text: Q:
  
  Would a spinning magnet generate a current?
  
  If you were to have a suspended magnet in the shape of an annulus with a wire going through the hole in the center:
  
  Would rotating the magnet like a wheel (staying in place but spinning) generate a current through the wire?
  If so, what formulas and theorems explain this phenomenon?
  
  A:
  
  Yes.  Faraday's Law: Whenever the flux linked or associated with a circuit changes, a voltage is induced in the voltage.
  $$V_{Ind} = N B l v$$
  N = Number of turns.  In your case, N = 1, since it is a straight conductor.
  B = flux density in T (Wb/m$^2$).
  l = length of coil cutting magnetic field in m.
  v - velocity in m/s.
  Overall, you get $\mu$V or mV, but it is Faraday's Law.
  
  Edit...
   
  Magnetic field of a Toroid is shown, with a red wire.
  No rotation = No voltage.
  Rotation but wire is perfectly at center = No voltage.
  But if the wire is not perfectly at center, the wire will cut the lines of flux. 
   The magnetic field changes and you will get you get $\mu$V or mV.
  
  