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PROBLEM 5.12
SOLUTION
Area mm2
, mmx
, mmy
3
, mmxA
3
, mmyA
3
1(200)(480) 32 10
6
11.52 10×
6
1.92 10
×
PROBLEM 5.13
The horizontal x-axis is drawn through the centroid C of the area shown,
and it divides the area into two component areas A1 and A2. Determine
the first moment of each component area with respect to the x-axis, and
explain the results obtained.
SOLUTION
Area above x-axis (Area A1):
1
33
(25)(20 80) (7.5)(15 20)
40 10 2.25 10
Q yA=Σ= ×+ ×
=×+ ×
33
1
42.3 10 mmQ= ×
Area below x-axis (Area A2):
2( 32.5)(65 20)Q yA=Σ=− ×
33
2
42.3 10 mmQ=−×
Dimensions in mm
PROBLEM 5.14
The horizontal x-axis is drawn through the centroid C of the area shown,
and it divides the area into two component areas A1 and A2. Determine the
first moment of each component area with respect to the x-axis, and
explain the results obtained.
PROBLEM 5.16
A composite beam is constructed by bolting
four plates to four 60 × 60 × 12-mm angles as
shown. The bolts are equally spaced along the
beam, and the beam supports a vertical load.
As proved in mechanics of materials, the
shearing forces exerted on the bolts at A and B
are proportional to the first moments with
respect to the centroidal x-axis of the red-
shaded areas shown, respectively, in parts a
and b of the figure. Knowing that the force
exerted on the bolt at A is 280 N, determine
the force exerted on the bolt at B.
SOLUTION
x
Therefore,
()
, or
() () ()
xB
AB BA
xA xB xA
Q
FF FF
QQ Q
= =
For the first moments,
3
3
12
( ) 225 (300 12)
2
831,600 mm
12
( ) ( ) 2 225 (48 12) 2(225 30)(12 60)
2
1,364,688 mm
xA
xB x
Q
Q QA
=+×
=
= + − ×+ − ×
=
Then
1,364,688 (280 N)
831,600
B
F=
or
459 N
B
F=
consent of McGraw-Hill Education.
PROBLEM 5.17
A thin, homogeneous wire is bent to form the perimeter of the figure indicated.
Locate the center of gravity of the wire figure thus formed.
SOLUTION
, mmL
, mmx
, mmy
2
,mmxL
2
,mmyL
PROBLEM 5.18
A thin, homogeneous wire is bent to form the perimeter of the figure indicated.
Locate the center of gravity of the wire figure thus formed.
SOLUTION
L, in.
, in.x
, in.y
2
, inxL
2
, inyL
1 4 0 2 0 8
2
22
3 6 6.7082+=
3 5.5 20.125 36.894
3 7 6 3.5 42 24.5
4 6 3 0 18 0
Σ
23.708 80.125 69.394
Then
(23.708) 80.125X L xL XΣ=Σ =
3.38 in.X=
(23.708) 69.394Y L yL YΣ=Σ =
2.93 in.Y=
consent of McGraw-Hill Education.
PROBLEM 5.19
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the center of gravity of the wire figure thus formed.
SOLUTION
Perimeter of Figure P5.4
L
x
y
2
, mmxL
2
, mmyL
I 30 0 15 0
3
0.45 10×
II 210 105 30
3
22.05 10×
3
6.3 10×
III 270 210 165
3
56.7 10×
3
44.55 10×
IV 30 225 300
3
6.75 10
×
3
9 10×
V 300 240 150
3
72 10×
3
45 10×
VII 240 120 0
3
28.8 10×
0
Σ
1080
3
186.3 10×
3
105.3 10×
32
(1080 mm) 186.3 10 mm
X L xL
X
Σ=Σ
= ×
172.5 mmX=
32
(1080 mm) 105.3 10 mm
Y L yL
Y
Σ=Σ
= ×
97.5 mmY=
Dimensions in mm
consent of McGraw-Hill Education.
PROBLEM 5.20
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the center of gravity of the wire figure thus formed.
SOLUTION
By symmetry,
.XY=
L, in.
, in.x
2
, inyL
1
1(10) 15.7080
2
π
=
2(10) 6.3662
π
=
100
2 5 0 0
3 5 2.5 12.5
4 5 5 25
5 5 7.5 37.5
Σ
35.708 175
Then
(35.708) 175X L xL XΣ=Σ =
4.90 in.XY= =
consent of McGraw-Hill Education.
