State and prove perpendicular axis theorem.

Explanation

Perpendicular axis theorem states that the moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moments of inertia about two perpendicular axes lying in the plane of the body such that all the three axes are mutually perpendicular and have a common point.
Let the X and Y-axes lie in the plane and Z-axis perpendicular to the plane of the laminar object. If the moments of inertia of the body about X and Y-axes and `I_(X)` and `I_(Y)` respectively and `I_(Z)` is the moment of inertia about Z-axis, then the perpendicular axis theorem could be expressed as,
`I_(Z)=I_(X)+I_(Y)`
To prove this theorem, consider a plane laminar object of negligibel thickness on which the origin (O) lies. The X and Y -axis lie onteh palne and z-axis is perpendicular to it as shown in the figure. Consider thelamina is made up of a lage number of particles of massm. one such particle is selected at a pointP that has coordinates (x,y) at a distance r from O.
The moment of inertia of the particle about Z-axis is `mr^(2)`.
The summation of the above expression gives the moment of inertia of a entire lamina about Z-axis as `I_(Z)=Sigma mr^(2)`.Then `I_(Z)=sigma m(x^(2)+y^(2))`
`I_(Z)=Sigma mx^(2)+Sigma my^(2)`
In the above expression, the moment of inertia of the body about the Y-axis is `Sigmamx^(2)`
Similarly, the moment of inertia about X-axis is `Sigmamy^(2)`
`:.I_(x)=Sigma my^(2)`
`I_(y)=Sigmjax^(2)`
Substitutig in the equation for `I_(z)` gives,
`I_(Z)=I_(X)+I_(Y)`
Thus, the perpendicular axis theorem is proved.