Theorems of Moment of Inertia

Theorems of Moment of Inertia

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. We have two important theorems to handle the case of shifting the axis of rotation.

(i) Parallel axis theorem:

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.

If IC 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,

I = IC + Md2       (5.46)

Let us consider a rigid body as shown in Figure 5.25. Its moment of inertia about an axis AB passing through the center of mass is IC. DE is another axis parallel to AB at a perpendicular distance d from AB. The moment of inertia of the body about DE is I. We attempt to get an expression for I in terms of IC. For this, let us consider a point mass m on the body at position x from its center of mass.

The moment of inertia of the point mass about the axis DE is, m( x + d)2.

The moment of inertia I of the whole body about DE is the summation of the above expression.

I = + ∑ m (x+d)2

This equation could further be written as,

Here, ∑mx2 is the moment of inertia of the body about the center of mass. Hence, IC = ∑mx2

The term, ∑mx = 0 because, x can take positive and negative values with respect to the axis AB. The summation ( ∑mx) will be zero.

Thus, I = IC + ∑md2 = IC + ( ∑m)d2

Here, Σm is the entire mass M of the object ( ∑m = M )

I = IC + Md2

Hence, the parallel axis theorem is proved.

(ii) Perpendicular axis theorem:

This perpendicular axis theorem holds good only for plane laminar objects.

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.

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 IX and IY respectively and IZ is the moment of inertia about Z-axis, then the perpendicular axis theorem could be expressed as,

IZ =IX +IY  (5.47)

To prove this theorem, let us consider a plane laminar 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 5.26. The lamina is considered to be made up of a large number of particles of mass m. Let us choose one such particle 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, mr2

The summation of the above expression gives the moment of inertia of the entire lamina about Z-axis as, IZ = ∑mr2

Here, r2 = x2 + y2

Then, IZ  = ∑m ( x2 + y2 )

IZ  = ∑m x2 + ∑m y2

IZ  = ∑m x2 + ∑m y2

In the above expression, the term Σmx 2 is the moment of inertia of the body about the Y-axis and similarly the term Σmy2 is the moment of inertia about X-axis. Thus,

IX =∑my2

and

IY =∑mx2

Substituting in the equation for Iz gives,

IZ = IX + IY

Thus, the perpendicular axis theorem is proved.