Chapter 2 Seawall of the figure below has a curved surface

Also,

To determine 1, consider a unit square that consist of a quarter circle and the remainder as

show in the figure. The centroids of areas (1) and (2) are as indicated.

(8) =

10.223m

0.5

I

(2)

C

2

C

Problem 2.123

The concrete (specific weight = 3

1

50 lb/ft ) seawall of the figure below has a curved surface

and restrains seawater at a depth of 24 ft . The trace of the surface is a parabola as

illustrated. Determine the moment of the fluid force (per unit length) with respect to an axis

through the toe (point A).

Solution 2.123

The components of the fluid force acting on the wall are 1

F and W as shown on the figure

where

y

Seawater

24 ft

A

x

y = 0.2x

2

15 ft

xc

D

C

y = 0.2x

2

Seawater

Volume = V

Thus,

and

CD

dx

y

Problem 2.124

A step-in viewing window having the shape of a half-cylinder is built into the side of a large

aquarium. See the figure below. Find the magnitude, direction, and location of the net

horizontal forces on the viewing window.

Solution 2.124

Due to symmetry, the net force parallel to the wall is zero or

The vertical location of x

Fis

x

y

z

25′

10′

Steps up to

window

R

= 5′

p

= 64 lbm/ft

3

2′

Problem 2.125

Find the magnitude, direction, and location of the net vertical force acting on the viewing

window in Problem 2.124.

Solution 2.125

The net vertical force must equal the weight of fluid inside the viewing window. Then

x

y

z

25′

10′

Steps up to

window

R

= 5′

p

= 64 lbm/ft

3

2′

Problem 2.126

A l0-m-long log is stuck against a dam, as shown in the figure below. Find the magnitudes

and locations of both the horizontal force and the vertical force of the water on the log in

terms of the diameter

D

. The center of the log is at the same elevation as the top of the dam.

Solution 2.126

Consider the water forces on the log as shown on the right.

The horizontal forces H

F is on the top portion only and is

The location of H

F is

Water

D

F

H

D

= 0.5 m

γ

Using Table B

and is the location of 1V

F

. The location of

x

2

x

1

F

V1

x

Problem 2.127

Find the net horizontal force on the 4.0–m -long log shown in the figure below.

Solution 2.127

The force L

Fon the left side of the log is the horizontal force on

the horizontally projected area of the log. This horizontally

projected area measures =1.0 mD by 4.0 m and gives

D = 1.0 m1.0 m

0.5 m

D

FL

FR

Problem 2.128

An open tank containing water has a bulge in its vertical side that is semicircular in shape

as shown in the figure below. Determine the horizontal and vertical components of the

force that the water exerts on the bulge. Base your analysis on a 1-ft length of the bulge.

Solution 2.128

~

H

F

horizontal force of wall on fluid

For equilibrium,

6 ft

3 ft

Water Bulge

F

1

F

H

𝒲

Problem 2.129

A closed tank is filled with water and has a 4-ft-diameter hemispherical dome as shown in

the figure below. A U-tube manometer is connected to the tank. Determine the vertical

force of the water on the dome if the differential manometer reading is 7ft and the air

pressure at the upper end of the manometer is 12.6 psi.

Solution 2.129

For equilibrium,

From the manometer,

() ()

γγ

+− =

2

7ft 4ft

Agf HO

p

20 psi 4-ft diameter

Air

Gage

ﬂuid

(

SG

= 3.0)

Water

2 ft

5 ft

p

A

F

D

Problem 2.130

A 3-m-diameter open cylindrical tank contains water and has a hemispherical bottom as

shown in the figure below. Determine the magnitude, line of action, and direction of the

force of the water on the curved bottom.

Solution 2.130

F

orce weight of water supported by hemispherical bottom

=

Water

8 m

3 m

Problem 2.131

Three gates of negligible weight are used to hold back water in a channel of width b as

shown in the figure below. The force of the gate against the block for gate (b) is R.

Determine (in terms of R) the force against the blocks for the other two gates.

Solution 2.131

For case (b)

Thus,

(a)(b)

Hinge

Block

(c)

hh

2

h

2

h

y

R

H

x

H

y

For case (c), for the free-body-diagram shown, the force 1

R

F on the curved section passes

through the hinge and therefore does not contribute to the moment around H. On bottom

part of gate

Thus,

F

B

Problem 2.133

An iceberg (specific gravity 0.917 ) floats in the ocean (specific gravity 1.025 ). What percent

of the volume of the iceberg is under water?

Solution 2.133

SG = 0.917

Problem 2.134

A floating 40-in.- thick piece of ice sinks 1 in. with a 500-lb polar bear in the center of the

ice. What is the area of the ice in the plane of the water level? For seawater, 1.03S=.

Solution 2.134

Without the polar bear on the ice, the submerged depth d of the ice is found by equating

the weight of the ice and the buoyant force. Denoting the pure water specific weight by

γ

and the ice area by A gives

Problem 2.135

A spherical balloon filled with helium at 40°F and 20 psia has a −25 ft diameter. What

load can it support in atmospheric air at 40°F and 14.696 psia ? Neglect the balloon’s

weight.

Solution 2.135

For static equilibrium, the buoyant force must equal the load. Neglecting the weight of the

balloon and assuming air and helium to be ideal gases, the load is

Problem 2.136

A river barge, whose cross section is approximately rectangular, carries a load of grain. The

barge is 28 ft wide and 90 ft long. When unloaded, its draft (depth of submergence) is 5 ft

and with the load of grain the draft is 7 ft . Determine: (a) the unloaded weight of the

barge, and (b) the weight of the grain.

Solution 2.136

(a) For equilibrium,

(b) =

0

vertical

F

Wb

Wb + Wg

Problem 2.137

A barge is 40 ft wide by 120 ft long. The weight of the barge and its cargo is denoted by

W

.

When in salt-free riverwater, it floats 0.25 ft deeper than when in seawater

()

3

64 lb/ft

γ

=.

Find the weight W.

Solution 2.137

In both cases, the weight W must equal the

weight of the displaced water or

Then

Problem 2.138

When the Tucurui Dam was constructed in northern Brazil, the lake that was created

covered a large forest of valuable hardwood trees. It was found that even after 15 years

underwater, the trees were perfectly preserved and underwater logging was started. During

the logging process, a tree is selected, trimmed, and anchored with ropes to prevent it from

shooting to the surface like a missile when cut. Assume that a typical large tree can be

approximated as a truncated cone with a base diameter of 8ft , a top diameter of 2ft, and a

height of

1

00 ft. Determine the resultant vertical force that the ropes must resist when the

completely submerged tree is cut. The specific gravity of the wood is approximately 0.6 .

Solution 2.138

~weight

W

, ~ buoyant force

B

F

, ~ tension in ropesT

Thus,

V

()( )

()( )()

π



=+×+=



22

3

100 ft 4ft 4ft 1ft 1ft 2200ft

3

tree

For buoyant force,

𝒲

Problem 2.140

An inverted test tube partially filled with air floats in a plastic water-filled soft drink bottle

as shown in the figure below. The amount of air in the tube has been adjusted so that it just

floats. The bottle cap is securely fastened. A slight squeezing of the plastic bottle will cause

the test tube to sink to the bottom of the bottle. Explain this phenomenon.

Solution 2.140

Air

Plastic bottle

Water

Test tube

pp

o

Volume

FB

h