need a deravation for parallel axis theorem
In physics, the parallel axis theorem or Huygens–Steiner theorem can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (r) between the axes.
The moment of inertia about the new axis z is given by:
- is the moment of inertia of the object about an axis passing through its centre of mass;
- is the object's mass;
- is the perpendicular distance between the two axes.
The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region D: