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Statics Note Statics Solution and Answer
Applied mathematics 1A (University of KwaZulu-Natal)
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4–1.
If A, B, and D are given vectors, prove the
distributive law for the vector cross product, i.e.,
A : (B + D) = (A : B) + (A : D).
SOLUTION
Consider the three vectors; with A vertical.
Note obd is perpendicular to A.
od = ƒ A * (B + D) ƒ = ƒ A ƒ ƒ B + D ƒ sin u3
ob = ƒ A * B ƒ = ƒ A ƒ ƒ B ƒ sin u1
bd = ƒ A * D ƒ = ƒ A ƒ ƒ D ƒ sin u2
Also, these three cross products all lie in the plane obd since they are all
perpendicular to A. As noted the magnitude of each cross product is proportional to
the length of each side of the triangle.
The three vector cross products also form a closed triangle o¿b¿d¿ which is similar to
triangle obd. Thus from the figure,
A * (B + D) = (A * B) + (A * D)
(QED)
Note also,
A = Ax i + Ay j + Az k
B = Bx i + By j + Bz k
D = Dx i + Dy j + Dz k
A * (B + D) = 3
i
Ax
Bx + Dx
j
Ay
By + Dy
k
Az 3
Bz + Dz
= [A y (Bz + Dz) - A z(By + Dy)]i
- [A x(Bz + Dz) - A z(Bx + Dx)]j
+ [A x(By + Dy) - A y(Bx + Dx)]k
= [(A y Bz - A zBy)i - (A x Bz - A z Bx)]j + (A x By - A y Bx)k
+ [(A y Dz - A z Dy)i - (A x Dz - A z Dx)j + (A x Dy - A y Dx)k
i
= 3 Ax
Bx
j
Ay
By
k
i
Az 3 + 3 Ax
Bz
Dx
j
Ay
Dy
k
Az 3
Dz
= (A * B) + (A * D)
(QED)
228
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4–2.
Prove
the
triple
scalar
A # (B : C) = (A : B) # C.
product
identity
SOLUTION
As shown in the figure
Area = B(C sin u) = |B * C|
Thus,
Volume of parallelepiped is |B * C||h|
But,
|h| = |A # u(B * C)| = ` A # a
B * C
b`
|B * C|
Thus,
Volume = |A # (B * C)|
Since |(A * B) # C| represents this same volume then
A # (B : C) = (A : B) # C
(QED)
Also,
LHS = A # (B : C)
= (A x i + A y j + A z k) # 3 Bx
i
Cx
j
By
Cy
k
Bz 3
Cz
= A x (ByCz - BzCy) - A y (BxCz - BzCx) + A z (BxCy - ByCx)
= A xByCz - A xBzCy - A yBxCz + A yBzCx + A zBxCy - A zByCx
RHS = (A : B) # C
i
= 3 Ax
Bx
j
Ay
By
k
A z 3 # (Cx i + Cy j + Cz k)
Bz
= Cx(A y Bz - A zBy) - Cy(A xBz - A zBx) + Cz(A xBy - A yBx)
= A xByCz - A xBzCy - A yBxCz + A yBzCx + A zBxCy - A zByCx
Thus, LHS = RHS
A # (B : C) = (A : B) # C
(QED)
229
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4–3.
Given the three nonzero vectors A, B, and C, show that if
A # (B : C) = 0, the three vectors must lie in the same
plane.
SOLUTION
Consider,
|A # (B * C)| = |A| |B * C | cos u
= (|A| cos u)|B * C|
= |h| |B * C|
= BC |h| sin f
= volume of parallelepiped.
If A # (B * C) = 0, then the volume equals zero, so that A, B, and C are coplanar.
230
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*4–4.
Determine the moment about point A of each of the three
forces acting on the beam.
F2 = 500 lb
F1 = 375 lb
5
A
4
3
8 ft
6 ft
SOLUTION
B
0.5 ft
5 ft
30˚
F3 = 160 lb
a + 1MF12A = - 375182
= - 3000 lb # ft = 3.00 kip # ft (Clockwise)
Ans.
4
a + 1MF22A = - 500 a b 1142
5
= -5600 lb # ft = 5.60 kip # ft (Clockwise)
Ans.
a + 1MF32A = - 1601cos 30°21192 + 160 sin 30°10.52
= - 2593 lb # ft = 2.59 kip # ft (Clockwise)
Ans.
Ans:
( MF1 ) A = 3.00 kip # ft (Clockwise)
( MF2 ) A = 5.60 kip # ft (Clockwise)
( MF3 ) A = 2.59 kip # ft (Clockwise)
231
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4–5.
Determine the moment about point B of each of the three
forces acting on the beam.
F2 = 500 lb
F1 = 375 lb
5
A
4
3
8 ft
SOLUTION
6 ft
B
0.5 ft
5 ft
30˚
F3 = 160 lb
a + 1MF12B = 3751112
= 4125 lb # ft = 4.125 kip # ft (Counterclockwise)
Ans.
4
a + 1MF22B = 500a b 152
5
= 2000 lb # ft = 2.00 kip # ft (Counterclockwise)
Ans.
a + 1MF32B = 160 sin 30°10.52 - 160 cos 30°102
= 40.0 lb # ft (Counterclockwise)
Ans.
Ans:
( MF1 ) B = 4.125 kip # ftd
( MF2 ) B = 2.00 kip # ftd
( MF3 ) B = 40.0 lb # ftd
232
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4–6.
The crowbar is subjected to a vertical force of P = 25 lb at the
grip, whereas it takes a force of F = 155 lb at the claw to pull
the nail out. Find the moment of each force about point A and
determine if P is suficient to pull out the nail. The crowbar
contacts the board at point A.
60
F
O
20
3 in.
P
A
14 in.
1.5 in.
SOLUTION
a + MP = 25 ( 14 cos 20° + 1.5 sin 20° ) = 341 in # lb (Counterclockwise)
c + MF = 155 sin 60°(3) = 403 in # lb (Clockwise)
Since MF 7 MP,
P = 25 lb is not suficient to pull out the nail.
Ans.
Ans:
MP = 341 in. # lbd
MF = 403 in. # lbb
Not sufficient
233
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4–7.
Determine the moment of each of the three forces about
point A.
F1
F2
250 N 30
300 N
60
A
2m
3m
4m
SOLUTION
The moment arm measured perpendicular to each force from point A is
d1 = 2 sin 60° = 1.732 m
B
4
5
3
d2 = 5 sin 60° = 4.330 m
F3
500 N
d3 = 2 sin 53.13° = 1.60 m
Using each force where MA = Fd, we have
a + 1MF12A = - 25011.7322
= - 433 N # m = 433 N # m (Clockwise)
Ans.
a + 1MF22A = - 30014.3302
= - 1299 N # m = 1.30 kN # m (Clockwise)
Ans.
a + 1MF32A = - 50011.602
= - 800 N # m = 800 N # m (Clockwise)
Ans.
Ans:
( MF1 ) A = 433 N # mb
( MF2 ) A = 1.30 kN # mb
( MF3 ) A = 800 N # mb
234
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*4–8.
Determine the moment of each of the three forces about point B.
F1
F2
250 N 30
300 N
60
A
2m
3m
SOLUTION
4m
The forces are resolved into horizontal and vertical component as shown in Fig. a.
For F1,
a + MB = 250 cos 30°(3) - 250 sin 30°(4)
= 149.51 N # m = 150 N # m d
Ans.
B
4
5
3
For F2,
F3
500 N
a + MB = 300 sin 60°(0) + 300 cos 60°(4)
= 600 N # m d
Ans.
Since the line of action of F3 passes through B, its moment arm about point B is
zero. Thus
MB = 0
Ans.
Ans:
MB = 150 N # md
MB = 600 N # md
MB = 0
235
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4–9.
Determine the moment of each force about the bolt located
at A. Take FB = 40 lb, FC = 50 lb.
0.75 ft
B
2.5 ft
30 FC
20
A
C
25
FB
SOLUTION
a +MB = 40 cos 25°(2.5) = 90.6 lb # ft d
Ans.
a +MC = 50 cos 30°(3.25) = 141 lb # ftd
Ans.
Ans:
MB = 90.6 lb # ftb
MC = 141 lb # ftd
236
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4–10.
If FB = 30 lb and FC = 45 lb, determine the resultant
moment about the bolt located at A.
0.75 ft
B
2.5 ft
A
C
30 FC
20
25
FB
SOLUTION
a +MA = 30 cos 25°(2.5) + 45 cos 30°(3.25)
= 195 lb # ft d
Ans:
MA = 195 lb # ftd
237
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4–11.
The towline exerts a force of P = 6 kN at the end of the
8-m-long crane boom. If u = 30°, determine the placement
x of the hook at B so that this force creates a maximum
moment about point O. What is this moment?
A
P 6 kN
8m
u
O
1m
B
x
SOLUTION
In order to produce the maximum moment about point O, P must act perpendicular
to the boom’s axis OA as shown in Fig. a. Thus
a+ (MO)max = 6 (8) = 48.0 kN # m (counterclockwise)
Ans.
Referring to the geometry of Fig. a,
x = x' + x" =
8
+ tan 30° = 9.814 m = 9.81 m
cos 30°
Ans.
Ans:
(MO)max = 48.0 kN # m d
x = 9.81 m
238
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*4–12.
The towline exerts a force of P = 6 kN at the end of the
8-m-long crane boom. If x = 10 m, determine the position u
of the boom so that this force creates a maximum moment
about point O. What is this moment?
A
P 6 kN
8m
u
O
1m
B
x
SOLUTION
In order to produce the maximum moment about point O, P must act perpendicular
to the boom’s axis OA as shown in Fig. a. Thus,
a+ (MO)max = 6 (8) = 48.0 kN # m (counterclockwise)
Ans.
Referring to the geometry of Fig. a,
x = x' + x";
10 =
8
+ tan u
cos u
10 =
8
sin u
+
cos u
cos u
10 cos u - sin u = 8
10
1
8
cos u sin u =
1101
1101
1101
(1)
From the geometry shown in Fig. b,
a = tan-1 a
sin a =
1
b = 5.711°
10
1
1101
cos a =
10
1101
Then Eq (1) becomes
cos u cos 5.711° - sin u sin 5.711° =
8
1101
Referring that cos (u + 5.711°) = cos u cos 5.711° - sin u sin 5.711°
cos (u + 5.711°) =
8
1101
u + 5.711° = 37.247°
u = 31.54° = 31.5°
Ans.
Ans:
(MO)max = 48.0 kN # m (counterclockwise)
u = 31.5°
239
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4–13.
z
The 20-N horizontal force acts on the handle of the socket
wrench. What is the moment of this force about point B.
Specify the coordinate direction angles a, b, g of the
moment axis.
20 N
200 mm
B
A
60
10 mm
50 mm
O
y
x
SOLUTION
Force Vector And Position Vector. Referring to Fig. a,
F = 20 (sin 60°i - cos 60°j) = {17.32i - 10j} N
rBA = { - 0.01i + 0.2j} m
Moment of Force F about point B.
MB = rBA * F
i
= † -0.01
17.32
k
0†
0
j
0.2
- 10
= { -3.3641 k} N # m
= { -3.36 k} N # m
Ans.
Here the unit vector for MB is u = - k. Thus, the coordinate direction
angles of MB are
a = cos-1 0 = 90°
Ans.
b = cos-1 0 = 90°
Ans.
-1
Ans.
g = cos
( - 1) = 108°
Ans:
MB = { - 3.36 k} N # m
a = 90°
b = 90°
g = 180°
240
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4–14.
z
The 20-N horizontal force acts on the handle of the socket
wrench. Determine the moment of this force about point O.
Specify the coordinate direction angles a, b, g of the
moment axis.
20 N
200 mm
B
A
60
10 mm
50 mm
O
y
x
SOLUTION
Force Vector And Position Vector. Referring to Fig. a,
F = 20 (sin 60°i - cos 60°j) = {17.32i - 10j} N
rOA = { - 0.01i + 0.2j + 0.05k} m
Moment of F About point O.
MO = rOA * F
i
= † - 0.01
17.32
j
0.2
-10
k
0.05 †
0
= {0.5i + 0.8660j - 3.3641k} N # m
= {0.5i + 0.866j - 3.36k} N # m
Ans.
The magnitude of MO is
MO = 2(MO)2x + (MO)2y + (MO)2z = 20.52 + 0.86602 + ( -3.3641)2
= 3.5096 N # m
Thus, the coordinate direction angles of MO are
a = cos-1 c
b = cos-1 c
g = cos-1 c
(MO)x
MO
(MO)y
MO
(MO)z
MO
d = cos-1 a
d = cos-1 a
d = cos-1 a
0.5
b = 81.81° = 81.8°
3.5096
0.8660
b = 75.71° = 75.7°
3.5096
-3.3641
b = 163.45° = 163°
3.5096
Ans.
Ans.
Ans.
Ans:
MO = {0.5i + 0.866j - 3.36k} N # m
a = 81.8°
b = 75.7°
g = 163°
241
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4–15.
Two men exert forces of F = 80 lb and P = 50 lb on the
ropes. Determine the moment of each force about A. Which
way will the pole rotate, clockwise or counterclockwise?
6 ft
P
F
45
3
B
12 ft
5
4
C
SOLUTION
A
4
c + (MA)C = 80a b (12) = 768 lb # ftb
5
Ans.
a + (MA)B = 50 (cos 45°)(18) = 636 lb # ftd
Ans.
Since (MA)C 7 (MA)B
Clockwise
Ans.
Ans:
(MA)C = 768 lb # ftb
(MA)B = 636 lb # ftd
Clockwise
242
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*4–16.
If the man at B exerts a force of P = 30 lb on his rope,
determine the magnitude of the force F the man at C must
exert to prevent the pole from rotating, i.e., so the resultant
moment about A of both forces is zero.
6 ft
P
F
45
3
B
SOLUTION
12 ft
5
4
C
a+
4
30 (cos 45°)(18) = F a b(12) = 0
5
A
F = 39.8 lb
Ans.
Ans:
F = 39.8 lb
243
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4–17.
The torque wrench ABC is used to measure the moment or
torque applied to a bolt when the bolt is located at A and a
force is applied to the handle at C. The mechanic reads the
torque on the scale at B. If an extension AO of length d is
used on the wrench, determine the required scale reading if
the desired torque on the bolt at O is to be M.
F
M
A
O
d
B
l
C
SOLUTION
Moment at A = m = Fl
Moment at O = M = (d + l)F
M = (d + l)
m = a
m
l
l
bM
d + l
Ans.
Ans:
m = a
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bM
d + l
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4–18.
The tongs are used to grip the ends of the drilling pipe P .
Determine the torque (moment) MP that the applied force
F = 150 lb exerts on the pipe about point P as a function of
u. Plot this moment MP versus u for 0 … u … 90°.
