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Problem 7.103 The mass of the bar per unit length is
2 kg/m. Choose the dimension bso that part BC of the
suspended bar is horizontal. What is the dimension b,
and what are the resulting reactions on the bar at A?
b
30⬚
1 m
A
B
C
Then
Fx:AxD0
Fy:Ayg⊲1.0m⊳g⊲b⊳ D0)
AxD0,A
yD61.6N,bD2.14 m
ρ
g(1.0 m)
ρ
gb
Problem 7.104 The semicircular part of the homoge-
neous slender bar lies in the x–zplane. Determine the
center of mass of the bar.
y
Solution: The bar is divided into three segments plus the com-
posite. The lengths and the centroids are given in the table: The
composite length is:
LD
3
Li.
Problem 7.105 The density of the cone is given by the
equation D0⊲1Cx/h⊳, where 0is a constant. Use
the procedure described in Example 7.17 to show that
the mass of the cone is given by mD⊲7/4⊳0V, where
Vis the volume of the cone, and that the xcoordinate
of the center of mass of the cone is xD⊲27/35⊳h.
x
z
y
R
h
Solution: Consider an element of volume dV of the cone in the
form of a “disk” of width dx. The radius of such a disk at position x
is (R/h)x,sodV D[⊲R/h⊳x]2dx.
The mass of the cone is
572
Problem 7.106 A horizontal cone with 800-mm length
and 200-mm radius has a built-in support at A. Its density
is D6000⊲1C0.4x2⊳kg/m3, where xis in meters.
What are the reactions at A?
y
A
800 mm
200 mm
x
Solution: The strategy is to determine the distance to the line of
action of the weight, from which to apply the equilibrium conditions.
y
200
Problem 7.107 In Active Example 7.18, suppose that
bar 1 is replaced by a bar with the same dimensions
that consists of aluminum alloy with a density of
2600 kg/m3. Determine the xcoordinate of the center
of mass of the machine part.
1
2
y
80 mm
240 mm
40 mm
The xcoordinate of the center of mass is
Problem 7.108 The cylindrical tube is made of
aluminum with mass density 2700 kg/m3. The cylin-
drical plug is made of steel with mass density
7800 kg/m3. Determine the coordinates of the center of
mass of the composite object.
y
x
z
y
z
A
A
20 mm
35 mm
Section A-A
x
y
100
mm
100
mm
Tube
Plug
Solution: The volume of the aluminum tube is
574
Problem 7.109 In Example 7.19, suppose that the
object is redesigned so that the radius of the hole in
the hollow cylinder is increased from 2 in to 3 in. What
is the xcoordinate of the center of mass of the object?
Front View
zx
y
y
Side View
10 in
2 in
4 in
12 in
5 in 5 in
Problem 7.110 A machine consists of three parts. The
Part Mass (kg) x(mm) y(mm) z(mm)
2 4.5 150 70 0
to position part 3 so that the center of mass location
Solution: The composite mass is mD2.0C4.5C2.5D9 kg.
2.5D82 mm
2.5D122 mm
2.5D16 mm
Problem 7.111 Two views of a machine element
are shown. Part 1 is aluminum alloy with density
2800 kg/m3, and part 2 is steel with density 7800 kg/m3.
Determine the coordinates of its center of mass.
60 mm
yy
18 mm
2
1
24 mm
8 mm
so their masses are
m1DS1V1D⊲2800⊳⊲17.92 ð105⊳D0.502 kg,
576
Problem 7.112 The loads F1DF2D25 kN. The
mass of the truss is 900 kg. The members of the truss are
homogeneous bars with the same uniform cross section.
(a) What is the xcoordinate of the center of mass of the
truss? (b) Determine the reactions at Aand G.
A
B
G
F2
F1
4 m
C
4 m
3 m
D
E
3 m
y
x
Solution:
(a) The center of mass of the truss is located at the centroid of
the composite line of the axes of the members. The lengths of
(b) The equilibrium equations for the truss are
Problem 7.113 With its engine removed, the mass of
the car is 1100 kg and its center of mass is at C. The
mass of the engine is 220 kg.
(a) Suppose that you want to place the center of mass
Eof the engine so that the center of mass of the
car is midway between the front wheels Aand the
rear wheels B. What is the distance b?
(b) If the car is parked on a 15°slope facing up the
slope, what total normal force is exerted by the
road on the rear wheels B?
1.14 m
A
b
2.60 m
B
0.6 m
0.45 m
C
E
xD2.6
2D1.3m,
mD0.475 m.
(b) Assume that the engine has been placed in the new position, as
given in Part (a). The sum of the moments about Bis
578
Problem 7.114 The airplane is parked with its landing
gear resting on scales. The weights measured at A,B, and
Care 30 kN, 140 kN, and 146 kN, respectively. After a
crate is loaded onto the plane, the weights measured at A,
B, and Care 31 kN, 142 kN, and 147 kN, respectively.
