A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.5 Centre of mass

where V is the volume of the object. According to Eq. (8.20), the centre of mass
of a body of uniform density is located at the geometric centre of that body.

geometric centre

a

a

a

Figure 72: Locating the geometric centre of a cube.

For many solid objects, the location of the geometric centre follows from sym-

metry. For instance, the geometric centre of a cube is the point of intersection

of the cube’s diagonals. See Fig. 72. Likewise, the geometric centre of a right

cylinder is located on the axis, half-way up the cylinder. See Fig. 73.

ometric centre

h

Figure 73: Locating the geometric centre of a right cylinder.

As an illustration of the use of formula (8.20), let us calculate the geometric

centre of a regular square-sided pyramid. Figure 74 shows such a pyramid. Let a
be the length of each side√. It follows, from simple trigonometry, that the height

of the pyramid is h = a/ 2. Suppose that the base of the pyramid lies on the

x-y plane, and the apex is aligned with the z-axis, as shown in the diagram. It

follows, from symmetry, that the geometric centre of the pyramid lies on the z-

axis. It only remains to calculate the perpendicular distance, zcm, between the
geometric centre and the base of the pyramid. This quantity is obtained from the

axis

ge

h/2