State and prove parallel axis theorem

Explanation

(i) Statement : The moment of inertia of a body about any axis is equal to the sum of its moment of inertia about a parallel axis through its center of mass and the product of the mass of the body and the square of the perpendicular distance between the two axes. To Prove : `Ic + Md^(2)`
(ii) Proof : Let us consider a rigid body as shown in figure.

(iii) Let `I_(C)` be the moment of inertia of the body about an axis AB, which passes through center of mass.
(iv) Consider I is the moment of inertia of the body to be found about an axis DE, which is parallel to AB. and d is the perpendicular distance between DE and AB.
(v) Let P be the point mass of mass m, which is located at a distance x from its center of mass.
(vi) The moment of inertia of the point mass about the axis DE is,
`dI = m (x + d)^(2)`
(vii) The moment of inertia of the whole body about the axis DE is,
`{:(I = sum m(x + d)^(2)),(I = sum m (x^(2) + d^(2) + 2xd)),(I = sum (mx^(2) + md^(2) + 2dmx)),(I = sum mx^(2) + sum md^(2) + 2d sum mx):}`
(viii) Here, `sum mx^(2) = I_(C)`, the moment of inertia of the body about the centre of mass and `sum mx^(2) = 0` (since x has +ve and -ve values about the axis AB).
(ix) Therefore, The moment of inertia of the whole body about the axis DE can be expressed as,
`I = I_(C) + sum Md^(2)`
(x) But `sum m = M`, mass of the whole body. Thus
`I = I_(C) + Md^(2)`
(xi) Hence, the parallel axis theorem is proved.