F
u
P
6 in.
SOLUTION
MP
MP = 150 cos u(43) + 150 sin u(6)
43 in.
= (6450 cos u + 900 sin u) lb # in.
= (537.5 cos u + 75 sin u) lb # ft
dMP
= - 537.5 sin u + 75 cos u = 0
du
tan u =
75
537.5
Ans.
u = 7.943°
At u = 7.943° , MP is maximum.
(MP)max = 538 cos 7.943° + 75 sin 7.943° = 543 lb # ft
Also (MP)max = 150 lb ¢ a
1
6 2 2
43 2
b + a b ≤ = 543 lb # ft
12
12
Ans:
MP = (537.5 cos u + 75 sin u) lb # ft
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4–19.
The tongs are used to grip the ends of the drilling pipe P . If
a torque (moment) of MP = 800 lb # ft is needed at P to
turn the pipe, determine the cable force F that must be
applied to the tongs. Set u = 30°.
F
u
P
6 in.
SOLUTION
MP = F cos 30°(43) + F sin 30°(6)
Set MP = 800(12) lb # in.
MP
43 in.
800(12) = F cos 30°(43) + F sin 30°(6)
F = 239 lb
Ans.
Ans:
F = 239 lb
246
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*4–20.
The handle of the hammer is subjected to the force of
F = 20 lb. Determine the moment of this force about the
point A.
F
30
5 in.
18 in.
SOLUTION
Resolving the 20-lb force into components parallel and perpendicular to the
hammer, Fig. a, and applying the principle of moments,
A
B
a +MA = - 20 cos 30°(18) - 20 sin 30°(5)
= -361.77 lb # in = 362 lb # in (Clockwise)
Ans.
Ans:
MA = 362 lb # in (Clockwise)
247
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4–21.
In order to pull out the nail at B, the force F exerted on the
handle of the hammer must produce a clockwise moment of
500 lb # in. about point A. Determine the required magnitude
of force F.
F
30
5 in.
18 in.
SOLUTION
Resolving force F into components parallel and perpendicular to the hammer, Fig. a,
and applying the principle of moments,
A
B
a + MA = - 500 = -F cos 30°(18) - F sin 30°(5)
F = 27.6 lb
Ans.
Ans:
F = 27.6 lb
248
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4–22.
y
Old clocks were constructed using a fusee B to drive the
gears and watch hands. The purpose of the fusee is to
increase the leverage developed by the mainspring A
as it uncoils and thereby loses some of its tension. The
mainspring can develop a torque (moment) Ts = ku,
where k = 0.015 N # m>rad is the torsional stiffness and u
is the angle of twist of the spring in radians. If the torque
Tf developed by the fusee is to remain constant as the
mainspring winds down, and x = 10 mm when u = 4 rad,
determine the required radius of the fusee when u = 3 rad.
x
A
B
y
t
x
12 mm
Ts
Tf
SOLUTION
When u = 4 rad, r = 10 mm
Ts = 0.015(4) = 0.06 N # m
F =
0.06
= 5N
0.012
Tf = 5(0.010) = 0.05 N # m (constant)
When u = 3 rad,
Ts = 0.015(3) = 0.045 N # m
F =
0.045
= 3.75 N
0.012
For the fusee require
0.05 = 3.75 r
r = 0.0133 m = 13.3 mm
Ans.
Ans:
r = 13.3 mm
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4–23.
4m
The tower crane is used to hoist the 2-Mg load upward at
constant velocity. The 1.5-Mg jib BD, 0.5-Mg jib BC, and
6-Mg counterweight C have centers of mass at G1 , G2 , and
G3 , respectively. Determine the resultant moment produced
by the load and the weights of the tower crane jibs about
point A and about point B.
G2
9.5m
B
D
C
G3
7.5 m
12.5 m
G1
23 m
SOLUTION
Since the moment arms of the weights and the load measured to points A and B are
the same, the resultant moments produced by the load and the weight about points
A and B are the same.
a + (MR)A = (MR)B = ©Fd;
A
(MR)A = (MR)B = 6000(9.81)(7.5) + 500(9.81)(4) - 1500(9.81)(9.5)
- 2000(9.81)(12.5) = 76 027.5 N # m = 76.0 kN # m (Counterclockwise)
Ans.
Ans:
(MR)A = (MR)B = 76.0 kN # md
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*4–24.
The tower crane is used to hoist a 2-Mg load upward at constant velocity. The 1.5-Mg jib BD and 0.5-Mg jib BC have
centers of mass at G1 and G2 , respectively. Determine the
required mass of the counterweight C so that the resultant
moment produced by the load and the weight of the tower
crane jibs about point A is zero. The center of mass for the
counterweight is located at G3 .
4m
G2
9.5m
B
D
C
G3
7.5 m
12.5 m
G1
23 m
SOLUTION
a + (MR)A = ©Fd;
A
0 = MC(9.81)(7.5) + 500(9.81)(4) - 1500(9.81)(9.5) - 2000(9.81)(12.5)
MC = 4966.67 kg = 4.97 Mg
Ans.
Ans:
MC = 4.97 Mg
251
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4–25.
If the 1500-lb boom AB, the 200-lb cage BCD, and the 175-lb
man have centers of gravity located at points G1 , G2 and G3 ,
respectively, determine the resultant moment produced by
each weight about point A.
G3
D
B
G2
C
2.5 ft 1.75 ft
20 ft
G1
SOLUTION
10 ft
75
Moment of the weight of boom AB about point A:
A
a + (MAB)A = - 1500(10 cos 75°) = - 3882.29 lb # ft
= 3.88 kip # ft (Clockwise)
Ans.
Moment of the weight of cage BCD about point A:
a + (MBCD)A = - 200(30 cos 75° + 2.5) = - 2052.91 lb # ft
= 2.05 kip # ft (Clockwise)
Ans.
Moment of the weight of the man about point A:
a + (Mman)A = - 175(30 cos 75° + 4.25) = - 2102.55 lb # ft
= 2.10 kip # ft (Clockwise)
Ans.
Ans:
( MAB ) A = 3.88 kip # ftb
( MBCD ) A = 2.05 kip # ftb
( Mman ) A = 2.10 kip # ftb
252
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4–26.
If the 1500-lb boom AB, the 200-lb cage BCD, and the
175-lb man have centers of gravity located at points G1 , G2
and G3 , respectively, determine the resultant moment produced by all the weights about point A.
G3
D
B
G2
C
2.5 ft 1.75 ft
20 ft
G1
SOLUTION
10 ft
75
Referring to Fig. a, the resultant moment of the weight about point A is given by
a + (MR)A = ©Fd;
A
(MR)A = - 1500(10 cos 75°) - 200(30 cos 75° + 2.5) - 175(30 cos 75° + 4.25)
= - 8037.75 lb # ft = 8.04 kip # ft (Clockwise)
Ans.
Ans:
(MR)A = 8.04 kip # ftb
253
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4–27.
z
Determine the moment of the force F about point O.
Express the result as a Cartesian vector.
F {–6i + 4 j 8k} kN
A
4m
P
3m
6m
O
1m
y
2m
SOLUTION
Position Vector. The coordinates of point A are (1, - 2, 6) m.
x
Thus,
rOA = {i - 2j + 6k} m
The moment of F About Point O.
MO = rOA * F
i
= † 1
-6
j
-2
4
k
6†
8
= { - 40i - 44j - 8k} kN # m
Ans.
Ans:
MO = { - 40i - 44j - 8k} kN # m
254
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*4–28.
z
Determine the moment of the force F about point P.
Express the result as a Cartesian vector.
F {–6i + 4 j 8k} kN
A
4m
P
3m
6m
O
1m
y
2m
SOLUTION
Position Vector. The coordinates of points A and P are A (1, - 2, 6) m and
P (0, 4, 3) m, respectively. Thus
x
rPA = (1 - 0)i + ( - 2 - 4)j + (6 - 3)k
= {i - 6j + 3k} m
The moment of F About Point P.
MP = rPA * F
i
= † 1
-6
j
-6
4
k
3†
8
= { - 60i - 26j - 32k} kN # m
Ans.
Ans:
MP = { -60i - 26j - 32k} kN # m
255
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4–29.
z
The force F = {400i - 100j - 700k} lb acts at the end of
the beam. Determine the moment of this force about
point O.
A
F
O
B
y
8 ft
1.5 ft
SOLUTION
Position Vector. The coordinates of point B are B(8, 0.25, 1.5) ft.
x
0.25 ft
Thus,
rOB = {8i + 0.25j + 1.5k} ft
Moments of F About Point O.
MO = rOB * F
i
= † 8
400
j
0.25
-100
k
1.5 †
- 700
= { -25i + 6200j - 900k} lb # ft
Ans.
Ans:
MO = { - 25i + 6200j - 900k} lb # ft
256
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4–30.
z
The force F = {400i - 100j - 700k} lb acts at the end of
the beam. Determine the moment of this force about
point A.
A
F
O
B
y
8 ft
1.5 ft
SOLUTION
Position Vector. The coordinates of points A and B are A (0, 0, 1.5) ft and
B (8, 0.25, 1.5) ft, respectively. Thus,
x
0.25 ft
rAB = (8 - 0)i + (0.25 - 0)j + (1.5 - 1.5)k
= {8i + 0.25j} ft
Moment of F About Point A.
MA = rAB * F
i
= † 8
400
j
0.25
- 100
k
0 †
- 700
= { - 175i + 5600j - 900k} lb # ft
Ans.
Ans:
MA = { -175i + 5600j - 900k} lb # ft
257
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4–31.
z
Determine the moment of the force F about point P.
Express the result as a Cartesian vector.
P
2m
2m
3m
y
O
3m
3m
SOLUTION
x
Position Vector. The coordinates of points A and P are A (3, 3, -1) m and
P ( - 2, -3, 2) m respectively. Thus,
1m
A
F {2i 4j 6k} kN
rPA = [3 - ( - 2)]i + [3 - ( - 3)] j + ( -1 - 2)k
= {5i + 6j - 3k} m
Moment of F About Point P.
MP = rAP * F
i
= †5
2
j
6
4
k
-3 †
-6
= { -24i + 24j + 8k} kN # m
Ans.
Ans:
MP = { -24i + 24j + 8k} kN # m
258
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*4–32.
z
The pipe assembly is subjected to the force of
F = {600i + 800j - 500k} N. Determine the moment of
this force about point A.
A
0.5 m
B
x
y
0.4 m
0.3 m
SOLUTION
0.3 m
Position Vector. The coordinates of point C are C (0.5, 0.7, - 0.3) m. Thus
C
rAC = {0.5i + 0.7 j - 0.3k} m
Moment of Force F About Point A.
F
MA = rAC * F
i
= † 0.5
600
j
0.7
800
k
- 0.3 †
-500
= { - 110i + 70j - 20k} N # m
Ans.
Ans:
MA = { -110i + 70j - 20k} N # m
259
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4–33.
z
The pipe assembly is subjected to the force of
F = {600i + 800j - 500k} N. Determine the moment of
this force about point B.
A
0.5 m
B
x
y
0.4 m
0.3 m
SOLUTION
Position Vector. The coordinates of points B and C are B (0.5, 0, 0) m and
C (0.5, 0.7, - 0.3) m, respectively. Thus,
0.3 m
C
rBC = (0.5 - 0.5)i + (0.7 - 0) j + ( - 0.3 - 0)k
= {0.7j - 0.3k} m
F
Moment of Force F About Point B. Applying Eq. 4
MB = rBC * F
i
= † 0
600
j
0.7
800
k
- 0.3 †
-500
= { - 110i - 180j - 420k} N # m
Ans.
Ans:
MB = { -110i - 180j - 420k} N # m
260
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4–34.
z
Determine the moment of the force of F = 600 N about
point A.
A
45
B
4m
4m
x
F
6m
C
SOLUTION
6m
Position Vectors And Force Vector. The coordinates of points A, B and C are
A (0, 0, 4) m, B (4 sin 45°, 0, 4 cos 45°) m and C (6, 6, 0) m, respectively. Thus
y
rAB = (4 sin 45° - 0)i + (0 - 0)j + (4 cos 45° - 4)k
= {2.8284i - 1.1716k} m
rAC = (6 - 0)i + (6 - 0)j + (0 - 4)k
= {6i + 6j - 4k} m
rBC = (6 - 4 sin 45°)i + (6 - 0)j + (0 - 4 cos 45°)k
= {3.1716i + 6j - 2.8284k} m
F = Fa
3.1716i + 6j - 2.8284k
rBC
b = 600£
≥
rBC
23.17162 + 62 + ( - 2.8284)2
= {258.82i + 489.63j - 230.81k} N
The Moment of Force F About Point A.
MA = rAB * F
i
= † 2.8284
258.82
j
0
489.63
k
- 1.1716 †
- 230.81
= {573.64i + 349.62j + 1384.89k} N # m
= {574i + 350j + 1385k} N # m
Ans.
OR
MA = rAC * F
= †
i
6
258.82
j
6
489.63
k
-4 †
- 230.81
= {573.64i + 349.62j + 1384.89k} N # m
= {574i + 350j + 1385k}
Ans.
Ans:
MA = {574i + 350j + 1385k} N # m
261
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4–35.
z
Determine the smallest force F that must be applied along
the rope in order to cause the curved rod, which has a radius
of 4 m, to fail at the support A. This requires a moment of
M = 1500 N # m to be developed at A.
A
45
B
4m
4m
x
F
6m
C
SOLUTION
6m
Position Vectors And Force Vector. The coordinates of points A, B and C are
A (0, 0, 4) m, B (4 sin 45°, 0, 4 cos 45°) m and C (6, 6, 0) m, respectively.
y
Thus,
rAB = (4 sin 45° - 0)i + (0 - 0)j + (4 cos 45° - 4)k
= {2.8284i - 1.1716k} m
rAC = (6 - 0)i + (6 - 0)j + (0 - 4)k
= {6i + 6j - 4k} m
rBC = (6 - 4 sin 45°)i + (6 - 0)j + (0 - 4 cos 45°)k
= {3.1716i + 6j - 2.8284k} m
F = Fa
3.1716i + 6j - 2.8284k
rBC
b = F£
≥
rBC
23.17162 + 62 + ( -2.8284)2
= 0.4314F i + 0.8161Fj - 0.3847F k
The Moment of Force F About Point A.
MA = rAB * F
i
= † 2.8284
0.4314F
j
0
0.8161F
k
- 1.1716 †
- 0.3847F
= 0.9561F i + 0.5827Fj + 2.3081F k
OR
MA = rAC * F
= †
i
6
0.4314F
j
6
0.8161F
k
-4
†
- 0.3847F
= 0.9561F i + 0.5827F j + 2.3081F k
The magnitude of MA is
MA = 2(MA)2x + (MA)2y + (MA)2z = 2(0.9561F)2 + (0.5827F)2 + (2.3081F)2
= 2.5654F
It is required that MA = 1500 N # m, then
1500 = 2.5654F
F = 584.71 N = 585 N
Ans.
