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Gravitation



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Gravitation



Physics Assignment Help & Live Online Tutoring From Our Expert Physics Tutors

Physics    Newtonian Mechanics    Gravitation

The Motions of Planets and Satellites:

          The motion of planets and satellites are governed by three laws given by Kepler. These three laws are given as follows:

1) The orbit of every planet is an ellipse with the sun at one of the foci. An ellipse is characterized
by its two focal points;

2) A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the
planet travels along its orbit.

3) The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (the "half-length" of the ellipse) of its orbit. That is:


The most general result is:



where: 

T = object's sidereal period
a = object's semimajor axis
G = 6.67 × 10−11 N • m²/kg² = the gravitational constant
M = mass of one object
m = mass of the other object 

            Let us assume that the body of mass M is at rest in an inertial frame of reference. Then the total mechanical energy E of two body system when the bodies are kept at a distance r from each other is the sum of kinetic energy of the body of mass m and the potential energy of the system is given by,


Thus, the total energy is given by,



When we have a system consisting a body of mass m moving in a circular orbit about a body of mass M >> m, then Newton’s second law applied to the body of mass m gives

Now the total mechanical energy of the object is given by,


Thus, the total mechanical energy is negative in case of circular motion. It is worth noting that the kinetic energy is positive and is equal to one-half of the potential energy.




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