Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?

Explanation

Suppose orbital radius of earth around sun is = r
Orbital radius of planet around sun is = r.
Mass of earth `= M_E` , Mass of sun = `M_s`
Angular speed of earth around sun = `omega`
Angular speed of planet around sun = `omega.`
`implies` Since earth is rotating around sun, centripetal force on earth provided by the gravitational attraction force between sun and earth.
`:. (M_Ev^2)/r=(GM_EM_s)/r^2`
`:. v^2 = (GM_s)/r`
`:. omega^2=(GM_s)/r^3 " "...(1)" " [ :. v = romega]`
and similarly for planet ,
`(omega.)^2=(GM_s)/((r.)^3) " "...(2)`
`:. omega^2/((omega.)^2)=((r.)/r)^3`
[ `:. ` Taking ratio of equ (1) and (2) ]
`:. (1/2)^2 =((r.)/r)^3 " "[ :. omega. = 2omega]`
`:. ((r.)/r)^3 =1/4`
`:. r. = r xx (1/4)^(1/3) = r xx 1/ (1.588) ~~ 0.63 r`