Ans:
F = 585 N
262
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*4–36.
z
Determine the coordinate direction angles a, b, g of force
F, so that the moment of F about O is zero.
O
y
0.4 m
A
0.5 m
0.3 m
x
SOLUTION
F
Position And Force Vectors. The coordinates of point A are A (0.4, 0.5, - 0.3) m.
Thus,
rOA = {0.4i + 0.5j - 0.3k} m
u OA =
0.4i + 0.5j - 0.3k
rOA
4
5
3
=
=
i +
j k
2
2
2
rOA
150
150
150
20.4 + 0.5 + ( - 0.3)
rAO = { - 0.4i - 0.5j + 0.3k} m
u AO =
- 0.4i - 0.5j + 0.3k
rAO
4
5
3
=
= i j +
k
2
2
2
rAO
150
150
150
2( - 0.4) + ( - 0.5) + 0.3
Moment of F About Point O. To produce zero moment about point O, the line of
action of F must pass through point O. Thus, F must directed from O to A (direction
deined by uOA). Thus,
cos a = -
4
;
150
a = 55.56° = 55.6°
Ans.
cos b = -
5
;
150
b = 45°
Ans.
cos g =
-3
;
150
g = 115.10° = 115°
Ans.
OR F must directed from A to O (direction deined by uAO). Thus
cos a = -
4
;
150
5
;
150
3
;
cos g =
150
cos b = -
a = 124.44° = 124°
Ans.
b = 135°
Ans.
g = 64.90° = 64.9°
Ans.
Ans:
a = 55.6°
b = 45°
g = 115°
OR
a = 124°
b = 135°
g = 64.9°
263
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4–37.
z
Determine the moment of force F about point O. The force
has a magnitude of 800 N and coordinate direction angles of
a = 60°, b = 120°, g = 45°. Express the result as a
Cartesian vector.
O
y
0.4 m
A
0.5 m
0.3 m
x
SOLUTION
F
Position And Force Vectors. The coordinates of point A are A (0.4, 0.5, -0.3) m.
Thus
rOA = {0.4i + 0.5j - 0.3k} m
F = FuF = 800 (cos 60°i + cos 120°j + cos 45°k)
= {400i - 400j + 565.69k} N
Moment of F About Point O.
MO = rOA * F
i
= † 0.4
400
j
0.5
-400
k
- 0.3 †
565.69
= {162.84 i - 346.27j - 360 k} N # m
= {163i - 346j - 360k} N # m
Ans.
Ans:
MO = {163i - 346j - 360k} N # m
264
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4–38.
z
Determine the moment of the force F about the door hinge
at A. Express the result as a Cartesian vector.
4 ft
7 ft
A
F 80 lb
C
1.5 ft
B
45
D
SOLUTION
Position Vectors And Force Vector. The coordinates of points A, C and D are
A ( - 6.5, -3, 0) ft, C [0, - (3 + 4 cos 45°), 4 sin 45°] ft and D ( -5, 0, 0) ft, respectively.
Thus,
rAC = [0 - ( - 6.5)]i + [ -(3 + 4 cos 45°) - ( -3)] j + (4 sin 45° - 0)k
1.5 ft
5 ft
3 ft
x
y
= {6.5i - 2.8284j + 2.8284k} ft
rAD = [ - 5 - ( - 6.5)]i + [0 - ( -3)]j + (0 - 0)k = {1.5i + 3j} ft
rCD = ( - 5 - 0)i + {0 - [ - (3 + 4 cos 45°)]} j + (0 - 4 sin 45°)k
= { - 5i + 5.8284j - 2.8284k} ft
F = Fa
-5i + 5.8284j - 2.8284k
rCD
b = 80 £
≥
rCD
2( - 5)2 + 5.82842 + ( - 2.8284)2
= { - 48.88 i + 56.98j - 27.65k} lb
Moment of F About Point A.
MA = rAC * F
= †
i
6.5
- 48.88
j
- 2.8284
56.98
k
2.8284 †
- 27.65
= { - 82.9496i + 41.47j + 232.10k} lb # ft
= { - 82.9i + 41.5j + 232k} lb # ft
Ans.
OR
MA = rAD * F
= †
i
1.5
- 48.88
j
3
56.98
k
0 †
- 27.65
= { - 82.9496i + 41.47j + 232.10 k} lb # ft
= { - 82.9i + 41.5j + 232k} lb # ft
Ans.
Ans:
MA = { -82.9i + 41.5j + 232k} lb # ft
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4–39.
z
Determine the moment of the force F about the door hinge
at B. Express the result as a Cartesian vector.
4 ft
7 ft
A
F 80 lb
C
1.5 ft
B
45
D
SOLUTION
Position Vectors And Force Vector. The coordinates of points B, C and D are
B ( -1.5, - 3, 0) ft, C [0, - (3 + 4 cos 45°), 4 sin 45°] ft and D ( - 5, 0, 0) ft, respectively.
Thus,
rBC = [0 - ( -1.5)]i + [ - (3 + 4 cos 45°) - ( - 3)] j + (4 sin 45° - 0)k
1.5 ft
5 ft
3 ft
x
y
= {1.5i - 2.8284j + 2.8284k} ft
rBD = [ -5 - ( - 1.5)]i + [0 - ( - 3)]j + (0 - 0)k = { -3.5i + 3j} ft
rCD = ( -5- 0)i + {0 - [ - (3 + 4 cos 45°)]} j + (0 - 4 sin 45°)k
= { -5i + 5.8284j - 2.8284k} ft
F = Fa
-5i + 5.8284j - 2.8284k
rCD
b = 80 £
≥
rCD
2( - 5)2 + 5.82842 + ( - 2.8284)2
= { -48.88 i + 56.98j - 27.65k} lb
Moment of F About Point B.
MB = rBC * F
= †
i
1.5
- 48.88
j
-2.8284
56.98
k
2.8284 †
- 27.65
= { -82.9496i - 96.77j - 52.78k} lb # ft
= { -82.9i - 96.8j - 52.8k} lb # ft
Ans.
or
MB = rBD * F
i
= † -3.5
- 48.88
j
3
56.98
k
0 †
- 27.65
= { -82.9496i - 96.77j - 52.78 k} lb # ft
= { -82.9i - 96.8j - 52.8k} lb # ft
Ans.
Ans:
MB = { -82.9i - 96.8j - 52.8k} lb # ft
266
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*4–40.
The curved rod has a radius of 5 ft. If a force of 60 lb acts at
its end as shown, determine the moment of this force about
point C.
z
C
5 ft
60°
A
5 ft
y
SOLUTION
60 lb
6 ft
Position Vector and Force Vector:
B
rCA = 515 sin 60° - 02j + 15 cos 60° - 52k6 m
7 ft
x
= 54.330j - 2.50k6 m
FAB = 60 ¢
16 - 02i + 17 - 5 sin 60°2j + 10 - 5 cos 60°2k
216 - 022 + 17 - 5 sin 60°22 + 10 - 5 cos 60°22
= 551.231i + 22.797j - 21.346k6 lb
≤ lb
Moment of Force FAB About Point C: Applying Eq. 4–7, we have
M C = rCA * FAB
i
0
=
51.231
=
j
4.330
22.797
k
- 2.50
- 21.346
- 35.4i - 128j - 222k lb # ft
Ans.
Ans:
MC = { -35.4i - 128j - 222k} lb # ft
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4–41.
z
Determine the smallest force F that must be applied along
the rope in order to cause the curved rod, which has a radius
of 5 ft, to fail at the support C. This requires a moment of
M = 80 lb # ft to be developed at C.
C
5 ft
60
A
5 ft
y
60 lb
SOLUTION
6 ft
B
Position Vector and Force Vector:
x
7 ft
rCA = {(5 sin 60° - 0)j + (5 cos 60° - 5)k} m
= {4.330j - 2.50 k} m
FAB = F a
(6 - 0)i + (7 - 5 sin 60°)j + (0 - 5 cos 60°)k
2(6 - 0)2 + (7 - 5 sin 60°)2 + (0 - 5 cos 60°)2
b lb
= 0.8539Fi + 0.3799Fj - 0.3558Fk
Moment of Force FAB About Point C:
M C = rCA * FAB
= 3
i
0
0.8539F
j
4.330
0.3799F
k
-2.50 3
- 0.3558F
= - 0.5909Fi - 2.135j - 3.697k
Require
80 = 2(0.5909)2 + ( - 2.135)2 + ( - 3.697)2 F
F = 18.6 lb.
Ans.
Ans:
F = 18.6 lb
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4–42.
A 20-N horizontal force is applied perpendicular to the
handle of the socket wrench. Determine the magnitude and
the coordinate direction angles of the moment created by
this force about point O.
z
200 mm
75 mm
A
20 N
SOLUTION
O
rA = 0.2 sin 15°i + 0.2 cos 15°j + 0.075k
15
= 0.05176 i + 0.1932 j + 0.075 k
x
F = -20 cos 15°i + 20 sin 15°j
= -19.32 i + 5.176 j
i
MO = rA * F = 3 0.05176
- 19.32
j
0.1932
5.176
k
0.075 3
0
= { -0.3882 i - 1.449 j + 4.00 k} N # m
MO = 4.272 = 4.27 N # m
Ans.
a = cos -1 a
- 0.3882
b = 95.2°
4.272
Ans.
b = cos -1 a
- 1.449
b = 110°
4.272
Ans.
g = cos -1 a
4
b = 20.6°
4.272
Ans.
Ans:
MO = 4.27 N # m
a = 95.2°
b = 110°
g = 20.6°
269
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4–43.
z
The pipe assembly is subjected to the 80-N force. Determine
the moment of this force about point A.
A
400 mm
B
x
SOLUTION
300 mm
Position Vector And Force Vector:
200 mm
rAC = {(0.55 - 0)i + (0.4 - 0)j + ( -0.2 - 0)k} m
200 mm
C
250 mm
= {0.55i + 0.4j - 0.2k} m
40
F = 80(cos 30° sin 40°i + cos 30° cos 40°j - sin 30°k) N
30
= (44.53i + 53.07j - 40.0k} N
F
80 N
Moment of Force F About Point A: Applying Eq. 4–7, we have
MA = rAC * F
i
= 3 0.55
44.53
j
0.4
53.07
k
- 0.2 3
- 40.0
= { -5.39i + 13.1j + 11.4k} N # m
Ans.
Ans:
MA = {- 5.39i + 13.1j + 11.4k} N # m
270
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*4–44.
z
The pipe assembly is subjected to the 80-N force. Determine
the moment of this force about point B.
A
400 mm
B
x
SOLUTION
300 mm
Position Vector And Force Vector:
200 mm
rBC = {(0.55 - 0) i + (0.4 - 0.4)j + ( - 0.2 - 0)k} m
200 mm
C
250 mm
= {0.55i - 0.2k} m
40
F = 80 (cos 30° sin 40°i + cos 30° cos 40°j - sin 30°k) N
30
= (44.53i + 53.07j - 40.0k} N
F
80 N
Moment of Force F About Point B: Applying Eq. 4–7, we have
MB = rBC * F
i
= 3 0.55
44.53
j
0
53.07
k
- 0.2 3
- 40.0
= {10.6i + 13.1j + 29.2k} N # m
Ans.
Ans:
MB = {10.6i + 13.1j + 29.2k} N # m
271
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4–45.
A force of F = 5 6i - 2j + 1k 6 kN produces a moment
of M O = 54i + 5j - 14k6 kN # m about the origin of
coordinates, point O. If the force acts at a point having an x
coordinate of x = 1 m, determine the y and z coordinates.
.
Note:The figure shows F and MO in an arbitrary position.
z
F
P
MO
z
d
y
O
1m
SOLUTION
y
MO = r * F
i
4i + 5j - 14k = 3 1
6
j
y
-2
x
k
z3
1
4 = y + 2z
5 = -1 + 6z
-14 = -2 - 6y
y = 2m
Ans.
z = 1m
Ans.
Ans:
y = 2m
z = 1m
272
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4–46.
The force F = 56i + 8j + 10k6 N creates a moment about
point O of M O = 5 - 14i + 8j + 2k6 N # m. If the force
passes through a point having an x coordinate of 1 m,
determine the y and z coordinates of the point.Also, realizing
that MO = Fd, determine the perpendicular distance d from
point O to the line of action of F. Note: The figure shows F
and MO in an arbitrary position.
z
F
P
MO
z
d
y
O
1m
SOLUTION
i
-14i + 8j + 2k = 3 1
6
j
y
8
y
k
z 3
10
x
-14 = 10y - 8z
8 = -10 + 6z
2 = 8 - 6y
y = 1m
Ans.
z = 3m
Ans.
MO = 2( -14)2 + (8)2 + (2)2 = 16.25 N # m
F = 2(6)2 + (8)2 + (10)2 = 14.14 N
d =
16.25
= 1.15 m
14.14
Ans.
Ans:
y = 1m
z = 3m
d = 1.15 m
273
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4–47.
A force F having a magnitude of F = 100 N acts along the
diagonal of the parallelepiped. Determine the moment of F
about point A, using M A = rB : F and M A = rC : F.
z
F
C
200 mm
rC
SOLUTION
F = 100 a
400 mm
B
- 0.4 i + 0.6 j + 0.2 k
b
0.7483
600 mm
F
F = 5 -53.5 i + 80.2 j + 26.7 k6 N
M A = rB * F = 3
rB
A
x
i
0
- 53.5
j
-0.6
80.2
k
0 3 = 5 -16.0 i - 32.1 k6 N # m
26.7
Ans.
i
- 0.4
- 53.5
j
0
80.2
k
0.2 =
26.7
Ans.
Also,
M A = rC * F =
-16.0 i - 32.1 k N # m
Ans:
MA = { -16.0i - 32.1k} N # m
274
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*4–48.
Force F acts perpendicular to the inclined plane. Determine
the moment produced by F about point A. Express the
result as a Cartesian vector.
z
3m
A
F
400 N
3m
B
SOLUTION
x
C
4m
F
orce Vector: Since force F is perpendicular to the inclined plane, its unit vector uF
is equal to the unit vector of the cross product, b = rAC * rBC, Fig. a. Here
rAC = (0 - 0)i + (4 - 0)j + (0 - 3)k = [4j - 3k] m
rBC = (0 - 3)i + (4 - 0)j + (0 - 0)k = [ - 3i + 4j] m
Thus,
i
b = rCA * rCB = 3 0
-3
j
4
4
k
-3 3
0
= [12i + 9j + 12k] m2
Then,
uF =
12i + 9j + 12k
b
=
= 0.6247i + 0.4685j + 0.6247k
b
212 2 + 92 + 12 2
And finally
F = FuF = 400(0.6247i + 0.4685j + 0.6247k)
= [249.88i + 187.41j + 249.88k] N
Vector Cross Product: The moment of F about point A is
M A = rAC * F = 3
i
0
249.88
j
4
187.41
k
-3 3
249.88
= [1.56i - 0.750j - 1.00k] kN # m
Ans.
