axis touching the edge and perpendicular to the plane of the disc and axis passing through the center and perpendicular to the plane of the disc, Solved Example Problems for Theorems of Moment of Inertia Example 5.16įind the moment of inertia of a disc of mass 3 kg and radius 50 cm about the following axes. Thus, the perpendicular axis theorem is proved. Substituting in the equation for I z gives, In the above expression, the term Σ mx 2 is the moment of inertia of the body about the Y-axis and similarly the term Σ my 2 is the moment of inertia about X-axis.
The summation of the above expression gives the moment of inertia of the entire lamina about Z-axis as, I Z = ∑ mr 2 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. Let this mass be at a distance x, y and z from the zy-plane, zx-plane, and xy-plane, respectively.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. This mass can be split into an infinite number of small parts each of mass d m. The centre of gravity of a body, or the system of particles rigidly connected together, is that point through which the line of action of the weight of the body always passes.įigure 6.1 shows a body of mass M. Such a point is called the centre of gravity of the body. the centre of the above-mentioned system of parallel forces. The weight of a body acts through a definite point in the body, viz. The resultant of these forces is called the weight of the body. Thus, they form a system of parallel forces. For different particles of a rigid body, these forces, which meet at the centre of the earth, may be considered parallel, as the distance to the centre is usually large in comparison to the size of the body. We know that earth attracts every particle towards its centre with a force that is proportional to the mass of the particle. 6 Centroid and Moment of Inertia CENTRE OF GRAVITY