ES-175 Parallel Axis Theorem
If the moment of inertia about an axis is known and is required about a parallel axis, the parallel axis theorem is employed.

The unprimed axis is a general axis and the primed axis is centroidal
    
    
(1) (2) (3)

The first integral represents Ix'x' the centroidal moment of inertia about the x-axis
    

The second integral is equal to zero because
    y'A = y'dA
where
    y' - distance from x' axis to centroid
but
    y' = 0 because the primed axis coincides with the centroid
hence
    y'A = 0
and
     ò y'dA = 0

The last integral
    
therefore
    
and
    
and
    J = J' + Ad2
where
    

The radius is given by
    k2 = k' 2 + d2
where
    k - radius of gyration in general coordinates
    k'- radius of gyration in centroidal coordinates

Note: The transfer of axis theorem may not be used between two non-centroidal parallel axes.
Two transfers, one from the original to the centroidal and one from the centroidal to the new axes must be performed.


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