Ans:
MA = [1.56i - 0.750j - 1.00k] kN # m
275
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4–49.
Force F acts perpendicular to the inclined plane. Determine
the moment produced by F about point B. Express the
result as a Cartesian vector.
z
3m
A
F
400 N
3m
B
SOLUTION
x
C
4m
Force Vector: Since force F is perpendicular to the inclined plane, its unit vector uF
is equal to the unit vector of the cross product, b = rAC * rBC, Fig. a. Here
rAC = (0 - 0)i + (4 - 0)j + (0 - 3)k = [4j - 3k] m
rBC = (0 - 3)i + (4 - 0)j + (0 - 0)k = [ - 3k + 4j] m
Thus,
i
b = rCA * rCB = 3 0
-3
j
4
4
k
- 3 3 = [12i + 9j + 12k] m2
0
Then,
uF =
12i + 9j + 12k
b
=
= 0.6247i + 0.4685j + 0.6247k
b
2122 + 92 + 122
And finally
F = FuF = 400(0.6247i + 0.4685j + 0.6247k)
= [249.88i + 187.41j + 249.88k] N
Vector Cross Product: The moment of F about point B is
MB = rBC * F = 3
i
-3
249.88
j
4
187.41
k
0 3
249.88
= [1.00i + 0.750j - 1.56k] kN # m
Ans.
Ans:
MB = {1.00i + 0.750j - 1.56k} kN # m
276
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4–50.
Strut AB of the 1-m-diameter hatch door exerts a force of
450 N on point B. Determine the moment of this force
about point O.
z
B
30°
0.5 m
F = 450 N
O
SOLUTION
y
0.5 m
Position Vector And Force Vector:
30°
rOB = 510 - 02i + 11 cos 30° - 02j + 11 sin 30° - 02k6 m
A
x
= 50.8660j + 0.5k6 m
rOA = 510.5 sin 30° - 02i + 10.5 + 0.5 cos 30° - 02j + 10 - 02k6 m
= 50.250i + 0.9330j6 m
F = 450 ¢
10 - 0.5 sin 30°2i + 31 cos 30° - 10.5 + 0.5 cos 30°24j + 11 sin 30° - 02k
210 - 0.5 sin 30°22 + 31 cos 30° - 10.5 + 0.5 cos 30°242 + 11 sin 30° - 022
= 5 - 199.82i - 53.54j + 399.63k6 N
≤N
Moment of Force F About Point O: Applying Eq. 4–7, we have
M O = rOB * F
= 3
i
0
- 199.82
j
0.8660
- 53.54
k
0.5 3
399.63
= 5373i - 99.9j + 173k6 N # m
Ans.
Or
M O = rOA * F
=
i
0.250
- 199.82
j
0.9330
- 53.54
k
0
399.63
=
373i - 99.9j + 173k N # m
Ans.
Ans:
MO = {373i - 99.9j + 173k} N # m
277
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4–51.
Using a ring collar the 75-N force can act in the vertical
plane at various angles u. Determine the magnitude of the
moment it produces about point A, plot the result of M
(ordinate) versus u (abscissa) for 0° … u … 180°, and
specify the angles that give the maximum and minimum
moment.
z
A
2m
1.5 m
SOLUTION
i
MA = 3 2
0
j
1.5
75 cos u
k
3
0
75 sin u
y
x
75 N
θ
= 112.5 sin u i - 150 sin u j + 150 cos u k
MA = 21112.5 sin u22 + 1 -150 sin u22 + 1150 cos u22 = 212 656.25 sin2 u + 22 500
dMA
1
1
= 112 656.25 sin2 u + 22 5002- 2 112 656.25212 sin u cos u2 = 0
du
2
sin u cos u = 0;
u = 0°, 90°, 180°
Ans.
umax = 187.5 N # m at u = 90°
umin = 150 N # m at u = 0°, 180°
Ans:
umax = 90°
umin = 0, 180°
278
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*4–52.
z
The lug nut on the wheel of the automobile is to be removed
using the wrench and applying the vertical force of F = 30 N
at A. Determine if this force is adequate, provided 14 N # m
of torque about the x axis is initially required to turn the nut.
If the 30-N force can be applied at A in any other direction,
will it be possible to turn the nut?
30 N
F
B
0.25 m
A
0.3 m
SOLUTION
Mx = 30 A 2(0.5) - (0.3) B = 12 N # m 6 14 N # m,
2
2
y
0.1 m
No
Ans.
x
For (Mx)max , apply force perpendicular to the handle and the x - axis.
(Mx)max = 30 (0.5) = 15 N # m 7 14 N # m,
0.5 m
Yes
Ans.
Ans:
No
Yes
279
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4–53.
Solve Prob. 4–52 if the cheater pipe AB is slipped over the
handle of the wrench and the 30-N force can be applied at
any point and in any direction on the assembly.
z
30 N
F
B
0.25 m
A
0.3 m
SOLUTION
0.5 m
y
4
Mx = 30 (0.75)a b = 18 N # m 7 14 N # m,
5
0.1 m
Yes
Ans.
x
(Mx)max occurs when force is applied perpendicular to both the handle and the x - axis.
(Mx)max = 30(0.75) = 22.5 N # m 7 14N # m,
Yes
Ans.
Ans:
Yes
Yes
280
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4–54.
z
The A-frame is being hoisted into an upright position by
the vertical force of F = 80 lb. Determine the moment of
this force about the y′ axis passing through points A and B
when the frame is in the position shown.
F
C
A
x¿
6 ft
x
SOLUTION
15
6 ft
30
y
B
y¿
Scalar analysis :
My′ = 80 (6 cos 15°) = 464 lb # ft
Vector analysis :
uAB = cos 60° i + cos 30° j
Coordinates of point C :
x = 3 sin 30° - 6 cos 15° cos 30° = - 3.52 ft
y = 3 cos 30° + 6 cos 15° sin 30° = 5.50 ft
z = 6 sin 15° = 1.55 ft
rAC = - 3.52 i + 5.50 j + 1.55 k
F = 80 k
sin 30°
My′ = † - 3.52
0
cos 30°
5.50
0
0
1.55 †
80
My′ = 464 lb # ft
Ans.
Ans:
My′ = 464 lb # ft
281
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4–55.
z
The A-frame is being hoisted into an upright position by the
vertical force of F = 80 lb. Determine the moment of this
force about the x axis when the frame is in the position
shown.
F
C
A
x¿
6 ft
x
SOLUTION
15
6 ft
30
y
B
y¿
Using x′, y′, z :
ux = cos 30° i′ + sin 30° j′
rAC = - 6 cos 15° i′ + 3 j′ + 6 sin 15° k
F = 80 k
cos 30°
Mx = † -6 cos 15°
0
Mx = 440 lb # ft
sin 30°
3
0
0
6 sin 15° † = 207.85 + 231.82 + 0
80
Also, using x, y, z,
Coordinates of point C :
x = 3 sin 30° - 6 cos 15° cos 30° = - 3.52 ft
y = 3 cos 30° + 6 cos 15° sin 30° = 5.50 ft
z = 6 sin 15° = 1.55 ft
rAC = - 3.52 i + 5.50 j + 1.55 k
F = 80 k
1
Mx = † - 3.52
0
0
5.50
0
0
1.55 † = 440 lb # ft
80
Ans.
Ans:
Mx = 440 lb # ft
282
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*4–56.
Determine the magnitude of the moments of the force F
about the x, y, and z axes. Solve the problem (a) using a
Cartesian vector approach and (b) using a scalar approach.
z
A
4 ft
y
SOLUTION
x
a) Vector Analysis
3 ft
C
PositionVector:
rAB = {(4 - 0) i + (3 - 0)j + ( - 2 - 0)k} ft = {4 i + 3j - 2k} ft
2 ft
Moment of Force F About x,y, and z Axes: T he unit vectors along x, y, and z axes are
i, j, and k respectively. Applying Eq. 4–11, we have
Mx = i # (rAB * F)
1
= 34
4
0
3
12
B
F
{4i
12j
3k} lb
0
-2 3
-3
= 1[3( - 3) - (12)( - 2)] - 0 + 0 = 15.0 lb # ft
Ans.
My = j # (rAB * F)
0
= 34
4
1
3
12
0
-2 3
-3
= 0 - 1[4( - 3) - (4)( - 2)] + 0 = 4.00 lb # ft
Ans.
Mz = k # (rAB * F)
0
= 34
4
0
3
12
1
-2 3
-3
= 0 - 0 + 1[4(12) - (4)(3)] = 36.0 lb # ft
Ans.
b) ScalarAnalysis
Mx = ©Mx ;
Mx = 12(2) - 3(3) = 15.0 lb # ft
Ans.
My = ©My ;
My = - 4(2) + 3(4) = 4.00 lb # ft
Ans.
Mz = ©Mz ;
Mz = - 4(3) + 12(4) = 36.0 lb # ft
Ans.
Ans:
Mx = 15.0 lb # ft
My = 4.00 lb # ft
Mz = 36.0 lb # ft
283
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4–57.
z
Determine the moment of the force F about an axis
extending between A and C. Express the result as a
Cartesian vector.
A
4 ft
y
SOLUTION
x
PositionVector:
3 ft
C
rCB = {- 2k} ft
rAB = {(4 - 0)i + (3 - 0)j + ( -2 - 0)k} ft = {4i + 3j - 2k} ft
2 ft
Unit Vector Along AC Axis:
uAC =
B
(4 - 0)i + (3 - 0)j
2(4 - 0)2 + (3 - 0)2
F
{4i
12j
3k} lb
= 0.8i + 0.6j
Moment of Force F About AC Axis: With F = {4i + 12j - 3k} lb, applying Eq. 4–7,
we have
MAC = uAC # (rCB * F)
0.8
= 3 0
4
0.6
0
12
0
-2 3
-3
= 0.8[(0)( -3) - 12( -2)] - 0.6[0(-3) - 4( -2)] + 0
= 14.4 lb # ft
Or
MAC = uAC # (rAB * F)
0.8
= 3 4
4
0.6
3
12
0
-2 3
-3
= 0.8[(3)( -3) - 12(- 2)] - 0.6[4(- 3) - 4( -2)] + 0
= 14.4 lb # ft
Expressing MAC as a Cartesian vector yields
M AC = MAC uAC
= 14.4(0.8i + 0.6j)
= {11.5i + 8.64j} lb # ft
Ans.
Ans:
M AC = {11.5i + 8.64j} lb # ft
284
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4–58.
z
The board is used to hold the end of a four-way lug wrench in
the position shown when the man applies a force of F =
100 N. Determine the magnitude of the moment produced by
this force about the x axis. Force F lies in a vertical plane.
F
60
250 mm
y
x
SOLUTION
250 mm
Vector Analysis
Moment About the x Axis: The position vector rAB, Fig. a, will be used to determine
the moment of F about the x axis.
rAB = (0.25 - 0.25)i + (0.25 - 0)j + (0 - 0)k = {0.25j} m
The force vector F, Fig. a, can be written as
F = 100(cos 60°j - sin 60°k) = {50j - 86.60k} N
Knowing that the unit vector of the x axis is i, the magnitude of the moment of F
about the x axis is given by
1
Mx = i # rAB * F = 3 0
0
0
0.25
50
0
0 3
- 86.60
= 1[0.25( - 86.60) - 50(0)] + 0 + 0
= - 21.7 N # m
Ans.
The negative sign indicates that Mx is directed towards the negative x axis.
Scalar Analysis
This problem can be solved by summing the moment about the x axis
Mx = ©Mx;
Mx = - 100 sin 60°(0.25) + 100 cos 60°(0)
= - 21.7 N # m Ans.
Ans:
Mx = 21.7 Ν # m
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4–59.
The board is used to hold the end of a four-way lug wrench
in position. If a torque of 30 N # m about the x axis is
required to tighten the nut, determine the required magnitude of the force F that the man’s foot must apply on the
end of the wrench in order to turn it. Force F lies in a vertical plane.
z
F
60
250 mm
y
x
250 mm
SOLUTION
Vector Analysis
Moment About the x Axis: The position vector rAB, Fig. a, will be used to determine the moment of F about the x axis.
rAB = (0.25 - 0.25)i + (0.25 - 0)j + (0 - 0)k = {0.25j} m
The force vector F, Fig. a, can be written as
F = F(cos 60°j - sin 60°k) = 0.5Fj - 0.8660Fk
Knowing that the unit vector of the x axis is i, the magnitude of the moment of F
about the x axis is given by
1
Mx = i # rAB * F = 0
0
0
0.25
0.5F
0
0
- 0.8660F
= 1[0.25( - 0.8660F) - 0.5F(0)] + 0 + 0
= - 0.2165F
Ans.
The negative sign indicates that Mx is directed towards the negative x axis. The
magnitude of F required to produce Mx = 30 N # m can be determined from
30 = 0.2165F
F = 139 N
Ans.
Scalar Analysis
This problem can be solved by summing the moment about the x axis
Mx = ©Mx;
- 30 = -F sin 60°(0.25) + F cos 60°(0)
F = 139 N
Ans.
Ans:
F = 139 Ν
286
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*4–60.
z
The A-frame is being hoisted into an upright position by the
vertical force of F = 80 lb. Determine the moment of this
force about the y axis when the frame is in the position shown.
F
C
SOLUTION
A
Using x¿ , y¿ , z:
x¿
6 ft
x
F = 80 k
- sin 30°
My = 3 - 6 cos 15°
0
cos 30°
3
0
6 ft
30
uy = - sin 30° i¿ + cos 30° j¿
rAC = - 6 cos 15°i¿ + 3 j¿ + 6 sin 15° k
15
y
B
y¿
0
6 sin 15° 3 = - 120 + 401.53 + 0
80
My = 282 lb # ft
Ans.
Also, using x,y,z:
Coordinates of point C:
x = 3 sin 30° - 6 cos 15° cos 30° = -3.52 ft
y = 3 cos 30° + 6 cos 15° sin 30° = 5.50 ft
z = 6 sin 15° = 1.55 ft
rAC = - 3.52 i + 5.50 j + 1.55 k
F = 80 k
0
My = 3 - 3.52
0
1
5.50
0
0
1.55 3 = 282 lb # ft
80
Ans.
Ans:
My = 282 lb # ft
287
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4–61.
