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 I_{x'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' + Ad ^{2}where |

The radius is given by k ^{2} = k'^{ 2} + d^{2}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. |