Question

Kepler's law of planetary motion and derive newtons law of gravitation from this

Answer 1

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You can start with Kepler’s third law that the square of the time-period varies directly with the cube of the distance.

T2∝r3

⟹T2r3=η(1)

Where η is just a constant.

For an object describing circular motion, you can write a couple equations:

T=2πrv(2)

v2r=a=Fm(3)

Putting (2) and (3) in (1):

m⋅v2r=4π2ηmr2

⟹ma=4π2ηmr2

⟹F=4π2ηmr2(4)

This is all you can get from Kepler’s laws of motion. From here, it’s good ol’ argumentation. Since momentum is conserved, according to Newton’s third law, the force acting on one mass by another would have to be equal and opposite to the force acting on the other mass by the first mass. Then, from (4) it means that 4π2η∝M

Answer 2

Answer:

Kepler's laws and Newton's laws taken together imply that the force that holds the planets in their orbits by continuously changing the planet's velocity so that it follows an elliptical path is

(1) directed toward the Sun from the planet,

(2) is proportional to the product of masses for the Sun and planet, and

(3) is inversely proportional to the square of the planet-Sun separation. This is precisely the form of the gravitational force, with the universal gravitational constant G as the constant of proportionality.

Thus, Newton's laws of motion, with a gravitational force used in the 2nd Law, imply Kepler's Laws, and the planets obey the same laws of motion as objects on the surface of the Earth.

Hope it helps you :)