Determine the magnitude of the moment of the force
F = {50i - 20j - 80k} N about the base line AB of the tripod.
z
F
D
4m
2m
1.5 m
SOLUTION
uAB =
C
A
{3.5i + 0.5j}
y
2(3.5)2 + (0.5)2
2.5 m
x
2m
uAB = {0.9899i + 0.1414j}
0.9899
MAB = uAB # ( rAD * F ) = † 2.5
50
1m
0.1414
0
- 20
0
4 †
- 80
0.5 m
MAB = 136 N # m
B
Ans.
Ans:
MAB = 136 N # m
288
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4–62.
Determine the magnitude of the moment of the force
F = {50i - 20j - 80k} N about the base line BC of the tripod.
z
F
D
4m
2m
1.5 m
SOLUTION
uBC =
C
A
{ - 1.5i - 2.5j}
y
2( - 1.5)2 + ( - 2.5)2
2.5 m
x
2m
uBC = { - 0.5145i - 0.8575j}
- 0.5145
MBC = uBC # ( rCD * F ) = † 0.5
50
1m
- 0.8575
2
- 20
0.5 m
0
4 †
- 80
MBC = 165 N # m
B
Ans.
Ans:
MBC = 165 N # m
289
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4–63.
Determine the magnitude of the moment of the force
F = {50i - 20j - 80k} N about the base line CA of the tripod.
z
F
D
4m
2m
1.5 m
SOLUTION
uCA =
C
A
{ -2i + 2j}
y
2( - 2)2 + (2)2
2.5 m
x
2m
uCA = { -0.707i + 0.707j}
- 0.707
MCA = uCA # ( rAD * F ) = † 2.5
50
1m
0.707
0
- 20
0
4 †
- 80
0.5 m
MCA = 226 N # m
B
Ans.
Ans:
MCA = 226 N # m
290
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*4–64.
A horizontal force of F = { - 50i} N is applied perpendicular
to the handle of the pipe wrench. Determine the moment
that this force exerts along the axis OA (z axis) of the pipe
assembly. Both the wrench and pipe assembly, OABC, lie in
the y-z plane. Suggestion: Use a scalar analysis.
z
B
0.8 m
0.2 m
C
A
135°
F
0.6 m
SOLUTION
O
Mz = 50(0.8 + 0.2) cos 45° = 35.36 N # m
x
Mz = {35.4 k} N # m
Ans.
y
Ans:
Mz = {35.4 k} N # m
291
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4–65.
Determine the magnitude of the horizontal force F = -F i
acting on the handle of the wrench so that this force
produces a component of the moment along the OA axis
(z axis) of the pipe assembly of Mz = {4k} N # m. Both the
wrench and the pipe assembly, OABC, lie in the y-z plane.
Suggestion: Use a scalar analysis.
z
B
0.8 m
0.2 m
C
A
135°
F
0.6 m
SOLUTION
O
Mz = F (0.8 + 0.2) cos 45° = 4
x
F = 5.66 N
Ans.
y
Ans:
F = 5.66 N
292
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4–66.
The force of F = 30 N acts on the bracket as shown.
Determine the moment of the force about the a-a axis of
the pipe if a = 60°, b = 60°, and g = 45°. Also, determine
the coordinate direction angles of F in order to produce the
maximum moment about the a-a axis.What is this moment?
z
F = 30 N
y
45••
60••
60••
50 mm
x
100 mm
100 mm
SOLUTION
F = 30 1cos 60° i + cos 60° j + cos 45° k2
a
= 515 i + 15 j + 21.21 k6 N
r = 5- 0.1 i + 0.15 k6 m
a
u = j
0
Ma = 3 -0.1
15
1
0
15
0
0.15 3 = 4.37 N # m
21.21
Ans.
F must be perpendicular to u and r.
uF =
0.1
0.15
i +
k
0.1803
0.1803
= 0.8321i + 0.5547k
a = cos-1 0.8321 = 33.7°
Ans.
b = cos-1 0 = 90°
Ans.
g = cos-1 0.5547 = 56.3°
Ans.
M = 30 0.1803 = 5.41 N # m
Ans.
Ans:
Ma = 4.37 N # m
a = 33.7°
b = 90°
g = 56.3°
M = 5.41 N # m
293
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4–67.
A clockwise couple M = 5 N # m is resisted by the shaft of
the electric motor. Determine the magnitude of the reactive
forces -R and R which act at supports A and B so that the
resultant of the two couples is zero.
M
150 mm
60
A
R
SOLUTION
60
B
R
a+MC = - 5 + R (2(0.15)>tan 60°) = 0
R = 28.9 N
Ans.
Ans:
R = 28.9 N
294
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*4–68.
A twist of 4 N # m is applied to the handle of the screwdriver.
Resolve this couple moment into a pair of couple forces F
exerted on the handle and P exerted on the blade.
–F
–P
P
5 mm
4 N·m
30 mm
F
SOLUTION
For the handle
MC = ©Mx ;
F10.032 = 4
F = 133 N
Ans.
For the blade,
MC = ©Mx ;
P10.0052 = 4
P = 800 N
Ans.
Ans:
F = 133 N
P = 800 N
295
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4–69.
If the resultant couple of the three couples acting on the
triangular block is to be zero, determine the magnitude of
forces F and P.
z
F
C
F
150 N
B
SOLUTION
300 mm
BA = 0.5 m
The couple created by the 150 - N forces is
x
400 mm
D
500 mm
150 N
A
P
600 mm
P
MC1 = 150 (0.5) = 75 N # m
Then
3
4
MC1 = 75 a b j + 75 a b k
5
5
= 45 j + 60 k
MC2 = - P (0.6) k
MC3 = - F (0.6) j
Require
MC1 + MC2 + MC3 = 0
45 j + 60 k - P (0.6) k - F (0.6) j = 0
Equate the j and k components
45 - F (0.6) = 0
F = 75 N
Ans.
60 - P (0.6) = 0
P = 100 N
Ans.
Ans:
F = 75 N
P = 100 N
296
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4–70.
200 lb
Two couples act on the beam. If F = 125 lb , determine the resultant couple moment.
F
30
1.25 ft
1.5 ft
SOLUTION
F
30
200 lb
125 lb couple is resolved in to their horizontal and vertical components as shown in
Fig. a.
2 ft
a + (MR)C = 200(1.5) + 125 cos 30° (1.25)
= 435.32 lb # ft = 435 lb # ftd
Ans.
Ans:
(MR)C = 435 lb # ft d
297
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4–71.
Two couples act on the beam. Determine the magnitude of
F so that the resultant couple moment is 450 lb # ft,
counterclockwise. Where on the beam does the resultant
couple moment act?
200 lb
F
30
1.25 ft
1.5 ft
F
30
200 lb
2 ft
SOLUTION
a +MR = ©M ;
450 = 200(1.5) + Fcos 30°(1.25)
F = 139 lb
Ans.
The resultant couple moment is a free vector. It can act at any point on the beam.
Ans:
F = 139 lb
298
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*4–72.
Determine the magnitude of the couple force F so that the
resultant couple moment on the crank is zero.
150 lb
–F
5 in.
30
30
30
5 in.
150 lb
4 in.
45
45
30
4 in.
SOLUTION
F
By resolving F and the 150-lb couple into components parallel and perpendicular to
the lever arm of the crank, Fig. a, and summing the moment of these two force
components about point A, we have
a + (MC)R = ©MA;
0 = 150 cos 15°(10) - F cos 15°(5) - F sin 15°(4) - 150 sin 15°(8)
F = 194 lb
Ans.
Note: Since the line of action of the force component parallel to the lever arm of the
crank passes through point A, no moment is produced about this point.
Ans:
F = 194 lb
299
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4–73.
The ends of the triangular plate are subjected to three
couples. Determine the magnitude of the force F so that the
resultant couple moment is 400 N # m clockwise.
F
600 N
F
600 N
40
40
SOLUTION
a + MR = ©M;
- 400 = 600 a
1m
0.5
0.5
b -F a
b -250(1)
cos 40°
cos 40°
F = 830 N
250 N
250 N
Ans.
Ans:
F = 830 N
300
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4–74.
The man tries to open the valve by applying the couple forces
of F = 75 N to the wheel. Determine the couple moment
produced.
150 mm
150 mm
F
SOLUTION
a + Mc = ©M;
Mc = - 75(0.15 + 0.15)
= -22.5 N # m = 22.5 N # m b
Ans.
F
Ans:
MC = 22.5 N # mb
301
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4–75.
If the valve can be opened with a couple moment of 25 N # m, determine
the required magnitude of each couple force which must be applied
to the wheel.
150 mm
150 mm
F
SOLUTION
a + Mc = ©M;
- 25 = -F(0.15 + 0.15)
F = 83.3 N
Ans.
F
Ans:
F = 83.3 N
302
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*4–76.
F
Determine the magnitude of F so that the resultant couple
moment is 12 kN # m, counterclockwise. Where on the beam
does the resultant couple moment act?
F
30 30
8 kN
0.3 m
1.2 m
0.4 m
SOLUTION
a + MR = ΣMC;
12 = (F cos 30°)(0.3) + 8(1.2)
8 kN
F = 9.238 kN = 9.24 kN
Ans.
Since the couple moment is a free vector, the resultant couple moment can act at
any point on or off the beam.
Ans:
F = 9.24 kN
303
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4–77.
Two couples act on the beam as shown. If F = 150 lb, determine
the resultant couple moment.
–F
5
3
4
200 lb
1.5 ft
200 lb
5
SOLUTION
F
150 lb couple is resolved into their horizontal and vertical components as shown in
Fig. a
4 ft
3
4
4
3
a + (MR)c = 150 a b (1.5) + 150 a b (4) - 200(1.5)
5
5
= 240 lb # ftd
Ans.
Ans:
(MR)C = 240 lb # ft d
304
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4–78.
Two couples act on the beam as shown. Determine the
magnitude of F so that the resultant couple moment is
300 lb # ft counterclockwise. Where on the beam does the
resultant couple act?
–F
5
3
4
200 lb
1.5 ft
200 lb
5
F
SOLUTION
a + (MC)R =
4 ft
4
3
F(4) + F(1.5) - 200(1.5) = 300
5
5
F = 167 lb
3
4
Ans.
Resultant couple can act anywhere.
Ans.
Ans:
F = 167 lb
Resultant couple can act anywhere.
305
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4–79.
Two couples act on the frame. If the resultant couple moment
is to be zero, determine the distance d between the 80-lb
couple forces.
y
2 ft
B
3 ft
50 lb
30
1 ft
5
d
50 lb
80 lb
SOLUTION
a + MC = -50 cos 30°(3) +
30
3
4
5
4
(80)(d) = 0
5
3
4
80 lb
d = 2.03 ft
Ans.
3 ft
A
x
Ans:
d = 2.03 ft
306
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*4–80.
Two couples act on the frame. If d = 4 ft, determine the
resultant couple moment. Compute the result by resolving
each force into x and y components and (a) finding the
moment of each couple (Eq. 4–13) and (b) summing the
moments of all the force components about point A.
y
2 ft
B
1 ft
5
5
M C = ©(r * F)
i
3
- 50 sin 30°
d
50 lb
80 lb
SOLUTION
= 3
30
3
4
(a)
30
3 ft
50 lb
3
4
j
0
-50 cos 30°
i
k
3
3
0
0 +
0
- 54(80)
j
4
- 53(80)
M C = {126k} lb # ft
80 lb
k
03
0
3 ft
A
x
Ans.
4
4
(b) a + MC = - (80)(3) + (80)(7) + 50 cos 30°(2) - 50 cos 30°(5)
5
5
MC = 126 lb # ft
Ans.
Ans:
MC = 126 lb # ft
307
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4–81.
Two couples act on the frame. If d = 4 ft, determine the
resultant couple moment. Compute the result by resolving
each force into x and y components and (a) finding the
moment of each couple (Eq. 4–13) and (b) summing the
moments of all the force components about point B.
y
2 ft
B
1 ft
5
d
5
MC = ©(r * F)
3
4
i
3
-50 sin 30°
50 lb
80 lb
SOLUTION
= 3
30
3
4
(a)
30
3 ft
50 lb
j
0
- 50 cos 30°
i
k
03 + 3 0
4
0
5 (80)
j
-4
3
5 (80)
k
03
0
MC = {126k} lb # ft
80 lb
3 ft
A
x
Ans.
4
4
(b) a + MC = 50 cos 30°(2) - 50 cos 30°(5)- (80)(1) + (80)(5)
5
5
MC = 126 lb # ft
Ans.
Ans:
MC = 126 lb # ftd
308
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4–82.
z
Express the moment of the couple acting on the pipe
assembly in Cartesian vector form. What is the magnitude
of the couple moment?
20 lb
x
A
3 ft
B
1 ft
1.5 ft
20 lb
y
2 ft
1 ft
SOLUTION
C
rCB = { - 3i - 2.5j} ft
MC = rCB * F
i
= † -3
0
j
- 2.5
0
k
0 †
20
MC = { - 50i + 60j} lb # ft
Ans.
MC = 2( - 50) + (60) = 78.1 lb # ft
Ans.
2
2
Ans:
MC = 78.1 lb # ft
309
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4–83.
If M1 = 180 lb # ft, M2 = 90 lb # ft, and M3 = 120 lb # ft,
determine the magnitude and coordinate direction angles
of the resultant couple moment.
z
150 lb ft
M3
1 ft
45
45
2 ft
2 ft
SOLUTION
x
Since the couple moment is a free vector, it can act at any point without altering its
effect. Thus, the couple moments M1, M2, M3, and M4 acting on the gear deducer can
be simplified, as shown in Fig. a. Expressing each couple moment in Cartesian
vector form,
2 ft
y
3 ft
M2
M1
M 1 = [180j]lb # ft
M 2 = [- 90i]lb # ft
M 3 = M3u = 120 C
(2 - 0)i + (- 2 - 0)j + (1 + 0)k
2(2 - 0)2 + ( -2 - 0)2 + (1 - 0)2
S = [80i - 80j + 40k]lb # ft
M 4 = 150[cos 45° sin 45°i - cos 45° cos 45°j - sin 45°k] = [75i - 75j - 106.07k]lb # ft
The resultant couple moment is given by
(M c)R = ©M;
(M c)R = M 1 + M 2 + M 3 + M 4
= 180j - 90i + (80i - 80j + 40k) + (75i - 75j - 106.07k)
= [65i + 25j - 66.07k]lb # ft
The magnitude of (M c)R is
(Mc)R = 2[(Mc)R]x 2 + [(Mc)R]y 2 + [(Mc)R]z 2
= 2(65)2 + (25)2 + (- 66.07)2
= 95.99 lb # ft = 96.0 lb # ft
Ans.
