Gravitational forces: the concept and features of the application of the formula for their calculation

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, they still distinguish electromagnetic, weak and nuclear (strong). Probably, it was their existence that humanity first realized. The force of gravity from the Earth was known from the earliest times. However, centuries passed before a man guessed that this kind of interaction takes place not only between the Earth and any body, but also between different objects. The first to understand how gravitational forces work was the English physicist I. Newton. It was he who brought out to everyone the now known law of universal gravitation.

The formula of gravitational force

Newton decided to analyze the laws by which the planets move in the system. As a result, he came to the conclusion that the rotation of the celestial bodies around the Sun is possible only if gravitational forces act between it and the planets themselves. Realizing that the celestial bodies from other objects differ only in their size and mass, the scientist derived the following formula:

F = fx (m ₁ xm 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 х sec ² .

Thus, it can be argued that any two objects are attracted to each other. The work of gravitational force in its magnitude is directly proportional to the masses of these bodies and inversely proportional to the distance between them, squared.

Features of application of the formula

At first glance, it seems that it is quite easy to use the mathematical description of the law of attraction. However, if you think about it, this formula makes sense only for two masses, whose dimensions are negligibly small in comparison with the distance between them. And so much that they can be taken as two points. And how then to be, when the distance is comparable with the sizes of bodies, and they have an irregular shape? Divide them into parts, determine the gravitational forces between them and calculate the resultant? If so, how many points should I take to calculate? As you can see, not everything is so simple. And if we take into account (from the point of view of mathematics) that there is no point of dimensions, then this situation seems hopeless at all. Fortunately, scientists have come up with a way to make calculations in this case. They use the apparatus of integral and differential calculus. The essence of the method is that the object is divided into an infinite number of small cubes, the masses of which are concentrated in their centers. Then a formula is made for finding the resultant force and a limiting transition is applied, whereby 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, some important conclusions have been obtained.

1. If the body is a sphere (sphere) whose density is uniform, then it attracts to itself any other object as if its entire mass is concentrated in its center. Therefore, with some error, this conclusion can also be applied to planets.
2. When the density of an object is characterized by central spherical symmetry, it interacts with other objects as if the entire mass of the object is at a symmetry point. Thus, if you take a hollow ball (for example, a soccer ball) or a few nested balls (like matryoshka dolls), they will attract other bodies in the same way as a material point with their total mass and located in Center.