State and prove perpendicular axis theorem.

Explanation

Perpendicular axis theorem : the perpendicular axis theorem holds good for plane laminar starts that the 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 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 are ` I_(X) and I_Y` respendicular and have a common point .
Let the X and Y axes lie in the plane and z - axis ,then the perpendicular axis theorem could be expressed as ,
`I_(Z) = I_(X ) +I_(Y)`
to prove this theorem . let us consider a plane laminer object of negligible thickness on which lies the origin (O ). The X and Y - axes lie on the plane and Z-axis is perpendicular to it as shown in figure .The lamina is considered to the made up of a large number of particles of mass m . Let us choose one such partivle at a point P which 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 hte above expression gives the moment of inertia of the lamina about Z- axis as , `L_(Z) = sum mr^2`
Here ,` r^2 =x^2 +y^2`
then ` I_(z) = sum m(x^2 +y^2)`
` I_(Z) = sum mx^2 + sum my^2`
in the above expression , the term ` sum mx^2` is the moment of inertia of the body about the Y -axis and similary the term ` sum my^2` is the moment of inertia about X - axis thus
` I_(X) = sum my^2 and I_(Y) = sum mx^2`
Substituting in the equation of `I_Z` gives
` I_(Z ) = I_(X ) +I_(Y)`
Thus the perpendicular axis theorem is proved .