Question

how T Square is directly proportional to R cube

Answer 1

Kepler's Third Law for Earth SatellitesThe velocity for a circular Earth orbit at any other distance r is similarly calculated, but one must take into account that the force of gravity is weaker at greater distances, by a factor (RE/r)2. We then getV2/r = g (RE r)2 = g RE2/r2Let T be the orbital period, in seconds. Then (as noted earlier), the distance 2 πrcovered in one orbit equals VT>VT= 2 π rV= 2 π r/TV2= 4 π2r2/T2V2/r= 4 π2 r/T2and by the earlier equality4 π2r/T2 = g RE2/r2Get rid of fractions by multiplying both sides by r2T24 π2r3 = g RE2 T2To better see what we have, divide both sides by g RE2, isolating T2:T2 = (4 π2/g RE2) r3What's inside the brackets is just a number. The rest tells a simple message--T2 is proportional to r3, the orbital period squared is proportional to the distance cubes. This is Kepler's 3rd law, for the special case of circular orbits around Earth.If you are not yet tired of the calculation, you may click here for turning the above into a practical formula.