You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Section 7.5 Applications from Science and Statistics 421<br />
This is a Riemann sum for the function 57p410 yy 2 on the interval from y 0<br />
to y 8. The work of pumping the oil to the rim is the limit of these sums as the<br />
norms of the partitions go to zero.<br />
57 p<br />
10 yy<br />
4<br />
2 dy 57 p<br />
10y<br />
4<br />
2 y 3 dy<br />
0<br />
0<br />
57 p<br />
4<br />
[ 10 y<br />
3<br />
y 4<br />
8<br />
3 4<br />
]<br />
30,561 ft • lb<br />
0<br />
Now try Exercise 17.<br />
W 8<br />
8<br />
Figure 7.43 To withstand the increasing<br />
pressure, dams are built thicker toward the<br />
bottom.<br />
Fluid Force and Fluid Pressure<br />
We make dams thicker at the bottom than at the top (Figure 7.43) because the pressure<br />
against them increases with depth. It is a remarkable fact that the pressure at any point on<br />
a dam depends only on how far below the surface the point lies and not on how much<br />
water the dam is holding back. In any liquid, the fluid pressure p (force per unit area) at<br />
depth h is<br />
lb<br />
b<br />
p wh, Dimensions check: f t2 f<br />
lt3 ft, for example<br />
where w is the weight-density (weight per unit volume) of the liquid.<br />
EXAMPLE 4 The Great Molasses Flood of 1919<br />
Typical Weight-densities (lb/ft 3 )<br />
Gasoline 42<br />
Mercury 849<br />
Milk 64.5<br />
Molasses 100<br />
Seawater 64<br />
Water 62.4<br />
90 ft<br />
1 ft<br />
SHADED BAND NOT TO SCALE<br />
90 ft<br />
Figure 7.44 The molasses tank of<br />
Example 4.<br />
At 1:00 P.M. on January 15, 1919 (an unseasonably warm day), a 90-ft-high, 90-footdiameter<br />
cylindrical metal tank in which the Puritan Distilling Company stored molasses at<br />
the corner of Foster and Commercial streets in Boston’s North End exploded. Molasses<br />
flooded the streets 30 feet deep, trapping pedestrians and horses, knocking down buildings,<br />
and oozing into homes. It was eventually tracked all over town and even made its<br />
way into the suburbs via trolley cars and people’s shoes. It took weeks to clean up.<br />
(a) Given that the tank was full of molasses weighing 100 lbft 3 , what was the total<br />
force exerted by the molasses on the bottom of the tank at the time it ruptured?<br />
(b) What was the total force against the bottom foot-wide band of the tank wall<br />
(Figure 7.44)?<br />
continued