The gravitational effects of the Moon on the Earth are verifiable in many different natural systems. Some have pronounced and visible effects, like tides, although others are more subtle in nature, like atmospheric tides or crustal tides. In this paper, the orbital period of the Moon is determined by studying one of the more subtle effects of the Earth-Moon-Sun gravitational interaction, the variation of local gravity at the Earth's surface as a function of the position of the Moon in its orbit. For this purpose, we use a set of remote controlled pendulums at different longitudes. The pendulums are part of the constellation displayed by the World Pendulum Alliance project. The spatial distribution of these accurate apparatus allows the experimental study of the variation of gravitational acceleration at the Earth's surface as a function of latitude. In this paper, we show the experimental verification of the variation of the period of the pendulum as a function of the phase of the Moon and, consequently, the determination of the period of the Moon's orbit around the Earth.