The coordinate angles of (M c)R are
a = cos - 1 ¢
b = cos - 1 ¢
g = cos - 1 ¢
[(Mc)R]x
65
b = 47.4°
≤ = cos a
(Mc)R
95.99
[(Mc)R]y
(Mc)R
[(Mc)R]z
(Mc)R
≤ = cos a
≤ = cos a
Ans.
25
b = 74.9°
95.99
Ans.
- 66.07
b = 133°
95.99
Ans.
Ans:
MR = 96.0 lb # ft, a = 47.4°, b = 74.9°, g = 133°
310
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*4–84.
Determine the magnitudes of couple moments
M1, M2, and M3 so that the resultant couple moment is zero.
z
150 lb ft
M3
1 ft
45
45
2 ft
2 ft
x
SOLUTION
2 ft
Since the couple moment is a free vector, it can act at any point without altering its
effect. Thus, the couple moments M1, M2, M3, and M4 acting on the gear deducer can
be simplified, as shown in Fig. a. Expressing each couple moment in Cartesian
vector form,
y
3 ft
M2
M1
M 1 = M1j
M 2 = -M2i
M 3 = M3u = M3 C
(2 - 0)i + ( -2 - 0)j + (1 + 0)k
2(2 - 0)2 + ( -2 - 0)2 + (1 - 0)2
S =
2
1
2
M i - M3j + M3k
3 3
3
3
M 4 = 150[cos 45° sin 45°i - cos 45° cos 45°j - sin 45°k] = [75i - 75j - 106.07k]lb # ft
The resultant couple moment is required to be zero. Thus,
(M c)R = ©M;
0 = M1 + M2 + M3 + M4
2
2
1
0 = M1j + (-M2i) + a M3i - M3j + M3kb + (75i - 75j - 106.07k)
3
3
3
0 = a -M2 +
2
2
1
M + 75 bi + aM1 - M3 - 75 b j + a M3 - 106.07b k
3 3
3
3
Equating the i, j, and k components,
0 = - M2 +
0 = M1 0 =
2
M + 75
3 3
(1)
2
M - 75
3 3
(2)
1
M - 106.07
3 3
(3)
Solving Eqs. (1), (2), and (3) yields
M3 = 318 lb # ft
Ans.
M1 = M2 = 287 lb # ft
Ans.
Ans:
M3 = 318 lb # ft, M1 = M2 = 287 lb # ft
311
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4–85.
The gears are subjected to the couple moments shown.
Determine the magnitude and coordinate direction angles
of the resultant couple moment.
z
M1 40 lb ft
20
M2 30 lb ft
30
y
15
SOLUTION
M1 = 40 cos 20° sin 15° i + 40cos 20° cos 15° j - 40 sin 20° k
x
= 9.728 i + 36.307 j - 13.681 k
M 2 = -30 sin 30° i + 30 cos 30° j
= - 15 i + 25.981 j
MR = M1 + M2 = - 5.272 i + 62.288 j - 13.681 k
MR = 2( - 5.272)2 + (62.288)2 + (- 13.681)2 = 63.990 = 64.0 lb # ft
Ans.
a = cos-1 a
- 5.272
b = 94.7°
63.990
Ans.
b = cos-1 a
62.288
b = 13.2°
63.990
Ans.
g = cos-1 a
- 13.681
b = 102°
63.990
Ans.
Ans:
MR = 64.0 lb # ft
a = 94.7°
b = 13.2°
g = 102°
312
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4–86.
Determine the required magnitude of the couple moments
M2 and M3 so that the resultant couple moment is zero.
M2
45
SOLUTION
M3
Since the couple moment is the free vector, it can act at any point without altering
its effect. Thus, the couple moments M1, M2, and M3 can be simplified as shown in
Fig. a. Since the resultant of M1, M2, and M3 is required to be zero,
(MR)y = ©My ;
M1 300 Nm
0 = M2 sin 45° - 300
M2 = 424.26 N # m = 424 N # m
(MR)x = ©Mx ;
Ans.
0 = 424.26 cos 45° - M3
M3 = 300 N # m
Ans.
Ans:
M2 = 424 N # m
M3 = 300 N # m
313
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4–87.
z
Determine the resultant couple moment of the two couples
that act on the assembly. Specify its magnitude and
coordinate direction angles.
60 lb
2 in.
2 in.
80 lb
30
y
x
4 in.
SOLUTION
i
MR = † 4 cos 30°
0
j
5
0
k
i
- 4 sin 30° † + † 4 cos 30°
60
0
j
0
80
k
-4 sin 30° †
0
80 lb
= 300 i - 207.85 j + 160 i + 277.13 k
3 in.
= {460 i - 207.85 j + 277.13 k} lb # in.
MR = 2(460)2 + ( - 207.85)2 + (277.13)2 = 575.85 = 576 lb # in.
a = cos-1 a
b = cos-1 a
g = cos-1 a
460
b = 37.0°
575.85
60 lb
Ans.
Ans.
-207.85
b = 111°
575.85
Ans.
277.13
b = 61.2°
575.85
Ans.
Ans:
MR = 576 lb # in.
a = 37.0°
b = 111°
g = 61.2°
314
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*4–88.
Express the moment of the couple acting on the frame in
Cartesian vector form. The forces are applied perpendicular
to the frame. What is the magnitude of the couple moment?
Take F = 50 N.
z
O
y
F
3m
30
1.5 m
SOLUTION
x
MC = 80(1.5) = 75 N # m
F
Ans.
MC = - 75(cos 30° i + cos 60° k)
= {- 65.0i - 37.5k} N # m
Ans.
Ans:
MC = { - 65.0i - 37.5k} N # m
315
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4–89.
In order to turn over the frame, a couple moment is applied
as shown. If the component of this couple moment along
the x axis is Mx = 5 -20i6 N # m, determine the magnitude
F of the couple forces.
z
O
y
F
3m
SOLUTION
MC = F (1.5)
30
1.5 m
Thus
x
F
20 = F (1.5) cos 30°
F = 15.4 N
Ans.
Ans:
F = 15.4 N
316
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4–90.
Express the moment of the couple acting on the pipe in
Cartesian vector form. What is the magnitude of the couple
moment? Take F = 125 N.
z
O
–F
y
150 mm
600 mm
A
SOLUTION
B
MC = rAB * (125 k)
200 mm
x
150 mm
MC = (0.2i + 0.3j) * (125 k)
F
MC = {37.5i - 25j} N # m
MC = 2(37.5)2 + ( -25)2 = 45.1 N # m
Ans.
Ans:
MC = 45.1 N # m
317
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4–91.
If the couple moment acting on the pipe has a magnitude of
300 N # m, determine the magnitude F of the forces applied
to the wrenches.
z
O
–F
y
150 mm
600 mm
A
SOLUTION
B
MC = rAB * (F k)
200 mm
x
150 mm
= (0.2i + 0.3j) * (F k)
F
= {0.2Fi - 0.3Fj} N # m
MC = F 2(0.2F)2 + ( - 0.3F) = 0.3606 F
300 = 0.3606 F
F = 832 N
Ans.
Ans:
F = 832 N
318
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*4–92.
If F = 80 N, determine the magnitude and coordinate
direction angles of the couple moment. The pipe assembly
lies in the x–y plane.
z
F
300 mm
300 mm
F
x
200 mm
SOLUTION
It is easiest to find the couple moment of F by taking the moment of F or –F about
point A or B, respectively, Fig. a. Here the position vectors rAB and rBA must be
determined first.
200 mm
300 mm
rAB = (0.3 - 0.2)i + (0.8 - 0.3)j + (0 - 0)k = [0.1i + 0.5j] m
rBA = (0.2 - 0.3)i + (0.3 - 0.8)j + (0 - 0)k = [ -0.1i - 0.5j] m
The force vectors F and –F can be written as
F = {80 k} N and - F = [-80 k] N
Thus, the couple moment of F can be determined from
i
M c = rAB * F = 3 0.1
0
j
0.5
0
i
Mc = rBA * -F = 3 -0.1
0
j
- 0.5
0
k
0 3 = [40i - 8j] N # m
80
or
k
0 3 = [40i - 8j] N # m
- 80
The magnitude of Mc is given by
Mc = 2Mx 2 + My 2 + Mz 2 = 2402 + ( -8)2 + 02 = 40.79 N # m = 40.8 N # m
Ans.
The coordinate angles of Mc are
a = cos - 1 ¢
b = cos - 1 ¢
g = cos - 1 ¢
Mx
40
≤ = cos ¢
≤ = 11.3°
M
40.79
My
M
Mz
M
≤ = cos ¢
≤ = cos ¢
Ans.
-8
≤ = 101°
40.79
Ans.
0
≤ = 90°
40.79
Ans.
Ans:
Mc = 40.8 N # m
a = 11.3°
b = 101°
g = 90°
319
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4–93.
If the magnitude of the couple moment acting on the pipe
assembly is 50 N # m, determine the magnitude of the couple
forces applied to each wrench. The pipe assembly lies in the
x–y plane.
z
F
300 mm
300 mm
F
x
200 mm
SOLUTION
It is easiest to find the couple moment of F by taking the moment of either F or –F
about point A or B, respectively, Fig. a. Here the position vectors rAB and rBA must
be determined first.
200 mm
300 mm
rAB = (0.3 - 0.2)i + (0.8 - 0.3)j + (0 - 0)k = [0.1i + 0.5j] m
rBA = (0.2 - 0.3)i + (0.3 - 0.8)j + (0 - 0)k = [ - 0.1i - 0.5j] m
The force vectors F and –F can be written as
F = {Fk} N and - F = [ -Fk]N
Thus, the couple moment of F can be determined from
i
M c = rAB * F = 3 0.1
0
j
0.5
0
k
0 3 = 0.5Fi - 0.1Fj
F
The magnitude of Mc is given by
Mc = 2Mx 2 + My 2 + Mz 2 = 2(0.5F)2 + (0.1F)2 + 02 = 0.5099F
Since Mc is required to equal 50 N # m,
50 = 0.5099F
F = 98.1 N
Ans.
Ans:
F = 98.1 N
320
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4–94.
z
Express the moment of the couple acting on the rod in
Cartesian vector form. What is the magnitude of the couple
moment?
F { 4i 3j 4k} kN
A
1m
x
2m
3m
1m
SOLUTION
Position Vector. The coordinates of points A and B are A (0, 0, 1) m and
B (3, 2, -1) m, respectively. Thus,
y
B
F {– 4i + 3j 4k} kN
rAB = (3 - 0)i + (2 - 0)j + ( - 1 - 1)k = {3i + 2j - 2k} m
Couple Moment.
MC = rAB * F
i
= † 3
-4
j
2
3
k
-2 †
-4
= { - 2i + 20j + 17k} kN # m
Ans.
The magnitude of MC is
MC = 2(MC)2x + (MC)2y + (MC)2z
= 2( - 2)2 + 202 + 172
= 26.32 kN # m = 26.3 kN # m
Ans.
Ans:
MC = { - 2i + 20j + 17k} kN # m
MC = 26.3 kN # m
321
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4–95.
If F1 = 100 N, F2 = 120 N and F3 = 80 N, determine the
magnitude and coordinate direction angles of the resultant
couple moment.
z
–F4
[ 150 k] N
0.3 m
0.2 m
0.2 m
0.3 m
0.2 m
F1
0.2 m – F1
x
30
F4
[150 k] N
y
SOLUTION
Couple Moment: The position vectors r1, r2 , r3 , and r4 , Fig. a, must be determined
first.
r1 = {0.2i} m
r2 = {0.2j} m
– F2 0.2 m
F2
– F3
r3 = {0.2j} m
0.2 m
From the geometry of Figs. b and c , we obtain
F3
r4 = 0.3 cos 30° cos 45°i + 0.3 cos 30° sin 45°j - 0.3 sin 30°k
= {0.1837i + 0.1837j - 0.15k} m
The force vectors F1 , F2 , and F3 are given by
F1 = {100k} N
F2 = {120k} N
F3 = {80i} N
Thus,
M 1 = r1 * F1 = (0.2i) * (100k) = {- 20j} N # m
M 2 = r2 * F2 = (0.2j) * (120k) = {24i} N # m
M 3 = r3 * F3 = (0.2j) * (80i) = { -16k} N # m
M 4 = r4 * F4 = (0.1837i + 0.1837j - 0.15k) * (150k) = {27.56i - 27.56j} N # m
Resultant Moment: The resultant couple moment is given by
(M c)R =
(M c)R = M 1 + M 2 + M 3 + M 4
M c;
= ( -20j) + (24i) + ( - 16k) + (27.56i -27.56j)
= {51.56i - 47.56j - 16k} N # m
The magnitude of the couple moment is
(M c)R = 2[(M c)R]x2 + [(M c)R]y 2 + [(M c)R]z 2
= 2(51.56)2 + ( -47.56)2 + ( -16)2
= 71.94 N # m = 71.9 N # m
Ans.
The coordinate angles of (Mc )R are
a = cos -1 a
[(Mc)R]x
51.56
b = 44.2°
b = cos a
(Mc)R
71.94
b = cos -1 a
[(Mc)R]y
g = cos -1 a
[(Mc)R]z
(Mc)R
(Mc)R
- 47.56
b = 131°
71.94
Ans.
- 16
b = 103°
71.94
Ans.
b = cos a
b = cos a
Ans.
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Ans:
(MC)R = 71.9 Ν # m
a = 44.2°
b = 131°
g = 103°
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*4–96.
Determine the required magnitude of F1 , F2 , and F3
so that the resultant couple moment is (Mc)R =
[50 i - 45 j - 20 k] N # m.
z
–F4
[ 150 k] N
0.3 m
SOLUTION
Couple Moment: The position vectors r1 , r2 , r3 , and r4 , Fig. a, must be determined
first.
r1 = {0.2i} m
0.2 m
0.2 m
r2 = {0.2j} m
0.3 m
0.2 m
F1
0.2 m – F1
x
30
F4
[150 k] N
y
r3 = {0.2j} m
– F2 0.2 m
F2
From the geometry of Figs. b and c, we obtain
– F3
r4 = 0.3 cos 30° cos 45°i + 0.3 cos 30° sin 45°j - 0.3 sin 30°k
0.2 m
= {0.1837i + 0.1837j - 0.15k} m
F3
The force vectors F1 , F2 , and F3 are given by
F1 = F1k
F2 = F2k
F3 = F3i
Thus,
M 1 = r1 * F1 = (0.2i) * (F1k) = - 0.2 F1j
M 2 = r2 * F2 = (0.2j) * (F2k) = 0.2 F2i
M 3 = r3 * F3 = (0.2j) * (F3i) = - 0.2 F3k
M 4 = r4 * F4 = (0.1837i + 0.1837j - 0.15k) * (150k) = {27.56i - 27.56j} N # m
Resultant Moment: The resultant couple moment required to equal
(M c)R = {50i - 45j - 20k} N # m. Thus,
(M c)R = ©M c;
(M c ) R = M 1 + M 2 + M 3 + M 4
50i - 45j - 20k = ( - 0.2F1j) + (0.2F2i) + ( - 0.2F3k) + (27.56i - 27.56j)
50i - 45j - 20k = (0.2F2 + 27.56)i + ( - 0.2F1 - 27.56)j - 0.2F3k
Equating the i, j, and k components yields
50 = 0.2F2 + 27.56
F2 = 112 N
Ans.
