The mathematical expression of centripetal force: f = mv2r = mω 2r.

The mathematical expression of the relationship between linear velocity and the period of uniform circular motion: v = 2π rt

The mathematical expression of the relationship between angular velocity and the period of uniform circular motion: ω = 2π t 。

When an object is on the surface of a planet, the gravity it receives is equal to gravity: mg = gmmr 2.

So the answer is: v = GMR = grF = mv2rf = mω2r；; v = 2πrT； ω= 2πT； mg=GMmR2。