468 15 Rotation of Rigid Bodies

(A)	z	(b)	z		(c)
m		m	m	m	m	m
					z
	O		O		O
	a			a		a

Fig. 15.8 A system of two identical point particles with masses m are attached with a rigid, massless rod of length a. a The rotation axis goes through the center of mass. b The rotation axis goes through the left-most particles. c The rotation axis goes through both particles

15.3.1 Example: Moment of Inertia of Two-Particle System

Problem: Find the moment of inertia around the indicated axes in Fig. 15.8.

Solution: In case (a) the moment of inertia around the axis z is:

2		−	2 a		2		2 a		2	2 ma2 .		(15.25)
Iz = i 1 mi ρi2 = m					+ m				=
			1				1			1
=
In case (b) the moment of inertia around the axis z is:
	2
		mi ρi2 = m02 + ma2 = ma2 .
	Iz =											(15.26)

i =1

Notice that the point on the axis does not give any contribution to Iz , but that the moment of inertia is larger, because the other particle is further away from the origin. In case (c), the moment of inertia around the axis z is:

2
Iz = mi ρi2 = m02 + m02 = 0 .	(15.27)
i =1

15.3.2 Example: Moment of Inertia of a Plate

Problem: Find the moment of inertia of a homogeneous plate with dimensions a and b and of thickness h around an axis z through the center of mass, as illustrated in Fig. 15.9.

Solution: The moment of inertia of a solid body around the axis O is defined as:

Iz =

ρ2 dm = ρM ρ2 d V .

(15.28)

M