Newton's Second Law $$m\ddot{x}(t)-F=0$$ Where we find the total external force from Coulomb's law $$F=kqQ\left(\frac{1}{(L-x)^2}-\frac{1}{(L+x)^2}\right)$$ Using the approximation for a small value $$x =\frac{h}{R} \ll 1; \quad(1+x)^\alpha\approx 1+\alpha x$$$$F=-\frac{4kqQLx}{(L^2-x^2)^2}\approx-\frac{4kqQx}{L^3}$$ Harmonic oscillation equation $$\ddot{x}(t)+\frac{4kqQx}{mL^3}x(t)$$ We obtain the required frequency of small oscillations $$\boxed{\omega =\sqrt{\frac{4kqQ}{mL^3}}=\sqrt{\frac{qQ}{m\pi\varepsilon_0L^3}}}$$
