# Sleeping Satellite

**Classical Mechanics**Level 5

Consider a small geostationary satellite consisting of four point masses \(m\) connected by light rigid rods of length \(a\) and \(b\) to a central mass \(M\) (\(M\gg m\)) as shown in the figure. In addition, four light rods of length \(\sqrt{a^{2}+b^{2}}\) ensure that the satellite behaves as rigid body. It turns out that if \(\frac{a}{b}>1\), the satellite can oscillate about its center of mass in the plane of the orbit. In other words, the configuration showed in the figure is stable. For what ratio \( \eta=\frac{a}{b}\) the period of small oscillations of the satellite equals its orbital period? Assume that the dimensions of the satellite are much smaller than the radius its orbit.