- Theorem of Perpendicular axes gives the moment of inertia of a body about an axis perpendicular to a body in terms of moment of inertia of two mutually perpendicular axes.
Contents
- We know that, the weight of the any body is due to its inertia. The moment of inertia of a body about any axis is nothing but the sum of product of mass of that each particle and square of the distance of each particle from that axis of rotation.
- That means, I = ∑ mi ri2
- The moment of inertia of the body depends on the mass, shape and size of the body.
- It also depends on the mass distribution of particles about the axis of rotation of the body.
- And also, on the position of the axis of rotation of the body.
- The SI unit of moment of inertia of the body is kg m2.
Statement:
- According to theorem of Perpendicular axes, the moment of inertia of a plane laminar body about an axis perpendicular to its plane is equal to the sum of moment of inertia about the two mutually perpendicular axes which are also concurring with the third perpendicular axis and that two mutually perpendicular axes must be in the plane of laminar body.
- The following neat labelled figure explains the theorem of perpendicular axes.
- Let Iz be the moment of inertia about an axis perpendicular to plane laminar body.
- And Ix and Iy are also the moment of inertia of about two mutually perpendicular axes lying in the plane of laminar body as shown in figure.
- Thus, mathematically from figure we can write the perpendicular axes theorem as,
- Iz = Ix + Iy
Applications:
- Theorem of perpendicular axes can be used to find moment of inertia in case if laminar bodies only.
- It is also used to find the moment of inertia of a ring about its diameter.
Example:
What is the moment of inertia of a ring about its diameter?
Solution:
- Let us consider R be the radius of the ring, Iz be the moment of inertia about an axis perpendicular to plane of ring and passing through centre of mass C.
Hence, here Ic = Iz
- And Ixand Iy be the moment of inertia about two mutually perpendicular axes which are lying in the plane of the ring as shown in figure below.
- Then by perpendicular axes theorem,
Iz = Ic = Ix + Iy
- But, here Ix and Iy both are the moment of inertia of ring about diameter.
Hence, Ix = Iy = Id
Thus, Ic = 2 Id
And we know that, for a ring Ic = MR2
Thus, MR2 = 2Id
And hence, Id = MR2/2
Hence, the moment of inertia of thin uniform ring about its diameter is 1/2MR2.