Determine the mass and the xand ycoordinates of the
center of mass of the crate.
x
10 m
6 m B
C
A
6 m
Solution: The weight of the airplane is WAD30 C140 C146 D
316 kN. The center of mass of the airplane:
Myaxis D30⊲10⊳xAWAD0,
Problem 7.115 A suitcase with a mass of 90 kg is
placed in the trunk of the car described in Example 7.20.
The position of the center of mass of the suitcase is
xsD0.533 m, ysD0.762 m, and zsD0.305 m. If
the suitcase is regarded as part of the car, what is the
new position of the car’s center of mass?
Solution: In Example 7.20, the following results were obtained
zcD0.769 m
The new center of mass is at
Problem 7.116 A group of engineering students
constructs a miniature device of the kind described in
Example 7.20 and uses it to determine the center of mass
of a miniature vehicle. The data they obtain are shown
in the following table:
Wheelbase =36 in
Track =30 in Measured Loads (lb)
˛D0˛D10°
Left front wheel, NLF 35 32
Right front wheel, NRF 36 33
Left rear wheel, NLR 27 34
Right rear wheel, NRR 29 30
Determine the center of mass of the vehicle. Use the
same coordinate system as in Example 7.20.
xD36⊲35 C36⊳
WD20.125 in.
Mxaxis DzW⊲Track⊳⊲NRF CNRR⊳D0,
from which
zD⊲30⊳⊲36 C29⊳
WD15.354 in.
from which
yDxWcos⊲10°⊳⊲36⊳⊲32 C33⊳
D8.034 in.
580
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.117 Determine the centroid of the area by
letting dA be a vertical strip of width dx.
x
y
y = x2
(1, 1)
Solution: The area: The length of the vertical strip is ⊲1y⊳,so
Problem 7.118 Determine the centroid of the area in
Problem 7.117 by letting dA be a horizontal strip of
height dy .
Solution: The area: The length of the horizontal strip is x, hence
the element of area is
Divide by the area: xD3
8
Problem 7.119 Determine the centroid of the area.
x
60 cm
60 cm
y
80 cm
0
The x-coordinate:
Divide by the area: xDw
2D40 cm
The y-coordinate:
1
h2dx D1
Dab
2CabDab
2D1800 cm2
Check: This is the familiar result. check.
The x-coordinate:
b
0a
bxCaxdx Dax3
3bCax2
2b
0
Dab2
6.
Divide by the area: xDb
3D20 cm
The y-coordinate:
A
ydAD1
2b
0a
bxCa2
dx
D b
6aa
bxCa3b
0
Dba2
6.
D56.36 cm
D27.27 cm
582
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.120 Determine the centroid of the area.
160 mm
20 mm
40 mm
y
40 mm
Solution: Divide the object into five areas:
(2) The rectangle 120 mm by 80 mm,
(4) The circle of 20 mm radius, and
(2) A2D9600 mm2,
x4D120 mm, y4D120 mm.
AD90.3mm .
yDA1y1CA2y2CA3y3A4y4
Problem 7.121 The cantilever beam is subjected to a
triangular distributed load. What are the reactions at A?
y
x
200 N/m
A
Solution: The load distribution is a straight line with intercept
wD200 N/m at xD0, and slope 200
10 D20 N/m2. The sum
of the moments is
MDMA10
⊲20xC200⊳x dx D0,
Problem 7.122 What is the axial load in member BD
of the frame?
5 m
10 m
100 N/m
5 m
A
BD
C
E
Solution: The distributed load is two straight lines: Over the
interval 0 y5 the intercept is wD0atyD0 and the slope is
C100
5D20.
MED5
⊲20y⊳y dy C10
100ydy,
d1D⊲2
3⊳5D3.333 m.
The centroid distance of the rectangle is d2D7.5 m. The moment
about Eis
MEDd1F1Cd2F2D4583.33 Nm check.
The Complete Structure: The sum of the moments about Eis
MD10ARCMED0,
where ARis the reaction at A, from which ARD458.33 N.
The element ABC : Element BD is a two force member, hence ByD0.
The sum of the moments about C:
MCD5Bx10AyD0,
where Ayis equal and opposite to the reaction of the support, from
which
BxD2AyD2ARD916.67 N.
Since the reaction in element BD is equal and opposite, BxD
916.67 N, which is a tension in BD.
Bx
By
Dy
Cx
Ex
Cy
Cy
Cx
584
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 7.123 An engineer estimates that the maxi-
mum wind load on the 40-m tower in Fig. a is described
by the distributed load in Fig. b. The tower is supported
by three cables A,B, and Cfrom the top of the tower to
equally spaced points 15 m from the bottom of the tower
(Fig. c). If the wind blows from the west and cables B
and Care slack, what is the tension in cable A? (Model
the base of the tower as a ball and socket support.)
N
15 m
A
B
C
(c)
40 m
(a)
200 N/m
400 N/m
(b)
Solution: The load distribution is a straight line with the intercept
wD400 N/m, and slope 5. The moment about the base of the tower
due to the wind load is
200 N/m
θ
