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Newton's force of gravity law as applied to earth mass, m, and sun mass, M, is; FGravitational=GMmr2F_{Gravitational}= G\frac{Mm}{r^2}FGravitational=Gr2Mm
Where r is the radius vector, the varying distance from earth to sun, and G is the gravitational constant.
FGravitational is the amount of force that acts in a straight line between the planet and sun. This force would place the planet in free fall toward the sun if it were not for the counteracting planet inertial force.
What is the inertial force on the planet?
Newton’s inertial force law states that the inertial force is equal to the acceleration of the planet times the mass of the planet
Finertia=maRadialF_{inertia}=ma_{Radial}Finertia=maRadial
The inertial and gravitational forces must be equal to each other in magnitude but opposite in direction, or else the planet would leave its orbit. With unequal forces, the planet would fall into the sun, or attain a different orbit in a new equilibriu m path, or go spinning off into space. Since the planet does maintain its orbit, the sum of the two forces must be zero.
FGravitation+Finertia=0F_{Gravitation} + F_{inertia} = 0 FGravitation+Finertia=0
GMmr2+maRadial=0G\frac{Mm}{r^2}+ ma_{Radial}=0Gr2Mm+maRadial=0
Divide through by m and obtain;
GMr2+aradial=0G\frac{M}{r^2}+ a_radial=0Gr2M+aradial=0
Then, GMr2=−aRadialG\frac{M}{r^2}=- a_{Radial}Gr2M=−aRadial
This is an important place in the proof where the inertial mass is assumed to be identical to gravity mass and the radial acceleration is shown opposite to the attraction of gravity. We must continue to be skeptical of these assumptions, including dist ance to the second power, until we derive the elliptical path of the planet around the sun.
This equation of the radial acceleration shows that aRadial is proportional to the inverse of distance squared.
By applying some mathematics we will modify the equation to obtain aRadial as a function of r and q. This radial acceleration equation is the basic equation that will evolve into the equation showing that the earth orbit is an ellipse.
Notice also that Newton's inertial force law can be considered simply as the definition of the unit of force. Once the standards of kilogram, meter and second are agreed upon, the unit of inertial force is established. We need a constant, (G), to make the gravitational units of force have the same dimensions and the same magnitude as inertial force units. But we have no reason (as yet), to believe that the inertial force, based on random but agreed upon standards, is directly proportional to the gravit ational force. We just assumed the equivalence when we canceled "m" in the above derivation. If the path of the earth around the sun is analytically determined to be an ellipse, then the assumption is correct