Using Newton's law of universal gravitation, show that T^2 is proportional to r^3 (where T is the orbital period of a planet around a star, and r is the distance between them).

(Lets consider a simple planetary system composed of a planet orbiting a star. the gravitational force between the two is given by F=(GMm)/(r²). Assuming the planet also moves in a circular orbit, we can consider the centripetal force, F=mω²r. As both gravitational and centripetal forces act in the same direction, we can equate them to find (GMm)/(r²)=mω²r.
We note that 'm' cancels and we can divide through by 'r' to arrive at GM/r³=ω². ω is simply angular frequency given by ω =2π/T. Substituting this into our expression we find that GM/r³= 4π²/T².After some simple rearranging, we note that  T² =(4π²r³)/(GM). So  T² is indeed proportional to  r3 . This simple statement is known as Kepler's third law of planetary motion.