The moment of inertia of the particle about Z-axis is, mr 2 Let us choose one such particle at a point P which has coordinates (x, y) at a distance r from O. The lamina is considered to be made up of a large number of particles of mass m. The X and Y-axes lie on the plane and Z-axis is perpendicular to it as shown in Figure 5.26. To prove this theorem, let us consider a plane laminar object of negligible thickness on which lies the origin (O). If the moments of inertia of the body about X and Y-axes are 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, Let the X and Y-axes lie in the plane and Z-axis perpendicular to the plane of the laminar object. The 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. This perpendicular axis theorem holds good only for plane laminar objects. Hence, the parallel axis theorem is proved. Here, Σ m is the entire mass M of the object ( ∑ m = M ) Thus, I = I C + ∑ md 2 = I C + ( ∑ m ) d 2 The term, ∑ mx = 0 because, x can take positive and negative values with respect to the axis AB. Here, ∑ mx 2 is the moment of inertia of the body about the center of mass. This equation could further be written as, The moment of inertia I of the whole body about DE is the summation of the above expression. The moment of inertia of the point mass about the axis DE is, m ( x + d ) 2. For this, let us consider a point mass m on the body at position x from its center of mass. We attempt to get an expression for I in terms of I C. The moment of inertia of the body about DE is I.

DE is another axis parallel to AB at a perpendicular distance d from AB. Its moment of inertia about an axis AB passing through the center of mass is I C. Let us consider a rigid body as shown in Figure 5.25. If I C is the moment of inertia of the body of mass M about an axis passing through the center of mass, then the moment of inertia I about a parallel axis at a distance d from it is given by the relation, 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 center of mass and the product of the mass of the body and the square of the perpendicular distance between the two axes. We have two important theorems to handle the case of shifting the axis of rotation. As the moment of inertia depends on the axis of rotation and also the orientation of the body about that axis, it is different for the same body with different axes of rotation.

Moment of inertia of a circle proof