= Solution
A <geostationary orbit> has the same <angular velocity> as the <Earth>. Let $M_E=4\pi\rho R^3/3$ and $r_g=\eta R$. Balancing the required <centripetal acceleration> with the exterior <gravitational field> gives
$$
\omega^2r_g=\frac{GM_E}{r_g^2},\qquad
\boxed{\eta=\left(\frac{GM_E}{\omega^2R^3}\right)^{1/3}=\left(\frac{4\pi G\rho}{3\omega^2}\right)^{1/3}.}
$$
The <orbital plane> must be equatorial and the motion prograde for the satellite to remain over one point. The uniform density assumption supplies $M_E$ in terms of the specified $\rho$ and $R$.
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