A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.5 Centre of mass

M

 v

P

 r^

O





Figure 71: Rigid rotation.

If the object under consideration is continuous, then

mi = ρ(ri) Vi, (8.18)

where ρ(r) is the mass density of the object, and Vi is the volume occupied by

the ith element. Here, it is assumed that this volume is small compared to the

total volume of the object. Taking the limit that the number of elements goes

to infinity, and the volume of each element goes to zero, Eqs. (8.17) and (8.18)

yield the following integral formula for the position vector of the centre of mass:

rcm =

1

∫∫∫

ρ r dV. (8.19)

Here, the integral is taken over the whole volume of the object, and dV =

dx dy dz is an element of that volume. Incidentally, the triple integral sign in-

dicates a volume integral: i.e., a simultaneous integral over three independent

Cartesian coordinates. Finally, for an object whose mass density is constant—

which is the only type of object that we shall be considering in this course—the

above expression reduces to

rcm =

V

∫∫∫

r dV, (8.20)

1