James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

560 Chapter 8 Further Applications of Integration
The total force is obtained by adding the forces on all the strips and taking the limit:
F − lim
n l` o n
62.5s7 2 y i *d 2s9 2 sy i *d 2 Dy
i−1
− 125 y 3 s7 2 yd s9 2 y 2 dy
23
− 125 ? 7 y 3 23 s9 2 y 2 dy 2 125 y 3 23
ys9 2 y 2 dy
The second integral is 0 because the integrand is an odd function (see Theorem 5.5.7).
The first integral can be evaluated using the trigonometric substitution y − 3 sin , but
it’s simpler to observe that it is the area of a semicircular disk with radius 3. Thus
F − 875 y 3 23 s9 2 y 2 dy − 875 ? 1 2 s3d2
− 7875
2
< 12,370 lb n
P
Moments and Centers of Mass
Our main objective here is to find the point P on which a thin plate of any given shape
bal ances horizontally as in Figure 5. This point is called the center of mass (or center of
grav ity) of the plate.
We first consider the simpler situation illustrated in Figure 6, where two masses m 1
and m 2 are attached to a rod of negligible mass on opposite sides of a fulcrum and at
distances d 1 and d 2 from the fulcrum. The rod will balance if
FIGURE 5
2
m 1 d 1 − m 2 d 2
d¡
d
m¡ m
fulcrum
FIGURE 6
This is an experimental fact discovered by Archimedes and called the Law of the Lever.
(Think of a lighter person balancing a heavier one on a seesaw by sitting farther away
from the center.)
Now suppose that the rod lies along the x-axis with m 1 at x 1 and m 2 at x 2 and the center
of mass at x. If we compare Figures 6 and 7, we see that d 1 − x 2 x 1 and d 2 − x 2 2 x
and so Equation 2 gives
m 1 sx 2 x 1 d − m 2 sx 2 2 xd
m 1 x 1 m 2 x − m 1 x 1 1 m 2 x 2
3
x − m 1x 1 1 m 2 x 2
m 1 1 m 2
The numbers m 1 x 1 and m 2 x 2 are called the moments of the masses m 1 and m 2 (with
respect to the origin), and Equation 3 says that the center of mass x is obtained by adding
the moments of the masses and dividing by the total mass m − m 1 1 m 2 .
⁄ –x ¤
0
m¡ –x-⁄
¤-x–
m
x
FIGURE 7
In general, if we have a system of n particles with masses m 1 , m 2 , . . . , m n located at
the points x 1 , x 2 , . . . , x n on the x-axis, it can be shown similarly that the center of mass
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