Theorem of parallel axis

Theorem of parallel axis: The moment of inertia of a body about an axis is equal to the sum of (i) its moment of inertia about a parallel axis through its centre of mass and (ii) the product of the mass of the body and the square of the distance between the two axes axes

 Let lo be the moment of inertia (MI) of a body of mass M about an axis through its centre of mass C, and I be its MI about a parallel axis through any point O Leth be the distance between the two axes Consider an infinitesimal mass element dm of the body at a point P. It is at a perpendicular of radius of gyrationed by expressingdistance (k) fromdefined as give distance CP from the rotation axis through C and a perpendicular distance OP from the parallel axis through O. The MI of the element about the axis through C is CP de. Therefore,