Existence and multiplicity of periodic solutions for a model of a tapping mode cantilever in an Atomic Force Microscope (AFM) with a Lennard-Jones force and considering an external harmonic excitation were studied. The analytical approach was carried out using the nonlinear technique of lower and upper solutions to prove the existence and multiplicity of periodic solutions for the nonlinear AFM diferential equation. Numerical solutions of the AFM equation using Poincaré map were implemented for the verification of theorems and allow established a method to find these periodic solutions.