Problem 10.55 The cable supports a railway bridge
between two tunnels. The distributed load is wD
1 MN/m, and hD40 m.
(a) What is the maximum tension in the cable?
(b) What is the length of the cable?
36 m 36 m
Problem 10.56 The cable in Problem 10.55 will safely
support a tension of 40 MN. What is the shortest cable
that can be used, and what is the corresponding value
of h?
w2x2. The terms on the right are known: T2
MAX D402⊲106⊳, and
Problem 10.57 An oceanographic research ship tows
an instrument package from a cable. Hydrodynamic drag
subjects the cable to a uniformly distributed force wD
2 lb/ft. The tensions in the cable at 1 and 2 are 800 lb
and 1300 lb, respectively. Determine the distance h.
h
2
300 ft
These are reduced to two equations in two unknowns:
a1Ca2⊲x1C300⊳2and solved by iteration using
Problem 10.58 Draw a graph of the shape of the cable
in Problem 10.57.
x1D287.5 ft. The value wis plotted on the abscissa, and zis
plotted on the ordinate. The result is a graph of the depth of the
cable against the horizontal extension.
–150
–200
depth, ft
830
Problem 10.59 The mass of the rope per unit length
is 0.10 kg/m. The tension at its lowest point is 4.6 N.
Using the approach described in Active Example 10.8,
determine
(a) the maximum tension in the rope
(b) the rope’s length.
x
y
12 m
Solution: The weight per unit length is
0.213
Problem 10.60 The stationary balloon’s tether is
horizontal at point Owhere it is attached to the truck.
The mass per unit length of the tether is 0.45 kg/m. The
tether exerts a 50-N horizontal force on the truck. The
horizontal distance from point Oto point Awhere
the tether is attached to the balloon is 20 m. What is
the height of point Arelative to point O?
O
A
Problem 10.61 In Problem 10.60, determine the mag-
nitudes of the horizontal and vertical components of the
force exerted on the balloon at Aby the tether.
Solution: From the solution to Problems 10.60, aD0.0883 m1.
TDT0cosh⊲ax⊳ D50 cosh[⊲0.0883⊳⊲20⊳]D150 N.
The slope at xD20 m (Equation 10.19) is
Dtan Dsinh⊲ax⊳ Dsinh[⊲0.0883⊳⊲20⊳]D2.84,
so Darctan 2.84 D70.6°. The horizontal and vertical compo-
nents are
TxD⊲150⊳cos D50 N
TyD⊲150⊳sin D142 N.
y
x
20 m
50 N
150 N
Problem 10.62 The mass per unit length of lines AB
and BC is 2 kg/m. The tension at the lowest point of
cable AB is 1.8 kN. The two lines exert equal horizontal
forces at B.
(a) Determine the sags h1and h2.
(b) Determine the maximum tensions in the two lines.
CBA h2
h1
40 m
60 m
Solution: The lines meet the condition for a catenary. (a) The line
AB. The weight density is
The sag is
1.8 kN. Thus the tension at the lowest point in BC is 1.8 kN, and the
parameter afor line BC is equal to a1. The sag is
832
Problem 10.63 The rope is loaded by 2-kg masses
suspended at 1-m intervals along its length. The mass
of the rope itself is negligible. The tension in the rope at
its lowest point is 100 N. Determine hand the maximum
tension in the rope.
Strategy: Obtain an approximate answer by modeling
the discrete loads on the rope as a load uniformly
distributed along its length.
10 m
h
Solution: The equivalent distributed load is
From Eq. (10.20),
Problem 10.64 In Active Example 10.9, what are the
tensions in cable segments 1 and 3?
1 m
1 m 1 m
Problem 10.65 Each lamp weighs 12 lb.
(a) What is the length of the wire ABCD needed to
suspend the lamps as shown?
(b) What is the maximum tension in the wire?
12 in
18 in 18 in
A
834
Problem 10.66 Two weights, W1DW2D50 lb, are
suspended from a cable. The vertical distance h1D4 ft.
(a) Determine the vertical distance h2.
(b) What is the maximum tension in the cable?
6 ft 10 ft
h1h2
3 ft
2 ft
Problem 10.67 In Problem 10.66, W1D50 lb, W2D
100 lb, and the vertical distance h1D4 ft.
(a) Determine the vertical distance h2.
(b) What is the maximum tension in the cable?
836
Problem 10.68 Three identical masses mD10 kg are
suspended from the cable. Determine the vertical
distances h1and h3and draw a sketch of the
configuration of the cable.
