Thursday 25 November 2010

Physics : Gravitation

Law of universal gravitation: F=GMm/r2, where M, m represents the mass of the two bodies (M is usually larger), r represents the distance, and G is the gravitation constant, g=6.67*10-11 with a unit of m3kg-1s-2.
Gravitational field strength is the gravitation force due to an object per unit mass. Mathematically g = F/m = GM/r2, with the unit of Nkg-1, the formula suggests that gravitational field strength is independent of the mass of object is feeling the gravitational force.
Gravitational field strength on Earth's surface is g0 = 9.81Nkg-1 (or 10), and the gravitational field strength above Earth's surface is given by g = g0r02/R2, where r0 is Earth's radius and R is the distance from the object to the center of Earth.
Gravitational field strength is often written in the form of g = GM/r2 = (GM/r0)(1+R/r0)-2 = g0(1+R/r0)-2, where this R is the distance from Earth's surface to the object. When R<<r0 (maybe a difference of thousands times), we estimate the field strength as g = g0(1-2R/r0).
It is suggested that outside the planet the field strength is proportional to r-2, but inside the planet the field strength is proportional to r (under the assumption of the perfectly mass-distributed and sphere planet) since M is proportional to V = kr3, then g = GM/r2 = Gkr.
Kepler's Laws on Planetary Motion
1.       The orbit of every planet is an ellipse with the Sun at one of the two foci.
2.       A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3.       The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit in the same solar system. i.e. T2 is proportional to r3.
The third law can be derived from circular motion, and it's suggested that T2 = 4π2r3/GM.
T2 = (2π/ω)2 = 4π22 = 4π2r/g = 4π2r3/GM.

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