The center of gravity between two celestial bodies, such as the Earth and the Moon, is known as the barycenter. In the case of the Earth-Moon system, the barycenter is the point around which both bodies orbit.
The barycenter of the Earth-Moon system is located within the Earth, but it is not at the exact center of the Earth due to the Moon's gravitational influence. The Moon's gravitational pull on the Earth causes the barycenter to be slightly offset from the Earth's center.
As the Earth and the Moon orbit each other, the barycenter remains relatively fixed because both bodies are attracted to each other by the force of gravity. Their orbits are governed by Newton's law of universal gravitation, which states that the gravitational force between two objects is proportional to the product of their masses and inversely proportional to the square of the distance between them. The Earth's much larger mass compared to the Moon's causes the barycenter to be located within the Earth and close to its center.
Although the Earth and the Moon are in motion, their respective masses and the relatively small separation between them ensure that the barycenter remains inside the Earth. The Moon's orbit around the barycenter is more pronounced due to its smaller mass, causing the Earth to undergo slight wobbles, but these movements are relatively negligible compared to the overall system.
In summary, the barycenter of the Earth-Moon system is the center of gravity between the two bodies. It remains fixed within the Earth due to the balance of gravitational forces acting on both objects, resulting in their stable orbits around the barycenter.