$ connected with another mass $$\mu$$ through a spring of constant $$k$$ ($$\mu \ll m$$). A third body of mass $$M$$ rests at a distance $$R$$ ($$r \ll R$$) and exerts gravitational force on the system which forces the system to rotate around it with the Keplerian angular velocity $$\omega = \sqrt{\frac{GM}{R^3}}$$. The mass $$m$$ is tidally locked around the mass $$M$$, meaning it rotates about itself with the same angular velocity $$\omega$$. Find the equation of motion that defines the position $$r$$ of the mass $$\mu$$ from the mass $$m$$.

What I did is I analyzed the rotating motion around the mass $$m$$ of the system $$m$$ - spring - $$\mu$$ and then I analyzed the motion of the mass $$m$$ around $$M$$. Since $$r\ll R$$ the system of the masses and the spring can be seen as a point mass rotating $$M$$. However, this doesn't feel complete. I think I'm missing something. Any help is appreciated. Thanks in advance!