A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.5 Centre of mass

a

̧ ̧ ̧

dx dy dz

± ±

̧

̧

̧

z dz

̧

dz

̧

dy

̧

dy

̧

̧

̧

z dz

y z

geometric centre

a x

a

h

zcm

x^

a

top view side view

Figure 74: Locating the geometric centre of a regular square-sided pyramid.

z-component of Eq. (8.20):

zcm =

̧ ̧ ̧

z dx dy dz

, (8.21)

where the integral is taken over the volume of the pyramid.

In the above integral, the limits of integration for z are z = 0 to z = h, respec-

tively (i.e., from the base to the apex of the pyramid). The corresponding limits of

integration for x and y are x, y = −a (1 − z/h)/2 to x, y = +a (1 − z/h)/2, respec-

tively (i.e., the limits are x, y = a/2 at the base of the pyramid, and x, y = 0

at the apex). Hence, Eq. (8.21) can be written more explicitly as

h

zcm = 0

0

+a (1−z/h)/2

−a (1−z/h)/2

+a (1−z/h)/2

−a (1−z/h)/2

+a (1−z/h)/2

−a (1−z/h)/2

+a (1−z/h)/2

−a (1−z/h)/2

. (8.22)

As indicated above, it makes sense to perform the x- and y- integrals before

the z-integrals, since the limits of integration for the x- and y- integrals are z-

dependent.^ Performing^ the^ x-integrals,^ we^ obtain^
h
zcm = 0
0

+a (1−z/h)/2

−a (1−z/h)/2

+a (1−z/h)/2

−a (1−z/h)/2

a (1 − z/h) dy

a (1 − z/h) dy

̧. (8.23)

dz

h

dx

dx

h