Electrostatic softening above a conducting wedge (${\omega }_y^2=(k-2\beta /y_e^3)/m$)

For an attractive force $-\beta /y^2$ and an otherwise linear vertical spring, differentiation about the charged equilibrium $y_e$ reduces the spring stiffness by $2\beta /y_e^3$. If the uncharged height is $y_0$ and the shift is small, the fractional frequency change is $-\beta /(k{y}_0^3)$. Using only the force value, without its derivative, would miss this frequency shift.