State and prove parallel axis theorem and perpendicular axis theorem.
Answers
The theorem of perpendicular axes for a plane laminar body states that the moment of inertia of a plane lamina about an axis perpendicular to the plane of the lamina is equal to the sum of the moments of inertia of the lamina about any two mutually perpendicular axes in its own plane and intersecting each other at the point where the perpendicular axis passes through it.
If Ix and Iy are the moments of inertia of a plane lamina (Fig) about the perpendicular axes x and y respectively which lie in the plane of the lamina and intersect each other at O, then the moment of inertia I of the lamina about an axis passing through O and perpendicular to its plane is given by
I = Ix + Iy
The two theorems described above help us in determining the moment of inertia of a body about any other axis
Explanation:
According to the parallel axis theorem, a moment of inertia of a body about any axis is equal to the sum of the moment of inertia about a parallel axis through the centre of gravity and the product of the mass of the body and the square of the perpendicular distance between the two axis.
let l
c
be the moment of inertia of a body of mass M about an axis (PQ) passing through the centre of gravity C. According to the theorem moment of inertia of a body an axis AB is I=I−c+Mr
2
According to the theorem on perpendicular axis, the moment of inertia of a plane about an axis perpendicular to the plane is equal to the sum of the moment of inertia about two perpendicular axes in the plane of the lamina such that the three mutually perpendicular axes have common point of
represent the moment of inertia of the body about the axes OX,OY & OZ, then according to perpendicular axes theorem I