I tied a stone with a piece of string and that is my pendulum which I had tested on the surface successfully, imagine I were to dig a hole to the center of the Earth to test out my pendulum... would it works suppose I'm experiencing the same pull of force everywhere? if so does it rocks to and from as on surface?
The gravitational pull on the bob of the pendulum will be zero as it gets pulled from all sides by an equal force. This means that the bob and the experimenter will just float at the center of earth. For a pendulum to go to and fro a net force must act on it, so no oscillations.
Going further you could remove the string and just displace the bob from the center of earth. It would go on oscillating about the center of earth.