Question

state Newton law of gravitation by using keplar's third law

Answer 1

Kepler's Third Law can be written as

P2 = kr3

or  P2 = (4π2/GM).r3  (1)

 

now, we know that the time period, P = 2πr/v, so (1) becomes

(2πr/v)2 =  (4π2/GM).r3

or  4π2 (r/v)2 = (4π2/GM).r3

or  (r/v)2 = (1/GM).r3 = r3/GM   (2)

 

we know that, 

centripetal acceleration a = v2/r  [from ma = mv2/r]

thus, equation (2) becomes

 

r/a = r3/GM

or  1/a = r2/GM

or  a = GM/r2    (3)

 

and we also know that F = ma, thus using equation (3) we would get

 

F = ma = m(GM/r2)

 

thus, we have derived the Newton's Law of Gravitation, which is

 

F =  GMm/r2

 

from Kepler's Third Law