Kepler's Laws

In the 17th century, Johannes Kepler formulated three laws, which are the principles of planetary motion, from previous observations made by Tycho Brahe. As Isaac Newton demonstrated, Kepler's three laws are the results of the universal physical laws. Therefore, Kepler's laws does not only apply to the motion of planets around the sun, it also applies to the motion of any object in orbit around any other object.

Law (1) Every planet revolves around the sun in an elliptical path, while the sun is located at one of the foci of the ellipse. From his observations, Tycho Brahe presented Kepler the problem of discovering a theory to explain the motion of mars. He tried using combinations of epicycles and eccentrics, but failed to obtain agreement between theory and observation. He finally guessed that the orbit would be an ellipse, and discovered that his theory would work as long as the sun was at one of the foci. He then proved that the orbits of all the planets are indeed ellipses. Venus is the most circular with an eccentricity of 0.0068. Pluto is the most eccentric, 0.2485.

Law (2) Law of Areas: The radius vector, or the line joining the sun and a planet, sweeps over equal areas in equal intervals of time. To prove and demonstrate the second law, Kepler performed an extremely large amount of computations using eight years worth of data. The nearer a planet is to the sun, the faster it moves in its orbit.

Law (3) Law of Harmonics: The cube of the average radius of a planet is proportional to the square of its period (time of one orbit). This law shows the relation of the distance of a planet to its period of revolution. It can be expressed using the formula below, where R is the radius and T is the period(time of one rotation).

Centripetal Force and Acceleration

Acceleration can be defined as the changing of an object's velocity, or speed. An object moving at constant speed in a circular path is said to be in uniform circular motion. Although the object is moving at constant speed, there is still centripetal acceleration. This acceleration is not changing the speed but it is necessary to change the direction of motion, keeping the object moving in its path, or orbit. Centripetal acceleration points towards the center of the circular path in which the object is moving. The magnitude of the centripetal acceleration can be found using the formula below, where v is the speed and r is the radius of the circular path.

From Newton's second law, Force equals mass times acceleration, we know that a certain force is causing centripetal acceleration and thus, creating circular motion. That force is called centripetal force and is also directed towards the center of the circular path. By simply using Ac instead of A in Newton's second law, we get the formula for centripetal force.

Newton's Law of Gravitation

Newton's Universal Law of Gravitation states that any two objects exerts a gravitational force of attraction on each other. The magnitude of the force, or gravitational pull on both objects, is proportional to the product of the masses of the objects, and inversely proportional to the square of the distance between them. Meaning that the greater the mass the greater the force and the greater the distance the smaller the force. That is why objects we encounter does not seem to have any attraction, because the masses are too small to be significant.

In this equation, there are two masses, m1 and m2, a separation distance of d, and a constant, G. By plugging in your own weight and the mass of the earth and the distance to the center of the earth the resulting force would be your exact weight.