A solid cylinder of mass \(M\) and cross-sectional area \(A\) floats vertically in a fluid whose density increases linearly with depth \(z\) according to \(\rho(z) = \rho_0 (1 + b z)\), where \(\rho_0\) and \(b\) are positive constants and \(z\) is measured downward from the fluid surface. The total upward buoyant force exerted on the cylinder when its submerged depth is \(z\) is given by \(F_B(z) = \rho_0 g A \left( z + \dfrac{1}{2} b z^2 \right)\). The cylinder rests in static equilibrium at depth \(z = z_0\). If the cylinder is displaced slightly from its equilibrium depth and released, what is the angular frequency \(\omega\) of its small vertical oscillations?