Alright, obviously for this problem we are using rotational kinetic energy, given by K = 1/2 * I * w^2 where I is the moment of inertia (angular mass) and w is the angular velocity.
We also know that for any solid, uniform disk on an axis of rotation through its center of mass the moment of inertia is I = 1/2 * M * R^2 , where M is the mass of the disk and R is the radius.
We also know from the given statements that the kinetic energy this flywheel must store is 1.80 * 10^9 J.
w = sqrt(2 * K / I ) = sqrt(4 * K / ( M * R^2) )= sqrt(4 * (1.80*10^9 J)/( [12kg]*[0.33m]^2))
= 74,230 rad/s
Now we simply have to convert this rad/s to rev/min:
1 rev = 2*pi rad, 1 min = 60 sec
w = (74,230 rad/s) * (1 rev/ 2*pi rad) * (60 sec/ 1min)
...= 708,840 rev/min
which is REALLY freakin' fast...
- but on a side note this could easily be decreased by limiting the range. I know that they have actually made successful vehicles using this technology, but they often go only 100 miles (also they are buses which can support a heavier flywheel or simply make the flywheel with a larger radius (this will have the greater impact))
EDIT: I accidentally left out a square root earlier, I fixed it now...sorry :P
Hope this is understandable :)
I'm a physics tutor at Iowa State University