Principles of Dairy Machine Design

Module 1. Statics and dynamics

Lesson 4
SECOND MOMENT OF INERTIA, PARALLEL AXIS THEOREM

4.1 Introduction

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

4.2 Moment of Inertia

The measure of an object's resistance to changes to its rotation is the inertia of a rotating body with respect to its rotation.

The moment of inertia is that property of a body which makes it reluctant to speed up or slow down in a rational manner. In fact moment of inertia means second moment of mass.

4.3 Law of Moment

It states that, if a number of coplanar forces acting at a point are in equilibrium, the sum of the clockwise moments must be equal to the sum of the anticlockwise moments about any point.

It is an important law in the field of statics and is used for finding out the reaction of forces in frames etc.

4.4 Parallel Axis Theorem

4.4.1 Statement

If the moment of inertia of a plane area about an axis through its center of gravity be denoted by I_g, the moment of inertia of the area about axis AB, parallel to the first, and at a distance h from the center of the gravity is given by

I_AB=I_G+Ah²

I_AB=M.I about AB

I_G= M.I about c.g

A=Area of the section.

h=distance between C.G of the section and the axis AB

Fig.4.2 Parallel axis theorem

Proof: Consider a strip of an area δa at a distance ‘y’ from the c.g of a section as shown in fig. 4.2.

Then

I_G= ∑ δay² (therefore I = mr²)

We also know that

I_AB=∑ δa (h+y)²

=∑ δa (h²+y²+2hy)

=∑h² δa + ∑ δay² +2h∑ δay

=Ah² + I_G + 0

∑ δay is the algebraic sum of the moments of all the areas of strips about the axis through c.g and is equal to ‘Ay’ where ‘y’ is the distance between the c.g of the section and axis through the c.g which is zero

Hence I_AB=I_G+Ah²

4.5 Moment of Inertia of a Rectangular Section

Consider a rectangular section ABCD as shown in fig. whose moment of inertia is required to be found out.

Let b= Width of the section and

d= Depth of the section

Now consider a strip PQ of thickness dy parallel to X-X axis and at a distance y from it as shown in the figure 4.3

Area of the strip

δa= b.dy

We know that moment of inertia of the strip about X-X axis,

= Area δy² = (b.dy) y²

Now moment of inertia of the whole section may be found out by integrating the above equation for the whole length of the lamina i.e. from –d/2 to +d/2,

Fig.4.3. Rectangular section

4.6 Moment of Inertia of a Circular Section

Find the moment of inertia of a rectangular section 30 mm wide and 40 mm about X-X axis and Y-Y axis.

Solution: Given: Width of the section (b) = 30 mm and depth of the section (d) = 40 mm. We know that moment of inertia of the section an axis passing through its centre of and parallel to X-X axis.