Physics : Gravitation

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 = 4π²/ω² = 4π²r/g = 4π²r³/GM.