Question

the distance from the earth to the sun is 1.5 times 10 to the 11 power m ( 93 million miles) and the time for one complete orbit of the earth about the sun is one year. how long would it take a planet located at twice this distance from the sun to complete one orbit?

Answer 1

F = m a

G M m/R^2 = m v^2/R
but T = 2 pi R/v time for circumference rounding
so
v^2 = (2 pi R/T)^2 = (2 pi)^2 R^2/T^2
so

G M/R^2 = (2 pi)^2 R/T^2

G M/(2pi)^2 = R^3/T^2 ( but Kepler knew that :)

= Rnew^3/Tnew^2 = (2R)^3/Tnew^2
= 8 R^3/Tnew^2
so
Tnew^2 = 8 T^2
Tnew = 2 sqrt 2 T
one year * 2 sqrt 2 = 2.83 years

Answer 2

The orbital period of Io, one of Jupiter,s moons, is 1.77 days ( 1.53 x 10 to the 5 power). If Io orbits Jupiter at a radius of 4.22 x 10 to the 8 power meters, what is the mass of Jupiter?