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Problem 7.38 If the cross-sectional area of the beam
shown in Problem 7.37 is 8400 mm2and the ycoordi-
nate of the centroid of the area is yD90 mm, what are
the dimensions band h?
Solution: From the solution to Problem 7.37
A1D120 b, A2D200 h
and yDy1A1Cy2A2
bD39.7mm
200 mm
120 mm
A2h
b
Problem 7.39 Determine the ycoordinate of the cen-
troid of the beam’s cross section.
x
y
2 in 5 in
8 in
Problem 7.40 Determine the coordinates of the cen-
troid of the airplane’s vertical stabilizer.
70°
48°
11 m
12.5 m
y
x
dD12.5mC11 m cot 70°D16.50 m
eD11 m tan 48°D12.22 m
11 m
48°
532
Problem 7.41 The area has elliptical boundaries. If
aD30 mm, bD15 mm, and εD6 mm, what is the x
coordinate of the centroid of the area?
y
b
Solution: The equation of the outer ellipse is
x2
⊲a Cε⊳2Cy2
⊲b Cε⊳2D1
and for the inner ellipse
x2
a2Cy2
b2D1
˛˛2
2
2D˛ˇ/4
(The area of a full ellipse is ˛ˇ so this checks.
Now for the composite area.
A1D2375 mm2A2D1414 mm2
Problem 7.42 By determining the xcoordinate of the
centroid of the area shown in Problem 7.41 in terms of
a,b, and ε, and evaluating its limit as ε!0, show that
the xcoordinate of the centroid of a quarter-elliptical
line is
3A2Dab
4
so x1A1D⊲a Cε⊳2⊲b Cε⊳
3
x2A2Da2b
3
C⊲2aCb⊳ε2Cε3⊳
Finally xDx1A1x2A2
A1A2
1
3⊲2ab Ca2⊳C⊲2aCb⊳ε Cε2ε
Problem 7.43 Three sails of a New York pilot
schooner are shown. The coordinates of the points are
(a)
Solution: Divide the object into three areas: (1) The triangle with
altitude 21 ft and base 20 ft. (2) The triangle with altitude 21 ft
(2) A2D42 ft2,
x2D16 C2
534
Problem 7.44 Determine the centroid of sail 2 in
Problem 7.43.
by 9.5 ft, (3) a triangle at the top with base of 9.5 ft and altitude of
(1) A1D3⊲20⊳
2D30 ft2,
y3D20 C2
33D22 ft
AD6.472 ft ,
yDA1y1CA2y2A3y3A4y4
Problem 7.45 Determine the centroid of sail 3 in
Problem 7.43.
Solution: Divide the object into six areas: (1) The triangle Oef,
(1) A1D36.75 ft2,
(4) A4D130.5ft
x4D17 ft,
y4D9.67 ft.
bga
AD10.877 ft
AD11.23 ft
Problem 7.46 In Active Example 7.5, suppose that the
distributed load is modified as shown. Determine the
reactions on the beam at Aand B.
Solution: We can treat the distributed load as two triangular distri-
buted loads. Using the area analogy, the magnitude of the left one
is 1
2⊲8m⊳⊲60 N/m⊳D240 N, and the magnitude of the right one is
Problem 7.47 Determine the reactions at Aand B.
200 lb/ft
4 ft
6 ft
6 ft
B
A
200 lb/ft
536
Problem 7.48 In Example 7.6, suppose that the
distributed loads are modified as shown. Determine the
reactions on the beam at Aand B.
600 N/m
400 N/m
6 m 6 m
6 m
AB
400 N/m
Problem 7.49 In Example 7.7, suppose that the
distributed load acting on the beam from xD0toxD
10 ft is given by wD350 C0.3x3lb/ft. (a) Determine
the downward force and the clockwise moment about
Aexerted by the distributed load. (b) Determine the
reactions at the fixed support.
10,000 ft-lb
2000 lb
w
y
A
x
10 ft 10 ft
Solution:
(a) The force and moment are
Problem 7.50 Determine the reactions at the fixed
support A.
5 m
= 3(1 – x2/25) kN/m
x
A
y
Solution: The free-body diagram of the beam is: The downward
From the equilibrium equations
FxDAxD0,
538
Problem 7.51 An engineer measures the forces
exerted by the soil on a 10-m section of a building
y
x
A
10 m2 m
Solution:
(a) The total force is
jFjD333.3kN
(b) The moment about the origin is
3x31
4x4C0.2
5x510
0
jMjD1833.33 kN.
