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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!

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