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328 CAM DESIGN HANDBOOK

mentally. While there are several ways to accomplish this, one of the more common
methods is to calculate the natural frequency of a pendulum made from the object (an inter-
esting application of this to moment of inertia calculation for passenger cars is described
by Garrott and Monk (1988). The following simple example describes the direct use of the
formulas for determining the mass, center of mass, and moment of inertia of a flat plate.

EXAMPLE Consider the square plate with sides of length s, a thickness t, and a density
r shown in Fig. 11.6. The mass of the plate is given by
s s
m = Ú r dV = r t Ú Ú dxdy = r txy = r s t.
2
=
rigid body x=0 y=0 xs
ys
=
The coordinates of the center of gravity of the plate are given by
1 s r st x 2 r s t r st s
3
3
()
r
y = x = Ú xst dx = = = =
cg
cg
m m 2 2 m 2r st 2
2
=
x=0 xs
which, of course, describe the center of the plate. The moment of inertia about the
center of the plate is
s 2 / s 2
J = Ú r dm = Ú Ú ( x + y ) r tdxdy
2
2
2
cg
cg
s
s
plate x=- 2 y=- 2
s 2 / s 2 s 2 / s 2
= t r Ú Ú x dxdy + r t Ú Ú y dxdy
2
2
s
s
s
x=- 2 y=- 2 x=- 2 y=- 2
s
s 2 / s 2 x 3 s 2 r ts 4 1
2
2
2
= r ts Ú x dx + r ts Ú y dy = 2r ts = = ms .
s
x x=- 2 y=- 2 3 x=- 2 6 6
s
s
y = s
y cg = s
2
O
x = 0,
y = 0
x cg = s x = s
2
FIGURE 11.6. Square plate of uniform density.