PROBLEM 5.12
SOLUTION
Area mm2
, mmx
, mmy
3
, mmxA
3
, mmyA
3
1(200)(480) 32 10
6
11.52 10×
6
1.92 10
×
PROBLEM 5.13
The horizontal x-axis is drawn through the centroid C of the area shown,
and it divides the area into two component areas A1 and A2. Determine
the first moment of each component area with respect to the x-axis, and
explain the results obtained.
SOLUTION
Area above x-axis (Area A1):
1
33
(25)(20 80) (7.5)(15 20)
40 10 2.25 10
Q yA=Σ= ×+ ×
=×+ ×
33
1
42.3 10 mmQ= ×
Area below x-axis (Area A2):
2( 32.5)(65 20)Q yA=Σ=− ×
33
2
42.3 10 mmQ=−×
Dimensions in mm
PROBLEM 5.14
The horizontal x-axis is drawn through the centroid C of the area shown,
and it divides the area into two component areas A1 and A2. Determine the
first moment of each component area with respect to the x-axis, and
explain the results obtained.
PROBLEM 5.16
A composite beam is constructed by bolting
four plates to four 60 × 60 × 12-mm angles as
shown. The bolts are equally spaced along the
beam, and the beam supports a vertical load.
As proved in mechanics of materials, the
shearing forces exerted on the bolts at A and B
are proportional to the first moments with
respect to the centroidal x-axis of the red-
shaded areas shown, respectively, in parts a
and b of the figure. Knowing that the force
exerted on the bolt at A is 280 N, determine
the force exerted on the bolt at B.
SOLUTION
x
Therefore,
()
, or
() () ()
xB
AB BA
xA xB xA
Q
FF FF
QQ Q
= =
For the first moments,
3
3
12
( ) 225 (300 12)
2
831,600 mm
12
( ) ( ) 2 225 (48 12) 2(225 30)(12 60)
2
1,364,688 mm
xA
xB x
Q
Q QA
=+×
=
= + − ×+ − ×
=
Then
1,364,688 (280 N)
831,600
B
F=
or
459 N
B
F=
consent of McGraw-Hill Education.
PROBLEM 5.17
A thin, homogeneous wire is bent to form the perimeter of the figure indicated.
Locate the center of gravity of the wire figure thus formed.
SOLUTION
, mmL
, mmx
, mmy
2
,mmxL
2
,mmyL
PROBLEM 5.18
A thin, homogeneous wire is bent to form the perimeter of the figure indicated.
Locate the center of gravity of the wire figure thus formed.
SOLUTION
L, in.
, in.x
, in.y
2
, inxL
2
, inyL
1 4 0 2 0 8
2
22
3 6 6.7082+=
3 5.5 20.125 36.894
3 7 6 3.5 42 24.5
4 6 3 0 18 0
Σ
23.708 80.125 69.394
Then
(23.708) 80.125X L xL XΣ=Σ =
3.38 in.X=
(23.708) 69.394Y L yL YΣ=Σ =
2.93 in.Y=
consent of McGraw-Hill Education.
PROBLEM 5.19
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the center of gravity of the wire figure thus formed.
SOLUTION
Perimeter of Figure P5.4
L
x
y
2
, mmxL
2
, mmyL
I 30 0 15 0
3
0.45 10×
II 210 105 30
3
22.05 10×
3
6.3 10×
III 270 210 165
3
56.7 10×
3
44.55 10×
IV 30 225 300
3
6.75 10
×
3
9 10×
V 300 240 150
3
72 10×
3
45 10×
VII 240 120 0
3
28.8 10×
0
Σ
1080
3
186.3 10×
3
105.3 10×
32
(1080 mm) 186.3 10 mm
X L xL
X
Σ=Σ
= ×
172.5 mmX=
32
(1080 mm) 105.3 10 mm
Y L yL
Y
Σ=Σ
= ×
97.5 mmY=
Dimensions in mm
consent of McGraw-Hill Education.
PROBLEM 5.20
A thin, homogeneous wire is bent to form the perimeter of the figure
indicated. Locate the center of gravity of the wire figure thus formed.
SOLUTION
By symmetry,
.XY=
L, in.
, in.x
2
, inyL
1
1(10) 15.7080
2
π
=
2(10) 6.3662
π
=
100
2 5 0 0
3 5 2.5 12.5
4 5 5 25
5 5 7.5 37.5
Σ
35.708 175
Then
(35.708) 175X L xL XΣ=Σ =
4.90 in.XY= =
consent of McGraw-Hill Education.
