derivation of Newton's law of gravitation using Kepler's third law
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Newton knew that the force of gravity was directly proportional to the product of the two masses. In order for this to hold true, a constant had to be added to the first equation. ... You can prove that Kepler's 3rd Law (period squared is proportional to semi-major axis cubed) is only true for a 1 / r 2 force
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it is called inverse square rule
From keplers 3rd law of planetary motion
T^2 directly proportional to r^3
so... T^2÷r^3 = n - - - - 1st eqn(where n is constant)
for object describing circular motion
T = 2pi r÷v----- 2nd eqn
also, v^2 ÷r =a=F÷m- - - - 3rd eqn
put 2 and 3 in 1 st eqn
m×v^2÷r = 4pi^2nm÷r^2
ma= 4 pi^2 nm÷r^2
f= 4pi^2nm÷r^2
From keplers 3rd law of planetary motion
T^2 directly proportional to r^3
so... T^2÷r^3 = n - - - - 1st eqn(where n is constant)
for object describing circular motion
T = 2pi r÷v----- 2nd eqn
also, v^2 ÷r =a=F÷m- - - - 3rd eqn
put 2 and 3 in 1 st eqn
m×v^2÷r = 4pi^2nm÷r^2
ma= 4 pi^2 nm÷r^2
f= 4pi^2nm÷r^2
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