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316 CHAPTER 10 Systems of Particles

1 n
x a m x (10.21)
CM i i
M
i 1
1 n
coordinates of center of mass y CM a m y (10.22)
M i 1 i i

1 n
z a m z (10.23)
CM i i
y M i 1
The position of the center of mass of a solid body can, in principle, be calculated
from Eqs. (10.21)–(10.23), since a solid body is a collection of atoms, each of which
23
can be regarded as a particle. However, it would be awkward to deal with the 10 or
so atoms that make up a chunk of matter the size of, say, a coin. It is more convenient
m i
y i
to pretend that matter in bulk has a smooth and continuous distribution of mass over
its entire volume. The mass in some small volume element at position x in the body
i
x is then m (see Fig. 10.9), and the x position of the center of mass is
O x i i
n
1
For a solid body, we weight x CM a x ¢m (10.24)
the position x by the mass M i 1 i i
i
m of a volume element.
i
In the limiting case of m S 0 (and n S
), this sum becomes an integral:
i
FIGURE 10.9 A small volume element of 1
the body at position x has a mass m . x x dm (10.25)
i i CM
M
Similar expressions are valid for the y and z positions of the center of mass:
1
y y dm (10.26)
CM
M
1
z z dm (10.27)
CM
The center of mass of a M
symmetric body is obvious
by inspection. Thus, the position of the center of mass is the average position of all the mass ele-
ments making up the body.
For a body of uniform density, the amount of mass dm in any given volume element
sphere dV is directly proportional to the amount of volume. For a uniform-density body, the
position of the center of mass is simply the average position of all the volume elements of the
body (in mathematics, this is called the centroid of the volume). If the body has a sym-
metric shape, this average position will often be obvious by inspection. For instance,
ring
a sphere of uniform density, or a ring, or a circular plate, or a cylinder, or a parallelepiped
will have its center of mass at the geometrical center (see Fig. 10.10). But for a less
symmetric body, the center of mass must often be calculated, either by considering
circular plate parts of the body (as in the next example) or by integrating over the entire body (as in
the two subsequent examples).

parallelepiped A meterstick of aluminum is bent at its midpoint so that the
EXAMPLE 5
two halves are at right angles (see Fig. 10.11). Where is the
center of mass of this bent stick?
FIGURE 10.10 Several bodies for which
the center of mass coincides with the geo- SOLUTION: We can regard the bent stick as consisting of two straight pieces, each
metrical center. of 0.500 m. The centers of mass of these straight pieces are at their midpoints,