Gravitational forces are one of the four main types of forces that manifest themselves in all their diversity between different bodies both on Earth and beyond. In addition to them, they also distinguish electromagnetic, weak and nuclear (strong). It is probably their existence that humanity first realized. From the side of the Earth has been known since ancient times. However, centuries passed before man realized that an interaction of this kind occurs not only between the Earth and any body, but also between different objects. The first person to understand how they work was the English physicist I. Newton. It was he who brought out the now well-known
Formula for gravitational force
Newton decided to analyze the laws according to which the planets move in the system. As a result, he came to the conclusion that the rotation of celestial bodies around the Sun is possible only if gravitational forces act between it and the planets themselves. Realizing that celestial bodies differ from other objects only in their size and mass, the scientist derived the following formula:
F = f x (m 1 x m 2) / r 2, where:
- m 1, m 2 are the masses of two bodies;
- r is the distance between them in a straight line;
- f is the gravitational constant, the value of which is 6.668 x 10 -8 cm 3 /g x sec 2.
Thus, it can be argued that any two objects are attracted to each other. The work done by the gravitational force is directly proportional in magnitude to the masses of these bodies and inversely proportional to the distance between them squared. 
Features of using the formula
At first glance, it seems that using the mathematical description of the law of attraction is quite simple. However, if you think about it, this formula makes sense only for two masses, the sizes of which are negligibly small compared to the distance between them. And so much so that they can be mistaken for two points. But what then to do when the distance is comparable to the size of the bodies, and they themselves have an irregular shape? Divide them into parts, determine the gravitational forces between them and calculate the resultant? If so, how many points should be taken for the calculation? As you can see, not everything is so simple. 
And if we take into account (from the point of view of mathematics) that a point has no dimensions, then this situation seems completely hopeless. Fortunately, scientists have come up with a way to make calculations in this case. They use the integral apparatus and the essence of the method is that the object is divided into an infinite number of small cubes, the masses of which are concentrated at their centers. Then a formula is drawn up to find the resultant force and a limiting transition is applied, through which the volume of each constituent element is reduced to a point (zero), and the number of such elements rushes to infinity. Thanks to this technique, it was possible to obtain some important conclusions.
- If the body is a ball (sphere), the density of which is uniform, then it attracts any other object to itself as if all its mass is concentrated at its center. Therefore, with some error, this conclusion can be applied to planets.
- When the density of an object is characterized by central spherical symmetry, it interacts with other objects as if all of its mass were located at the point of symmetry. Thus, if you take a hollow ball (for example, or several balls nested inside each other (like nesting dolls), then they will attract other bodies in the same way as a material point that has their common mass and is located in the center would do.
