NO ME SALEN
APUNTES TEÓRICOS Y EJERCICIOS RESUELTOS DE FÍSICA DEL CBC
Ley de Gravitación Universal

Gravitational forces

The Gravitational Law was attributed to Isaac Newton (1643-1727), being the culmination of a cientific revolution. It states something pretty easy to understand, difficult to believe, and catastrophically revolutionary.

It says that two bodies are attracted mutually with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. All these can be resumed into this equation:

In which F_G is the gravitatory force (always attraction, never a repulsive one); m₁ and m₂ are the masses of the bodies which are being attracted; d₁₂ is the distance between the centers of the bodies (later we will reach a better definition of center); and G  is the constant of proportionality, which allows us to transform the law of proportionality into a law of equality.   

In this mutual attraction, as in all attractions, two forces appear with the same modulus and direction, but in opposite sense, one in each body: F_G12 = F_G21 (the action-reaction law is always present). In a gravitatory interaction we will refer to any of the two and will call it F_G.                        

Why I say, it's easy to understand:

What means that the force of gravitatory attraction is directly propotional to the product of the mass which are being attracted? If in identical conditions there was an attraction between these two bodies with mass:

m₁ = 2 kg and m₂ = 10 kg,

would attract themselves with less intensity than these other two bodies, with mass:

m'₁ = 3 kg and m'₂ = 7 kg,

The second pair attracts themselves more easily, because the product of the last two is greater that the one from the first two. Easy.

¿Y qué significa que es inversamente proporcional al cuadrado de las distancia que las separa? Supongamos que tenemos un par de cuerpos de masa unidad... distanciados una distancia d. Si las separo una distancia 10 d... la fuerza con la que se atraerán será 100 veces menor. Y si ahora las acerco para que queden separadas una distancia d/2... entonces la fuerza con la que se atraerán será 4 veces mayor que la inicial. Fácil, ¿no?

And what does it mean that it is inversely proportional to the square of the distances that separate them? Lets suppose we had a pair of bodies with the same mass... distanced a distance d. If I separate them a distance 10 d... the force of attraction will be 100 times smaller. If I bring them closer so as to keep them separated a distance d/2... then the force of attraction will be 4 times more stronger that the first one. Easy, right?

Why it is difficult to believe

The constasnt of proportionality, G, known as the Constant of Universal Gravitation, only depends of our universe... its value is independent of any time and place, any circuntance and material enviroment... and its value is...

G = 6,67 x 10^-11 Nm²/kg²

A reeeally tiiiny value:

G = 0,000 000 000 066 7 Nm²/kg²

Almost -I would say- nothing. The constant's units are neccesary so as to the gravity force to me measured in the units of force, N. That's why the numerator has the unit of measurement to the square, m², to cancel out the distance to the square from the Law; in the denominator units of mass to the square, kg², to cancel out the mass from the Law.

Newton died without knowing the value of G. He knew is was reeeeally small... but we had to wait almost 200 years to know its value with precision, and when sir Henry Cavedish achieved to measure it, he didn't realize (neither his co-workers) what was actually calculated. Its a really curious stroy, if it is of your interest, I will tell you about ir here.

The thing is that the gravitatory forces are insignificant when we consider the size of the bodies: apples, bibles, cars, airplanes... and almost impossible to measure with common intruments. If we were talking about bodies and told us that there are force of attraction, we would never guess that it was really gravitational.

Why I say is catastrophically revolutionary.

The key is in the word universal. Since Aristoteles onwards it was thought that there were two physics: one to explain the celestial universe and the other one to the earthly one. The Law of Universal Gravitation comes to tell us the universe is only one and the Physics that describes it also: only one.

The force that maked the bodies fall (the old and familiar weight), Isn't another thing that the force of gravitational attraction between the one that falls and the Earth. Nowadays the weight is the name we give to the gravitatory force, here, in the Earth's surface.

The same law that describes the fall of two bodies, describes the Moon's, planet's, stars', galaxies' orbits. The mith about the apple falling over Newton's head exactly when he realizes that the Moon also has the same law, is hugely descriptive about the unification of the universe. I tell you about it here. A catastrophy of the aristotelic physics, which stopped being valued in the stock exchange.

• Se cuenta que la ley inversa al cuadrado de la distancia pertenece a Cristian Huygens, y Newton no se lo reconoció nunca. Más aún, la célebre frase "a hombros de gigantes" era una alusión irónica y ofensiva a Huygens, que era bastante petiso.
• Newton calló sus convicciones por más de 15 años debido a que no existía en aquella época una matemática suficientemente potente como para sostener sus afirmaciones que, suponía, serían fuertemente resistidas. En el interín desarrolló (paralelemente con Gottfried Leibniz) el análisis matemático. Cuando el cálculo numérico (un sinónimo) fue un hecho entre la comunidad científica, presentó la Ley... y no hubo con qué darle.
• La parte de la teoría más difícil de sostener era que los cuerpos se comportaban (para la Ley de Gravitación) como si toda su masa estuviera reunida un un sólo punto, su centro de masa (generalmente el centro geométrico del cuerpo), y las distancias entre los cuerpos había que tomarlas desde ahí. Ese asunto espinoso cedió al ser abordado con una herramienta matemática del cálculo numérico: la integral.
• En 1915 Albert Einstein enuncia la Teoría General de la Relatividad que reemplaza la gravedad por la deformación del espacio producida por las masas. La pista que condujo a Einstein hasta la Relatividad fue ésta: la masa a la que alude la Ley de Gravitación no era la misma masa a la que alude la Segunda Ley de la dinámica... es totalmente cierto, y a todos se les había pasado por alto. (Actualmente se las distingue llamándolas masa gravitatoria y masa inercial respectivamente). Pero vos no tenés que preocuparte. Aunque sean cosas distintas no está mal que no las distingamos, ya que a los efectos de la dinámica traen las mismas consecuencias. Existe un principio que reconoce ésto, y se llama principio de correspondencia.

• Johannes Kepler llegó a formular tres leyes celestiales (Leyes de Kepler) bastante antes de que naciera Newton, por la simple observación del cielo (el observador no fue él sino su antesesor, Tycho Brade). Las tres leyes dejaron de ser leyes para convertirse en derivaciones de una ley anterior: la ley de Gravitación Universal. ¿Qué dice cada una de las 3 Leyes de Kepler?
• ¿En qué situaciones deja de ser verdadera la Ley de Gravitación Universal?