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 interia 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 interia 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 interia 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 interia of the body about the center of mass and `sum mx^(2)=0` (since `x` has `+ve` and `-ve` values about the axis `AB`)
`(ix)` Therefore ,The moment of interia of the whole body about the axis `DE` can be expressed as,
`I=I_(c )+sumMd^(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 .