2 m
h1
2 m
m
1
2
m
h3
3
4
1 m 3 m 1 m
m
Solution: We make 3 cuts and then draw one diagram of the
whole system
TV
2 m
98.1 N
TV
A3
A2
A1
THh1h3
1 m
2 m 3 m
98.1 N
98.1 N
TV
T5
1 m
1 m
2 m 3 m
Problem 10.69 In Problem 10.68, what are the
tensions in cable segments 1 and 2?
98.1 N
Problem 10.70 Three masses are suspended from the
cable, where mD30 kg, and the vertical distance h1D
400 mm. Determine the vertical distances h2and h3.h1h2h3
m
200 mm
300 mm300 mm700 mm
500 mm
1
2
m
3
4
2m
Solution: Cutting to the right of the left mass,
Th
h2
Cutting to the right of the right mass, we obtain
0.2 m
Th
A4
838
Problem 10.71 In Problem 10.70, what is the maxi-
mum tension in the cable, and where does it occur?
Problem 10.72 Each suspended object has the same
weight W. Determine the vertical distances h2and h3.
4 ft
h2
W
TBC
B
TH
840
Problem 10.73 An engineer planning a water system
for a new community estimates that at maximum
expected usage, the pressure drop between the central
system and the farthest planned fire hydrant will be
25 psi. Fire fighting personnel indicate that a gage
pressure of 40 psi at the fire hydrant is required. The
weight density of the water is D62.4 lb/ft3. How tall
would a water tower at the central system have to be to
provide the needed pressure?
62.4ft3
lb ⊲65⊳lb
in212 in
1ft 2
Problem 10.74 A cube of material is suspended below
the surface of a liquid of weight density . By calcu-
lating the forces exerted on the faces of the cube by
pressure, show that their sum is an upward force of
Problem 10.75 The area shown is subjected to a uniform
pressure patm D1ð105Pa.
(a) What is the total force exerted on the area by the
pressure?
(b) What is the moment about the yaxis due to the
pressure?
y
Problem 10.76 The area shown is subjected to a
uniform pressure. Determine the coordinates of the
center of pressure.
Solution:
RD0.6m
Problem 10.77 The area shown is subjected to a
uniform pressure patm D14.7 psi.
(a) What is the total force exerted on the area by the
pressure?
(b) What is the moment about the yaxis due to the
pressure on the area?
x
y
10 in
20 in
Solution: (a) The total force is
FDA
Patm dA DPatm A
dA DPatmA
D14.7lb
in21
2⊲20⊳21
2⊲10⊳2in2D6930 lb.
(b) We can represent the pressure by an equivalent force Facting at
842
Problem 10.78 In Active Example 10.10, suppose that
the water depth relative to point Ais increased from
2 ft to 3 ft. Determine the reactions on the gate at the
supports Aand B.
3 ft
B
Problem 10.79 The top of the rectangular plate is
2 m below the surface of a lake. Atmospheric pressure
patm D1ð105Pa and the mass density of water is
D1000 kg/m3.
(a) What is the maximum pressure exerted on the plate
by the water?
(b) Determine the force exerted on a face of the plate
by the pressure of the water.
(See Example 10.11.)
2 m
3 m
2 m
Solution:
(a) The maximum pressure occurs at the bottom of the plate:
(b)
2 m
3 m
x
y
dx
844
Problem 10.80 In Problem 10.79, how far below the
top of the plate is the center of pressure located?
Problem 10.81 The width of the dam (the dimension
into the page) is 100 m. The mass density of the water
is D1000 kg/m3. Determine the force exerted on the
dam by the gage pressure of the water (a) by integration;
Problem 10.82 In Problem 10.81, how far down from
the surface of the water is the center of pressure due to
the gage pressure of the water on the dam?
Problem 10.83 The width of the gate (the dimension
into the page) is 3 m. Atmospheric pressure patm D
1ð105Pa and the mass density of the water is D
1000 kg/m3. Determine the horizontal force and couple
AxDFD58.9kN,
846
Problem 10.84 The homogenous gate weighs 100 lb,
and its width (the dimension into the page) is 3 ft. The
weight density of the water is D62.4 lb/ft3, and the
atmospheric pressure is patm D2120 lb/ft2. Determine
the reactions at Aand B.
B
Thus the “volume” is FD1
2⊲62.4⊳⊲2⊳⊲6.92⊳D432.32 lb. This
force acts normally to the surface of the gate, or at an angle of D210°
relative to the positive xaxis. The centroid of the pressure is
dD2
256.7 lb, to the right, and AyD346.3 sin⊲210°⊳C86.6 sin⊲120°⊳D