FD5.5m,
Problem 7.52 Determine the reactions on the beam at
Aand B.
3 kN/m
2 kN/m
Solution: Replace the distributed load with three equivalent single
forces.
The equilibrium equations
Fx:AxD0
2 kN
Problem 7.53 The aerodynamic lift of the wing is
described by the distributed load
wD300p10.04x2N/m.
The mass of the wing is 27 kg, and its center of mass is
located 2 m from the wing root R.
y
x
R
jFjD1178.1N.
The moment about the root due to the lift is
MD300 5
0
⊲10.04x2⊳1/2xdx,
540
Problem 7.54 Determine the reactions on the bar at A
and B.
x
2 ft
2 ft
y
400 lb/ft
4 ft 4 ft
600 lb/ft
B
A
400 lb/ft
Solution: First replace the distributed loads with three equiva-
lent forces.
The equilibrium equations
Solving:
Problem 7.55 Determine the reactions on member AB
at Aand B.
6 ft
6 ft 6 ft
300 lb/ft
300 lb/ft
AB
C
542
Problem 7.56 Determine the axial forces in members
BD, CD, and CE of the truss and indicate whether they
are in tension (T) or compression (C). ADFH
G
E
C
2 m
2 m 2 m 2 m 2 m
8 kN/m
4 kN/m
B
Problem 7.57 Determine the reactions on member
ABC at Aand B.
160
mm
160
mm
160
mm
160 mm
240 mm
400 N/m
400 N/m
200 N/m
A
B
C
D
E
Solution: Work on the entire structure first to find the reactions
at A. Replace the distributed forces with equivalent concentrated forces
Fx:AxC160 N D0
MB:Ax⊲0.24 m⊳C⊲160 N⊳⊲0.04 m⊳
⊲48 N⊳⊲0.16 m⊳⊲96 N⊳⊲0.24 m⊳
7
p65 CD⊲0.32 m⊳C4
p65 CD⊲0.16 m⊳D0
Fx:AxCBxC160 N 4
p65 CD D0
Fy:AyCBy48 N 96 N 7
p65 CD D0
Solving:
AxD160 N,A
yD52 N
E
Ax
Ay
48 N
96 N
96 N
48 N
544
Problem 7.58 Determine the forces on member ABC
of the frame.
2 m
A
C
2 m 1 m
1 m
1 m 3 kN/m
B
Solution: The free body diagram of the member on which the
distributed load acts is
From the equilibrium equations
we obtain CyD16 kN. Then from the middle free body diagram,
we write the equilibrium equations
BX
(4 m)(3 kN/m) = 12 kN
DY
2 m 2 m
Problem 7.59 Use the method described in Active
Example 7.8 to determine the centroid of the truncated
cone.
y
Solution: Just as in Active Example 7.8, the volume of the disk
Problem 7.60 A grain storage tank has the form of a
surface of revolution with the profile shown. The height
of the tank is 7 m and its diameter at ground level is
10 m. Determine the volume of the tank and the height
above ground level of the centroid of its volume.
y
x
10 m
7 m
y = ax1/2
Solution:
0
0
xDx3/37
0
x2/27
The volume is
VD7
a2xdx Da249
546
Problem 7.61 The object shown, designed to serve
as a pedestal for a speaker, has a profile obtained by
revolving the curve yD0.167x2about the xaxis. What
is the xcoordinate of the centroid of the object?
x
z
y
0.75 m
0.75 m
Solution:
y = 0.167 x2
dV = y2dx
π
Problem 7.62 The volume of a nose cone is generated
by rotating the function yDx0.2x2about the xaxis.
(a) What is the volume of the nose cone?
(b) What is the xcoordinate of the centroid of
the volume?
y
x
z
Problem 7.63 Determine the centroid of the hemisphe-
rical volume.
zR
y
x
Solution: The equation of the surface of a sphere is x2Cy2C
z2DR2.
4R4.
Divide by the volume:
y
548
Problem 7.64 The volume consists of a segment of a
sphere of radius R. Determine its centroid.
R
R
2
x
y
z
Solution: The volume: The element of volume is a disk of radius
and thickness dx. The area of the disk is 2, and the element of
x
R
Problem 7.65 A volume of revolution is obtained
by revolving the curve x2
a2Cy2
b2D1 about the xaxis.
Determine its centroid.
x2
y2
y
550
