Physics For Scientists And Engineers 6E - part 113
Problems
449
h
d
Figure P14.74
Figure P14.75
75.
The hull of an experimental boat is to be lifted above the
water by a hydrofoil mounted below its keel, as shown in
Figure P14.75. The hydrofoil has a shape like that of an
airplane wing. Its area projected onto a horizontal surface
is A. When the boat is towed at sufficiently high speed, wa-
ter of density $ moves in streamline flow so that its average
speed at the top of the hydrofoil is n times larger than its
speed v
b
below the hydrofoil. (a) Neglecting the buoyant
force, show that the upward lift force exerted by the water
on the hydrofoil has a magnitude given by
(b) The boat has mass M. Show that the liftoff speed is
given by
(c) Assume that an 800-kg boat is to lift off at 9.50 m/s.
Evaluate the area A required for the hydrofoil if its design
yields n ! 1.05.
v
%
√
2Mg
(n
2
&
1)A$
F
%
1
2
(n
2
&
1)$v
b
2
A
Answer to Quick Quizzes
14.1 (a). Because the basketball player’s weight is distributed
over the larger surface area of the shoe, the pressure
(F/A) that he applies is relatively small. The woman’s
lesser weight is distributed over the very small cross-
sectional area of the spiked heel, so the pressure is high.
14.2 (a). Because both fluids have the same depth, the one with
the smaller density (alcohol) will exert the smaller pressure.
14.3 (c). All barometers will have the same pressure at the bot-
tom of the column of fluid—atmospheric pressure. Thus,
the barometer with the highest column will be the one
with the fluid of lowest density.
14.4 (d). Because there is no atmosphere on the Moon, there
is no atmospheric pressure to provide a force to push the
water up the straw.
14.5 (b). For a totally submerged object, the buoyant force
does not depend on the depth in an incompressible fluid.
14.6 (c). The ice cube displaces a volume of water that has a
weight equal to that of the ice cube. When the ice cube
melts, it becomes a parcel of water with the same weight
and exactly the volume that was displaced by the ice cube
before.
14.7 (b) or (c). In all three cases, the weight of the treasure chest
causes a downward force on the raft that makes it sink into
the water. In (b) and (c), however, the treasure chest also
displaces water, which provides a buoyant force in the up-
ward direction, reducing the effect of the chest’s weight.
14.8 (b). The liquid moves at the highest speed in the straw
with the smaller cross sectional area.
14.9 (a). The high-speed air between the balloons results in low
pressure in this region. The higher pressure on the outer
surfaces of the balloons pushes them toward each other.
Calvin and Hobbes © 1992 Watterson. Reprinted with permission of Universal Press
Syndicate. All rights reserved.