- 45 = - 0.2F1 - 27.56
F1 = 87.2 N
Ans.
- 20 = - 0.2F3
F3 = 100 N
Ans.
Ans:
F2 = 112 N
F1 = 87.2 N
F3 = 100 N
323
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4–97.
y
Replace the force system by an equivalent resultant force
and couple moment at point O.
455 N
12
5
13
2m
2.5 m
O
x
0.75 m
0.75 m
60
SOLUTION
P
1m
Equivalent Resultant Force And Couple Moment At O.
+ (FR)x = ΣFx;
S
+ c (FR)y = ΣFy;
600 N
12
(FR)x = 600 cos 60° - 455 a b = -120 N = 120 N d
13
(FR)y = 455 a
5
b - 600 sin 60° = - 344.62 N = 344.62 NT
13
As indicated in Fig. a
FR = 2 (FR)2x + (FR)2y = 21202 + 344.622 = 364.91 N = 365 N
Ans.
u = tan-1 c
Ans.
And
Also,
(FR)y
(FR)x
d = tan-1 a
a+(MR)O = ΣMO; (MR)O = 455 a
344.62
b = 70.80° = 70.8° d
120
12
b(2) + 600 cos 60° (0.75) + 600 sin 60° (2.5)
13
= 2364.04 N # m
= 2364 N # m (counterclockwise)
Ans.
Ans:
FR = 365 N
u = 70.8° d
(MR)O = 2364 N # m (counterclockwise)
324
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4–98.
y
Replace the force system by an equivalent resultant force
and couple moment at point P.
455 N
12
5
13
2m
2.5 m
O
x
0.75 m
0.75 m
60
SOLUTION
P
1m
Equivalent Resultant Force And Couple Moment At P.
(FR)x = 600 cos 60° - 455 a
+ (FR)x = ΣFx;
S
+ c (FR)y = ΣFy;
(FR)y = 455 a
12
b = - 120 N = 120 N d
13
600 N
5
b - 600 sin 60° = -344.62 N = 344.62 N T
13
As indicated in Fig. a,
FR = 2 (FR)2x + (FR)2y = 21202 + 344.622 = 364.91 N = 365 N
Ans.
And
u = tan-1 c
(FR)y
(FR)x
Also,
a+ (MR)P = ΣMP;
d = tan-1 a
344.62
b = 70.80° = 70.8° d
120
(MR)P = 455 a
Ans.
12
5
b(2.75) - 455 a b(1) + 600 sin 60° (3.5)
13
13
= 2798.65 N # m
= 2799 N # m (counterclockwise)
Ans.
Ans:
FR = 365 N
u = 70.8° d
(MR)P = 2799 N # m (counterclockwise)
325
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4–99.
Replace the force system acting on the beam by an
equivalent force and couple moment at point A.
3 kN
2.5 kN 1.5 kN 30
5
3
4
B
A
2m
4m
2m
SOLUTION
4
FRx = 1.5 sin 30° - 2.5a b
5
+ F = ©F ;
:
Rx
x
= - 1.25 kN = 1.25 kN ;
3
FRy = - 1.5 cos 30° - 2.5 a b - 3
5
+ c FRy = ©Fy ;
= - 5.799 kN = 5.799 kN T
Thus,
FR = 2F 2Rx + F 2Ry = 21.252 + 5.7992 = 5.93 kN
Ans.
and
u = tan - 1 ¢
a + MRA = ©MA ;
FRy
FRx
≤ = tan - 1 a
5.799
b = 77.8° d
1.25
Ans.
3
MRA = - 2.5a b (2) - 1.5 cos 30°(6) - 3(8)
5
= - 34.8 kN # m = 34.8 kN # m (Clockwise)
Ans.
Ans:
FR = 5.93 kN
u = 77.8° d
MRA = 34.8 kN # m b
326
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*4–100.
Replace the force system acting on the beam by an
equivalent force and couple moment at point B.
3 kN
2.5 kN 1.5 kN 30
5
3
4
B
A
2m
4m
2m
SOLUTION
4
FRx = 1.5 sin 30° - 2.5a b
5
+ F = ©F ;
:
Rx
x
= - 1.25 kN = 1.25 kN ;
3
FRy = - 1.5 cos 30° - 2.5a b - 3
5
+ c FRy = ©Fy ;
= - 5.799 kN = 5.799 kN T
Thus,
FR = 2F 2Rx + F 2Ry = 21.252 + 5.7992 = 5.93 kN
and
u = tan - 1 ¢
FRy
FRx
a + MRB = ©MRB ;
≤ = tan - 1 a
5.799
b = 77.8° d
1.25
Ans.
Ans.
3
MB = 1.5cos 30°(2) + 2.5a b(6)
5
= 11.6 kN # m (Counterclockwise)
Ans.
Ans:
FR = 5.93 kN
u = 77.8° d
MB = 11.6 kN # m (Counterclockwise)
327
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4–101.
y
450 N
Replace the loading system acting on the beam by an
equivalent resultant force and couple moment at point O.
30
200 N m
0.2 m
x
O
1.5 m
2m
1.5 m
200 N
SOLUTION
+
d FRx = ΣFx ;
FRx = 450 sin 30° = 225.0
+ T FRy = ΣFy ;
FRy = 450 cos 30° - 200 = 189.7
FR = 2(225)2 + (189.7)2 = 294 N
Ans.
u = tan-1 a
Ans.
189.7
b = 40.1° d
225
c+MRO = ΣMO ; MRO = 450 cos 30° (1.5) - 450 (sin 30°)(0.2) - 200 (3.5) + 200
MRO = 39.6 N # m b
Ans.
Ans:
FR = 294 N
u = 40.1° d
MRO = 39.6 N # m b
328
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4–102.
650 N
30
Replace the loading system acting on the post by an
equivalent resultant force and couple moment at point A.
500 N
300 N
1500 N m
60 B
A
3m
5m
2m
SOLUTION
Equivalent Resultant Force And Couple Moment at Point A.
+ (FR)x = ΣFx ;
S
(FR)x = 650 sin 30° - 500 cos 60° = 75 N S
+ c (FR)y = ΣFy ;
(FR)y = - 650 cos 30° - 300 - 500 sin 60°
= - 1295.93 N = 1295.93 NT
As indicated in Fig. a,
FR = 2(FR)2x + (FR)2y = 2752 + 1295.932 = 1298.10 N = 1.30 kN Ans.
And
u = tan-1 c
(FR)y
(FR)x
d = tan-1 a
1295.93
b = 86.69° = 86.7° c
75
Ans.
Also,
a+ (MR)A = ΣMA;
(MR)A = 650 cos 30° (3) + 1500 - 500 sin 60° (5)
= 1023.69 N # m
= 1.02 kN # m (counter clockwise)
Ans.
Ans:
FR = 1.30 kN
u = 86.7° c
(MR)A = 1.02 kN # m (counterclockwise)
329
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4–103.
650 N
30
Replace the loading system acting on the post by an
equivalent resultant force and couple moment at point B.
500 N
300 N
1500 N m
60 B
A
3m
5m
2m
SOLUTION
Equivalent Resultant Force And Couple Moment At Point B.
+ (FR)x = ΣFx;
S
(FR)x = 650 sin 30° - 500 cos 60° = 75 N S
+ c (FR)y = ΣFy;
(FR)y = - 650 cos 30° - 300 - 500 sin 60°
= - 1295.93 N = 1295.93 NT
As indicated in Fig. a,
FR = 2 (FR)2x + (FR)2y = 2752 + 1295.932 = 1298.10 N = 1.30 kN Ans.
And
u = tan-1 c
(FR)y
(FR)x
d = tan-1 a
1295.93
b = 86.69° = 86.7° c
75
Ans.
Also,
a+ (MR)B = ΣMB;
(MR)B = 650 cos 30° (10) + 300(7) + 500 sin 60°(2) + 1500
= 10,095.19 N # m
= 10.1 kN # m (counterclockwise)
Ans.
Ans:
FR = 1.30 kN
u = 86.7° c
(MR)B = 1.01 kN # m (counterclockwise)
330
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*4–104.
Replace the force system acting on the post by a resultant
force and couple moment at point O.
300 lb
30
150 lb
3
2 ft
5
4
SOLUTION
Equivalent Resultant Force: Forces F1 and F2 are resolved into their x and y components, Fig. a. Summing these force components algebraically along the x and y
axes, we have
+ ©(F ) = ©F ;
:
R x
x
4
(FR)x = 300 cos 30° - 150 a b + 200 = 339.81 lb :
5
+ c (FR)y = ©Fy;
3
(FR)y = 300 sin 30° + 150 a b = 240 lb c
5
2 ft
200 lb
2 ft
O
The magnitude of the resultant force FR is given by
FR = 2(FR)x2 + (FR)y2 = 2339.812 + 2402 = 416.02 lb = 416 lb
Ans.
The angle u of FR is
u = tan-1 c
(FR)y
(FR)x
d = tan-1 c
240
d = 35.23° = 35.2° a
339.81
Ans.
Equivalent Resultant Couple Moment: Applying the principle of moments,
Figs. a and b, and summing the moments of the force components algebraically
about point A, we can write
a + (MR)A = © MA;
4
(MR)A = 150 a b (4) - 200(2) - 300 cos 30°(6)
5
= -1478.85 lb # ft = 1.48 kip # ft (Clockwise) Ans.
Ans:
FR = 416 lb
u = 35.2° a
( MR ) A = 1.48 kip # ft (Clockwise)
331
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4–105.
Replace the force system acting on the frame by an
equivalent resultant force and couple moment acting at
point A.
A
300 N
0.5 m
30
1m
500 N
SOLUTION
Equivalent Resultant Force And Couple Moment At A.
0.5 m
+ (FR)x = ΣFx;
S
(FR)x = 300 cos 30° + 500 = 759.81 N
+c (FR)y = ΣFy;
(FR)y = -300 sin 30° - 400 = -550 N = 550 N
S
0.3 m
400 N
T
As indicated in Fig. a,
FR = 2(FR)2x + (FR)2y = 2759.812 + 5502 = 937.98 N = 938 N
Ans.
u = tan-1 c
Ans.
And
Also;
(FR)y
(FR)x
a+ (MR)A = ΣMA;
d = tan-1a
550
b = 35.90° = 35.9° c
759.81
(MR)A = 300 cos 30°(0.5) + 500(1.5) - 400(0.5)
= 679.90 N # m
= 680 N # m (counterclockwise)
Ans.
Ans:
FR = 938 N
u = 35.9° c
(MR)A = 680 N # m (counterclockwise)
332
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4–106.
z
The forces F1 = { - 4i + 2j - 3k} kN and F2 = {3i - 4j 2k} kN act on the end of the beam. Replace these forces by
an equivalent force and couple moment acting at point O.
F1
150 mm
F
150 mm 2
O
250 mm
y
4m
SOLUTION
x
FR = F1 + F2 = { - 1i - 2j - 5k} kN
Ans.
MRO = r1 * F1 + r2 * F2
i
= † 4
-4
j
- 0.15
2
k
i
0.25 † + † 4
-3
3
j
0.15
-4
k
0.25 †
-2
= ( -0.05i + 11j + 7.4k) + (0.7i + 8.75j - 16.45k)
= (0.65i + 19.75j - 9.05k)
MRO = {0.650i + 19.75j - 9.05k} kN # m
Ans.
Ans:
MRO = {0.650i + 19.75j - 9.05k} kN # m
333
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4–107.
A biomechanical model of the lumbar region of the human
trunk is shown. The forces acting in the four muscle groups
consist of FR = 35 N for the rectus, FO = 45 N for the
oblique, FL = 23 N for the lumbar latissimus dorsi, and
FE = 32 N for the erector spinae. These loadings are
symmetric with respect to the y–z plane. Replace this system
of parallel forces by an equivalent force and couple moment
acting at the spine, point O. Express the results in Cartesian
vector form.
z
FR
FO
FR
FE
FL
FE
FO
FL
O
75 mm
15 mm
45 mm
SOLUTION
50 mm
30 mm
40 mm
x
FR = © Fz ;
FR = {2(35 + 45 + 23 + 32)k } = {270k} N
MROx = © MOx ;
MR O = [ -2(35)(0.075) + 2(32)(0.015) + 2(23)(0.045)]i
MR O = { -2.22i} N # m
Ans.
Ans.
Ans:
FR = 5270k6 N
MRO = 5 - 2.22i6 N # m
334
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*4–108.
z
Replace the force system by an equivalent resultant
force and couple moment at point O. Take
F3 = { -200i + 500j - 300k} N.
F1 = 300 N
O
2m
x
F3
1.5 m
y
SOLUTION
1.5 m
Position And Force Vectors.
F2 = 200 N
r1 = {2j} m
r2 = {1.5i + 3.5j}
r3 = {1.5i + 2j} m
F1 = { -300k} N
F2 = {200j} N
F3 = { -200i + 500j - 300k} N
Equivalent Resultant Force And Couple Moment At Point O.
FR = ΣF;
FR = F1 + F2 + F3
= ( -300k) + 200j + ( -200i + 500j - 300k)
= { - 200i + 700j - 600k} N
(MR)O = ΣMO;
Ans.
(MR)O = r1 * F1 + r2 * F2 + r3 * F3
i
= 30
0
j
2
0
k
i
0 3 + 3 1.5
- 300
0
j
3.5
200
k
i
0 3 + 3 1.5
0
- 200
j
2
500
k
0 3
- 300
= ( -600i) + (300k) + ( - 600i + 450j + 1150k)
= { - 1200i + 450j + 1450k} N # m
Ans.
Ans:
FR = { -200i + 700j - 600k} N
(MR)O = { - 1200i + 450j + 1450k} N # m
335
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z
4–109.
O
Replace the loading by an equivalent resultant force and
couple moment at point O.
x
0.5 m
y
0.7 m
F2 = {–2 i + 5 j – 3 k} kN
0.8 m
F1 {8 i – 2 k} kN
SOLUTION
Position Vectors. The required position vectors are
r1 = {0.8i - 1.2k} m
r2 = { - 0.5k} m
Equivalent Resultant Force And Couple Moment At Point O.
FR = ΣF;
FR = F1 + F2
= (8i - 2k) + ( -2i + 5j - 3k)
= {6i + 5j - 5k} kN
(MR)O = ΣMO;
Ans.
