Kepler's third law

Question

Sir/madam as F=GMM/R2 and by Kepler's third law it is T2 proportion to R3

HENCE can I say F=GMMK/T4\3 where K is arbitrary constant after removing proportionality sign ?

Answer 1

Hi Prashant!

 

So, by Kepler's third law (if we make the not-bad assumption that, though the planets are in elliptical orbits, they are pretty nearly circular, and so the semi-major axis can be thought of as the orbital "radius" R):

 

R = K'T^2/3, where K' is the cube root of the standard Kepler constant

 

Then you can say:

 

R² =K"T^4/3  where K" = (K')²

 

And then substitute it into the gravitational law.  Within the approximation that the planet's elliptical orbit is circular, it is a fair statement to say that:

 

F = GMM/R² = GMMK/T^4/3 (where K = 1/K")

 

though you could also lump GM_sm_p into your constant and make the force proportional to just 1/T^4/3 in this case. (M_s = solar mass; M_p = planet mass)

 

You can also make several other relationships by noting that, for circular motion, we also have:

 

T = (2πr)/v  

F_g = F_c = m_pv²/R