Understanding the Law of Universal Gravitation
Thanks to the studies of scientists, it has been possible to understand the phenomena of nature, and make technological advances over the years. Newton, based on Galileo's studies of the laws governing the motion of projectiles on Earth, and Kepler's studies of the laws of motion of planets in the solar system, concludes that the force necessary to keep a planet in an orbit depends on the masses and the separation distance. The law of universal gravitation, published in 1687 by Isaac Newton, allows us to determine the force with which two objects with mass are attracted, being very useful in the study of the orbits of comets, the discovery of other planets, the tides, the movement of satellites, among other phenomena.
Basic Concepts to understand "Law of Universal Gravitation"
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Centripetal force:
Force that forces the mobile to bend its trajectory making it describe a circular motion. The centripetal force acts on a body directed towards the center of the circular path. The body experiences a centripetal acceleration since the velocity, of constant modulus, changes direction as it moves. See figure 1.
The centripetal force can be calculated using Newton's second law [1], where the centripetal acceleration can be expressed as a function of angular velocity, linear velocity, or as a function of the period of the body in circular motion. See figure 2.
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Kepler's Laws
The astronomer Johannes Kepler explained the movement of the planets of the solar system, by means of three laws: the law of orbits, areas and periods. [two].
Kepler's first law, or law of orbits:
all the planets in the solar system revolve around the sun in an elliptical orbit. The sun is in one of the two foci of the ellipse. See figure 3.
Kepler's second law, or law of areas:
The radius that joins a planet to the sun describes equal areas in equal times. The (imaginary) line that goes from the sun to a planet, sweeps equal areas in equal times; that is, the rate at which the area changes is constant. See figure 4.
Kepler's third law, or law of periods:
For all planets, the relationship between the cube of the radius of the orbit and the square of its period is constant. The major axis of the ellipse cubed and divided by the period (time to make a complete revolution) is the same constant for the different planets. The kinetic energy of a planet decreases as the inverse of its distance from the sun. See figure 5.
Law of Universal Gravitation
The law of universal gravitation, published in 1687 by Isaac Newton, allows us to determine the force with which two objects with mass are attracted. Newton concluded that:
- Bodies are attracted by the mere fact of having mass.
- The force of attraction between the bodies is only noticeable when at least one of the interacting bodies is enormously large, like a planet.
- There is an interaction at a distance, therefore, it is not necessary for the bodies to be in contact for the attractive force to act.
- The gravitational interaction between two bodies always manifests itself as a pair of forces equal in direction and modulus, but in the opposite direction.
Statement of the Law of Universal Gravitation
The force of attraction between two masses is directly proportional to the product of the masses and inversely proportional to the square of the distance that separates them. The attractive force has a direction that coincides with the line that joins them. [3]. See figure 6.
The constant of proportionality G between the quantities is known as the universal constant of gravitation. In the international system it is equivalent to:
Exercise 1. Determine the force with which the bodies in figure 7 are attracted in a vacuum.
Solution
In figure 8 there are two bodies with masses m1 = 1000 kg and m2 = 80 kg, separated by a distance of 2 meters. Applying the universal law of gravitation, the force of attraction between them can be determined, as shown in figure 8.
Deduction of the Law of Universal Gravitation
Starting from Kepler's third law that relates the radius to the period of an orbiting planet, the centripetal acceleration experienced by a planet is inversely proportional to the square of the radius of its orbit. To find the centripetal force that acts on the planet, Newton's second law [] is used, considering the centripetal acceleration it experiences, expressed as a function of the period. See figure 9.
The value of the universal constant of gravitation was determined by Henry Cavendish many years after Newton's law of gravitation was established. The constant G is considered "universal" since its value is the same in any part of the known universe, and it is independent of the environment in which the objects are found.
Exercise 2. Determine the mass of planet Earth, knowing that the radius is 6380 km
Solution
The bodies located on the surface of the earth are attracted towards its center, this force is known as the weight of a body (force with which the Earth attracts it). On the other hand, Newton's second law can be applied expressing the weight of the body as a function of gravity, thus the mass of the Earth, known as its radius, can be obtained. See figure 11.
Application of the law of universal gravitation
The law of universal gravitation is useful to explain the orbit of comets, the discovery of other planets, the tides, the movement of satellites, among other phenomena.
Newton's laws are fulfilled exactly, when it is observed that some star does not comply with it, it is because some other non-visible star disturbs the movement, thus the existence of planets has been discovered from the disturbance that they produce in the orbits of known planets .
Satellites:
A satellite is an object that orbits around another larger object with a greater gravitational field, for example, you have the moon, the natural satellite of planet Earth. A satellite experiences a centripetal acceleration because it is subjected to an attractive force in the gravitational field.
Exercise 3. Determine the speed of a satellite orbiting the earth at 6870 km from the center of the earth. See figure 12
Solution
Artificial satellites are kept in orbit around the Earth due to the force of attraction that the Earth exerts on it. Using the universal law of gravitation and Newton's second law, the speed of the satellite can be determined. See figure 13.
CONCLUSIONS
Every material particle attracts any other material particle with a force directly proportional to the product of the masses of both and inversely proportional to the square of the distance that separates them.
The gravitational interaction between two bodies always manifests itself as a pair of forces equal in direction and modulus, but in the opposite direction.
Newton's law of universal gravitation allows us to determine the force with which two objects with mass are attracted, knowing that the force of attraction between two masses is directly proportional to the product of the masses and inversely proportional to the square of the distance that separates them.