8 ROTATIONAL MOTION 8.6 Moment of inertia(^) ̧ (^0). (8.24)
2
i
0
i
1
h
0 a
(^2) (1 − z/h) (^2) dz
Performing the y-integrals, we obtain
̧h
a^2 z (1 − z/h)^2 dz
Finally, performing the z-integrals, we obtain
a^2
h
z^2 / 2 − 2 z^3 /( 3 h) + z^4 /( 4 h^2 )
ih
(^)
a^2 h^2 /12 h
zcm =
a^2 [z − z^2 /(h) + z^3 /(3 h)]h
(^0) =
a^2 h/3
=. (8.25)
4
Thus, the geometric centre of a regular square-sided pyramid is located on the
symmetry axis, one quarter of the way from the base to the apex.
8.6 Moment of inertia
Consider an extended object which is made up of N elements. Let the ith element
possess mass mi, position vector ri, and velocity vi. The total kinetic energy of
the object is written
K =
X (^1)
m^ v^2.^ (8.26)^
Suppose that the motion of the object consists merely of rigid rotation at angular
velocity ω. It follows, from Sect. 8.4, that
vi = ω × ri. (8.27)
Let us write
ω = ω k, (8.28)
where k is a unit vector aligned along the axis of rotation (which is assumed
to pass through the origin of our coordinate system). It follows from the above
equations that the kinetic energy of rotation of the object takes the form
K =
X
m |k × r |^2 ω^2 , (8.29)
i=1,N 2 i^ i^
or
K =
1
I ω^2. (8.30)
2
z (^) cm =
i=1,N