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: F_G=(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 F_C=mω²r (from circular motion). Since this centripetal force F_C and the gravitational force F_G 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.