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).
Newton's law of gravitation is: FG=(GMm)/(r2).First of all, it's a good idea to draw a diagram of the planet and star, labelling the directions of the centripetal force and and the planet's velocity in particular, along with anything else that helps visualise the question. We know that the equation for centripetal force is FC=mω2r (from circular motion). Since this centripetal force FC and the gravitational force FG point in the same direction (from the planet to the star), we can equate them!
This gives us: (GMm)/(r2) = mω2rSubstituting in ω=2π/T, we get: (GMm)/(r2) = (4π2mr)/(T2)We can see that the two 'm's cancel out, and the 'r's combine to make r3.Do a bit of rearranging: T2 =(4π2r3)/(GM)There it is! T2 is proportional to r3; this is known as Kepler's 3rd Law of planetary motion.
