Moment Of Inertia

1
CHAPTER 2
MOMENT OF INERTIA

2.1 Definition of Moment of Inertia
Consider a straight line (the "axis") and a set of point masses m1 , m2 , m3 , K such that the
distance of the mass mi from the axis is ri . The quantity mi ri 2 is the second moment of the i th
mass with respect to (or "about") the axis, and the sum

∑m r

ii

2

is the second moment of mass of

all the masseswith respect to the axis.
Apart from some subtleties encountered in general relativity, the word "inertia" is synonymous
with mass - the inertia of a body is merely the ratio of an applied force to the resulting
acceleration. Thus ∑ mi ri 2 can also be called the second moment of inertia. The second moment
of inertia is discussed so much in mechanics that it is usually referred to as just "the"moment of
inertia.
In this chapter we shall consider how to calculate the (second) moment of inertia for different
sizes and shapes of body, as well as certain associated theorems. But the question should be
asked: "What is the purpose of calculating the squares of the distances of lots of particles from
an axis, multiplying these squares by the mass of each, and adding them all together? Isthis
merely a pointless make-work exercise in arithmetic? Might one just as well, for all the good it
does, calculate the sum ∑ mi2 ri ? Does ∑ mi ri 2 have any physical significance?"

2.2 Meaning of Rotational Inertia.
If a force acts of a body, the body will accelerate. The ratio of the applied force to the resulting
acceleration is the inertia (or mass) of the body.
If a torque acts ona body that can rotate freely about some axis, the body will undergo an
angular acceleration. The ratio of the applied torque to the resulting angular acceleration is the
rotational inertia of the body. It depends not only on the mass of the body, but also on how that
mass is distributed with respect to the axis.
Consider the system shown in figure II.1.

2

Fa

r
O

m

τ

&&
θr
O

m
FIGURE II.1

A particle of mass m is attached by a light (i.e. zero or negligible mass) arm of length r to a
point at O, about which it can freely rotate. A force F is applied, and the mass consequently
undergoes a linear acceleration a =F/m. The angular acceleration is then && = F .mr . Also, the
θ
torque is τ = Fr. The ratio of the applied torque to the angular accelerationis therefore mr2 .
Thus the rotational inertia is the second moment of inertia. Rotational inertia and (second)
moment of inertia are one and the same thing, except that rotational inertia is a physical concept
and moment of inertia is its mathematical representation.
2.3 Moments of inertia of some simple shapes.
A student may well ask: "For how many different shapes of body must I commit tomemory
the formulas for their moments of inertia?" I would be tempted to say: "None". However, if any
are to be committed to memory, I would suggest that the list to be memorized should be limited
to those few bodies that are likely to be encountered very often (particularly if they can be used
to determine quickly the moments of inertia of other bodies) and for which it is easier to
rememberthe formulas than to derive them. With that in mind I would recommend learning no
more than five. In the following, each body is supposed to be of mass m and rotational inertia I.
1. A rod of length 2l about an axis through the middle, and at right angles to the rod:
I = 1 ml 2
3

2.3.1

2. A uniform circular disc of radius a about an axis through the centre and perpendicular to the

3plane of the disc:
1
I = 2 ma 2

2.3.2

3. A uniform right-angled triangular lamina about one of its shorter sides - i.e. not the
hypotenuse. The other not-hypotenuse side is of length a:
I = 1 ma 2
6

2.3.3

4. A uniform solid sphere of radius a about an axis through the centre.
I = 2 ma 2
5

2.3.4

5. A uniform spherical shell of radius a about an axis through the centre....