The term "center of mass" is used not only in mechanics and in calculating the motion of celestial bodies, but also in everyday life. It's just that people do not always think about what kind of laws of nature are manifested in a particular situation. For example, skaters in pair skating actively use the center of mass of the system when they unwind, holding hands.
Previously, before Newton discovered the laws of gravity, it was believed that the Earth and all the planets of the system revolve around a central star - the Sun. That is, the location of the central point coincides with the physical center of the star. Theoretically, it was quite possible. But then Newton proved that objects are mutually attracted to each other - not only does the Sun affect the planet, but the latter also affects the star. Since not only the square of the distance, but also the mass appears in the formula, the interaction forces are unequal. As a result, it turns out that the center of mass of the Earth-Sun mechanical system is not located in the middle of the star, but is shifted relative to the physical center. Therefore, both the Earth and the Sun itself revolve around this point.
The center of mass allowed first to suggest, and then brilliantly confirm the existence of a subtle satellite in the Sirius star system. Watching the movement of the star, the astronomer drew attention to its strange undulating displacements. It turned out that Sirius is a double star. They rotate around the center of mass, which causes the observed oscillations.
The center of mass is completely independent of the type of connection of objects: magnetic fields, rope, rod. Imagine two balls with masses m1 and m2, connected to each other by a light metal rod of length L. Due to this, part of the momentum imparted to the ball m1 will also be transmitted by m2. If we now throw this bunch of balls at an angle into the air, then the flight only at first glance will seem messy: although the balls “tumble”, changing their position, a certain common point on the rod moves as it should - along a parabolic path. This is the center of mass. It is not necessarily located in the object itself. For example, in the example with stars, it is located in the void. When performing the calculations, it is believed that the entire total mass of objects is concentrated at this particular point, and vectors of external forces are applied to it. If you give an impulse to any body of the system, then in full accordance with the laws of conservation of energy of an impulse, its value at the point of the center of mass will be zero. If we consider a system isolated from external influences, then the movement of its center of mass occurs at a constant speed. Accordingly, in this case we can talk about an inertial reference system.
The concept of the center of mass is also applied in the design of ships. It is necessary to take into account not just two bodies, but their huge number and bring everything to a common denominator. Errors in the calculations mean the lack of stability of the ship: in one case, it will be excessively immersed in water, risking sinking at the most insignificant waves; and in the other too elevated above sea level, creating the danger of a coup on its side. By the way, that is why every thing on board should be in its place, as provided for by the calculations: the most massive ones at the very bottom.
The center of mass is used not only in relation to celestial bodies and the design of mechanisms, but also in the study of the "behavior" of microworld particles. For example, many of them are born in pairs (electron-positron). Having the initial rotation and obeying the laws of attraction / repulsion, they can be considered as a system with a common center of mass.
For simple systems, calculating the position of the center of mass point is fairly easy. To do this, sum the products of the mass of each body by the radius vector, and divide the resulting value by the total mass.