State and prove parallel axis theorem

Explanation

Parallel axis theorem. Parallel axis theorem states that 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 centre of mass and the product of the mass of the body and the square of the perpendicular distance betwen the two axes.

If `I_(C)` is the moment of inertia of the body of mass M about an axis passing through the centre of mass, then the moment of inertia I about a parallel axis at a distance d from it is given by:
`I=I_(C)+Md^(2)`
Let us consider a rigid body as shown in the figure. Its moment of inertia about an axis AB passing thorugh the centre of mass is `I_(C)`. Let DE be another axis parallel to AB at a perpendicular distance d from AB. The moment fo inertia of the body about DE is I. Let us attempt to get an express for moment of inertia of the body in tems of `I_(C)`. For this, let us consider a point mass m on the body at position x from its centre of mass.
The moment of inertia of the point mass baout the axis DE is `m(x+d)^(2)`.
The summation of the above expression given by the moment of inertia I of the whole body about DE is
`I=Sigmam(x+d)^(2)`
`I=Sigmam(x^(2)+d^(2)+2xd)`
`1=Sigma(mx^(2)+md^(2)+2dmx)`
`1=Sigma x^(2)+Sigma d^(2)+2dSigma mx)`
Here `Sigma mx^(2)` is the moment of inertia of the body about the centre of mass.
`:.I_(C)=Sigmamx^(2)`
The term `Sigma m=0`, because x can take positive and negative values with respect to the axis AB.
Thus `I=I_(C)+Sigma md^(2)=I_(C)=(Sigma m)d^(2)`
Here `Sigma m` is the entire mass M of the object `(Sigma m=M)`
`I=I_(C)+md^(2)`
Hence, the parallel axis theorem is proved.