• Mechanics
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• Dynamics
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• Universal gravitation

The simple fact of having mass causes objects to attract each other. In some cases, you can consider to be constant the gravitational force that a planet exerts on objects. However, for the celestial bodies, such as the Earth-Sun system, it is necessary to use the Law of Universal Gravitation.

## Law of Universal Gravitation

Considering the two bodies represented in the above figure: one with a mass $$M$$ and another with mass $$m$$ , separated by a distance $$d$$ , a gravitational attraction will occur between them, whose intensity is: $$F = G\frac{ M m }{d^2},$$ where $$G$$ is a constant of modulus $$6.67 \times 10^{-11} N(\frac{m}{kg})^2$$. The directions of forces are in a straight line connecting the centers of objects, and the direction is from one object to another, as shown in the figure.

### Kepler's Laws

As a result of the attraction that the celestial bodies exert on each other, the following laws can be deduced. They were discovered by Kepler.

$$1^a$$ - Orbits Law
The planets describe elliptical paths, where the Sun occupies one focus of the ellipse.
$$2^a$$ - Areas of Law
The area swept by the radius vector of a planet fills out equal areas in equal times (see figure above).
$$3^a$$ - Law of Periods
The cubes of the average radius of the planets around the sun are proportional to the squares of the periods of revolution. Or mathematically: $$R^3 = k \tau^2,$$ where $$R$$ is the average radius, $$\tau$$ is the period of rotation around the sun and $$k$$ is a constant.

Note: The farther a planet is from the Sun (larger radius), the greater its period of rotation around the Sun ( the greater the $$\tau$$ ).

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