Answer to Question #163646 in Field Theory for Hanna

Question

State in words, Newton’s law of Gravitation.

By considering the centripetal force which acts on a planet in a circular orbit, show that T²  is directly proportional to R³ , where T is the time taken for one orbit around the Sun and R is the radius of the orbit.

Answer 1

Answer

Newton's law of gravitation, statement that any particle of matter in the universe attracts any other with a force varying directly as the product of the masses and inversely as the square of the distance between them as

$F=\frac{Gmm'}{r^2}$

According to kepler law

$\frac{dA}{dt}=\frac{\pi ab}{T}=\frac{L}{2m}$

$T=\frac{2\pi abm}{L}$

$T=\frac{2\pi a^{\frac{3}{2}}}{\sqrt{Gm}}$

This implies

$T^2\propto a^3$