How to find Kepler's law from Newton's laws

[Grimlock_smash]

We were asked to do an experiment where we had to prove the equation:
T²=(4π²m)/Mgr
Where M is the mass of the mass stack in kg (0.3kg), m is the mass of the rubber bung in kg (0.0226kg), T is the time taken for one rotation in seconds, r is the horizontal radius of rotation in meters, and g is the strength of gravity (9.8Nkg-1).

And I tried to relate this to Newton's laws in my report by using:
F=Mgm/r²
and F=mv²/r

Which gave me:
T²=(4π²r³)/Mg

How do I reach the original formula using Newton's laws?

 

[Comeback City]

Make sure you know there is a difference between "g" and "G".
g is the acceleration of gravity (about 9.8 m/s²)
G is the Gravitational Constant (6.67 × 10-11 m³/kg⋅s²)