(MR)O = r1 * F1 + r2 * F2
i
= 3 0.8
8
j
0
0
k
i
- 1.2 3 + 3 0
-2
-2
j
0
5
k
- 0.5 3
-3
= ( -8j) + (2.5i + j)
= {2.5i - 7j} kN # m
Ans.
Ans:
FR = {6i + 5j - 5k} kN
(MR)O = {2.5i - 7j} kN # m
336
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z
4–110.
Replace the force of F = 80 N acting on the pipe assembly
by an equivalent resultant force and couple moment at
point A.
A
400 mm
B
[
300 mm
y
200 mm
SOLUTION
200 mm
250 mm
FR = ΣF ;
40
FR = 80 cos 30° sin 40° i + 80 cos 30° cos 40° j - 80 sin 30° k
30
= 44.53 i + 53.07 j - 40 k
F
= {44.5 i + 53.1 j - 40 k} N
MRA = ΣMA ;
i
MRA = † 0.55
44.53
80 N
Ans.
j
0.4
53.07
k
- 0.2 †
-40
= { - 5.39 i + 13.1 j + 11.4 k} N # m
Ans.
Ans:
FR = {44.5 i + 53.1 j + 40 k} N
MRA = { -5.39 i + 13.1 j + 11.4 k} N # m
337
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4–111.
The belt passing over the pulley is subjected to forces F1 and
F2, each having a magnitude of 40 N. F1 acts in the - k direction.
Replace these forces by an equivalent force and couple
moment at point A. Express the result in Cartesian vector form.
Set u = 0° so that F2 acts in the - j direction.
z
r 80 mm
y
300 mm
A
SOLUTION
x
FR = F1 + F2
FR = {- 40j - 40 k} N
Ans.
F2
M RA = ©(r * F)
i
= 3 -0.3
0
j
0
- 40
F1
k
i
0.08 3 + 3 - 0.3
0
0
j
0.08
0
k
0 3
- 40
MRA = {- 12j + 12k} N # m
Ans.
FR = 5 -40j - 40k 6 N
Ans:
MRA = 5 - 12j + 12k6 N # m
338
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*4–112.
z
The belt passing over the pulley is subjected to two forces F1
and F2, each having a magnitude of 40 N. F1 acts in the - k
direction. Replace these forces by an equivalent force and
couple moment at point A. Express the result in Cartesian
vector form. Take u = 45°.
r
80 mm
y
300 mm
A
SOLUTION
x
FR = F1 + F2
= - 40 cos 45°j + ( - 40 - 40 sin 45°)k
F2
FR = { - 28.3j - 68.3k} N
Ans.
F1
rAF1 = { - 0.3i + 0.08j} m
rAF2 = - 0.3i - 0.08 sin 45°j + 0.08 cos 45°k
= {- 0.3i - 0.0566j + 0.0566k} m
MRA = (rAF1 * F1) + (rAF2 * F2)
i
= 3 - 0.3
0
j
0.08
0
i
k
0 3 + 3 - 0.3
0
- 40
j
- 0.0566
- 40 cos 45°
k
0.0566 3
- 40 sin 45°
MRA = { - 20.5j + 8.49k} N # m
Ans.
Also,
MRAx = ©MAx
MRAx = 28.28(0.0566) + 28.28(0.0566) - 40(0.08)
MRAx = 0
MRAy = ©MAy
MRAy = - 28.28(0.3) - 40(0.3)
MRAy = - 20.5 N # m
MRAz = ©MAz
MRAz = 28.28(0.3)
MRAz = 8.49 N # m
MRA = { - 20.5j + 8.49k} N # m
Ans.
FR = 5-28.3j - 68.3k6 N
Ans:
MRA = 5- 20.5j + 8.49k6 N # m
339
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4–113.
The weights of the various components of the truck are
shown. Replace this system of forces by an equivalent
resultant force and specify its location measured from B.
+ c FR = ©Fy;
FR = - 1750 - 5500 - 3500
= - 10 750 lb = 10.75 kip T
a +MRA = ©MA ;
3500 lb
B
SOLUTION
5500 lb
14 ft
3 ft
A
1750 lb
6 ft
2 ft
Ans.
-10 750d = - 3500(3) - 5500(17) - 1750(25)
d = 13.7 ft
Ans.
Ans:
FR = 10.75 kip T
d = 13.7 ft
340
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exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
4–114.
The weights of the various components of the truck are
shown. Replace this system of forces by an equivalent
resultant force and specify its location measured from
point A.
5500 lb
14 ft
Equivalent Force:
+ c FR = ©Fy ;
3500 lb
B
SOLUTION
3 ft
A
1750 lb
6 ft
2 ft
FR = -1750 - 5500 - 3500
= - 10 750 lb = 10.75 kip T
Ans.
Location of Resultant Force From Point A:
a + MRA = ©MA ;
10 750(d) = 3500(20) + 5500(6) - 1750(2)
d = 9.26 ft
Ans.
Ans:
FR = 10.75 kip T
d = 9.26 ft
341
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4–115.
Replace the three forces acting on the shaft by a single
resultant force. Specify where the force acts, measured from
end A.
5 ft
3 ft
2 ft
4 ft
A
B
5
13
12
3
4
5
500 lb
200 lb
260 lb
SOLUTION
+ FR = ΣFx;
S
x
+ c FRy = ΣFy;
5
4
FRx = -500 a b + 260 a b = - 300 lb = 300 lb d
5
13
12
3
FRy = -500 a b - 200 - 260 a b = -740 lb = 740 lb T
5
13
F = 2( - 300)2 + ( - 740)2 = 798 lb
u = tan-1 a
Ans.
740
b = 67.9° d
300
c + MRA = ΣMA;
Ans.
12
3
740(x) = 500 a b(5) + 200(8) + 260 a b(10)
5
13
740(x) = 5500
x = 7.43 ft
Ans.
Ans:
F = 798 lb
67.9° d
x = 7.43 ft
342
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*4–116.
Replace the three forces acting on the shaft by a single
resultant force. Specify where the force acts, measured from
end B.
5 ft
3 ft
2 ft
4 ft
A
B
5
13
12
3
4
5
500 lb
200 lb
260 lb
SOLUTION
+ ΣFR = ΣFx;
S
x
4
5
FRx = - 500 a b + 260 a b = - 300 lb = 300 lb d
5
13
+ c FRy = ΣFy;
3
12
FRy = - 500 a b - 200 - 260 a b = - 740 lb = 740 lb T
5
13
F = 2( -300)2 + ( -740)2 = 798 lb
u = tan-1 a
Ans.
740
b = 67.9° d
300
a+ MRB = ΣMB;
Ans.
3
12
740(x) = 500 a b(9) + 200(6) + 260 a b(4)
5
13
x = 6.57 ft
Ans.
Ans:
F = 798 lb
u = 67.9° d
x = 6.57 ft
343
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4–117.
700 N
Replace the loading acting on the beam by a single resultant
force. Specify where the force acts, measured from end A.
450 N
30
300 N
60
B
A
2m
4m
3m
1500 N m
SOLUTION
+ F = ©F ;
:
Rx
x
FRx = 450 cos 60° - 700 sin 30° = - 125 N = 125 N
+ c FRy = ©Fy ;
FRy = - 450 sin 60° - 700 cos 30° - 300 = - 1296 N = 1296 N
;
T
F = 2( - 125)2 + ( - 1296)2 = 1302 N
Ans.
1296
b = 84.5°
125
Ans.
u = tan-1 a
c + MRA = ©MA ;
d
1296(x) = 450 sin 60°(2) + 300(6) + 700 cos 30°(9) + 1500
x = 7.36 m
Ans.
Ans:
F = 1302 N
u = 84.5° d
x = 7.36 m
344
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4–118.
Replace the loading acting on the beam by a single resultant
force. Specify where the force acts, measured from B.
700 N
450 N
30
300 N
60
B
A
2m
4m
3m
1500 N m
SOLUTION
+ F = ©F ;
:
Rx
x
FRx = 450 cos 60° - 700 sin 30° = - 125 N = 125 N
+ c FRy = ©Fy ;
FRy = - 450 sin 60° - 700 cos 30° - 300 = - 1296 N = 1296 N
F = 2( - 125)2 + ( - 1296)2 = 1302 N
u = tan-1 a
T
Ans.
1296
b = 84.5° d
125
c + MRB = ©MB ;
;
Ans.
1296(x) = - 450 sin 60°(4) + 700 cos 30°(3) + 1500
x = 1.36 m (to the right)
Ans.
Ans:
F = 1302 N
u = 84.5° d
x = 1.36 m (to the right)
345
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4–119.
Replace the loading on the frame by a single resultant force.
Specify where its line of action intersects a vertical line
along member AB, measured from A.
400 N
200 N
200 N
0.5 m
0.5 m
600 N
B
C
1.5 m
SOLUTION
A
Equivalent Resultant Force. Referring to Fig. a,
+ (FR)x = ΣFx;
S
(FR)x = 600 N S
+ c (FR)y = ΣFy;
(FR)y = - 200 - 400 - 200 = -800 N = 800 NT
As indicated in Fig. a,
FR = 2(FR)2x + (FR)2y = 26002 + 8002 = 1000 N
Ans.
And
u = tan-1 c
(FR)y
(FR)x
d = tan-1a
800
b = 53.13° = 53.1°
600
Ans.
c
Location of Resultant Force. Along AB,
a+ (MR)B = ΣMB;
600(1.5 - d) = -400(0.5) - 200(1)
d = 2.1667 m = 2.17 m
Ans.
Ans:
FR = 1000 N
u = 53.1° c
d = 2.17 m
346
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*4–120.
y
Replace the loading on the frame by a single resultant force.
Specify where its line of action intersects a vertical line
along member AB, measured from A.
1m
600 N
0.5 m
B
0.5 m
3
400 N
1.5 m
4
5
900 N
SOLUTION
1m
S
3
4
(FR)y = 600 + 400 a b - 400 - 900 a b
5
5
= -280 N = 280 N T
As indicated in Fig. a,
FR = 2(FR)2x + (FR)2y = 22202 + 2802 = 356.09 N = 356 N
Ans.
u = tan-1 c
Ans.
And
(FR)y
(FR)x
d = tan-1a
280
b = 51.84° = 51.8°
220
Location of Resultant Force. Referring to Fig. a
a+ (MR)A = ΣMA;
400 N
x
3
4
(FR)x = 900 a b - 400 a b = 220 N
5
5
+ c (FR)y = ΣFy;
5
4
A
Equivalent Resultant Force. Referring to Fig. a
+ (FR)x = ΣFx;
S
3
3
280 a - 220 b = 400(1.5) - 600(0.5) - 900 a b(2.5)
5
4
+ 400 a b(1)
5
220 b - 280 a = 730
(1)
347
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*4–120. Continued
Along AB, a = 0. Then Eq (1) becomes
220 b - 280(0) = 730
b = 3.318 m
Thus, the intersection point of line of action of FR on AB measured upward from
point A is
d = b = 3.32 m
Ans.
Ans:
FR = 356 N
u = 51.8°
d = b = 3.32 m
348
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4–121.
y
Replace the loading on the frame by a single resultant force.
Specify where its line of action intersects a horizontal line
along member CB, measured from end C.
1m
600 N
0.5 m
B
0.5 m
3
400 N
1.5 m
4
5
900 N
SOLUTION
1m
+ c (FR)y = ΣFy;
3
4
(FR)x = 900 a b - 400 a b = 220 N
5
5
S
3
4
(FR)y = 600 + 400 a b - 400 - 900 a b
5
5
As indicated in Fig. a,
FR = 2(FR)2x + (FR)2y = 22202 + 2802 = 356.09 N = 356 N
Ans.
u = tan-1 c
Ans.
And
(FR)x
d = tan-1a
280
b = 51.84° = 51.8°
220
Location of Resultant Force. Referring to Fig. a
a+ (MR)A = ΣMA;
400 N
x
= - 280 N = 280 N T
(FR)y
5
4
A
Equivalent Resultant Force. Referring to Fig. a
+ (FR)x = ΣFx;
S
3
3
280 a - 220 b = 400(1.5) - 600(0.5) - 900 a b(2.5)
5
4
+ 400 a b(1)
5
220 b - 280 a = 730
(1)
349
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4–121. Continued
Along BC, b = 3 m. Then Eq (1) becomes
220(3) - 280 a = 730
a = -0.25 m
Thus, the intersection point of line of action of FR on CB measured to the right of
point C is
d = 1.5 - ( -0.25) = 1.75 m
Ans.
Ans:
FR = 356 N
u = 51.8°
d = 1.75 m
350
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4–122.
Replace the force system acting on the post by a resultant
force, and specify where its line of action intersects the post
AB measured from point A.
0.5 m
B
1m
500 N
5
3
0.2 m
4
250 N
30
1m
SOLUTION
300 N
Equivalent Resultant Force: Forces F1 and F2 are resolved into their x and y
components, Fig. a. Summing these force components algebraically along the x and
y axes,
+ (F ) = ©F ;
:
R x
x
4
(FR)x = 250 a b - 500 cos 30° - 300 = - 533.01 N = 533.01 N ;
5
+ c (FR)y = ©Fy;
3
(FR)y = 500 sin 30° - 250a b = 100 N c
5
1m
A
The magnitude of the resultant force FR is given by
FR = 2(FR)x 2 + (FR)y 2 = 2533.012 + 1002 = 542.31 N = 542 N
Ans.
The angle u of FR is
u = tan - 1 B
(FR)y
(FR)x
R = tan - 1 c
100
d = 10.63° = 10.6° b
533.01
Ans.
Location of the Resultant Force: Applying the principle of moments, Figs. a and b,
and summing the moments of the force components algebraically about point A,
a + (MR)A = ©MA;
4
3
533.01(d) = 500 cos 30°(2) - 500 sin 30°(0.2) - 250 a b (0.5) - 250 a b (3) + 300(1)
5
5
d = 0.8274 mm = 827 mm
Ans.
Ans:
FR = 542 N
u = 10.6° b
d = 0.827 m
351
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4–123.
Replace the force system acting on the post by a resultant
force, and specify where its line of action intersects the post
AB measured from point B.
0.5 m
B
1m
500 N
5
3
0.2 m
4
250 N
30
1m
SOLUTION
300 N
Equivalent Resultant Force: Forces F1 and F2 are resolved into their x and y
components, Fig. a. Summing these force components algebraically along the x and
y axes,
+ © (F ) = ©F ;
:
R x
x
4
(FR)x = 250 a b - 500 cos 30° - 300 = - 533.01N = 533.01 N ;
5
+ c (FR)y = ©Fy;
3
(FR)y = 500 sin 30° - 250 a b = 100 N c
5
1m
A
The magnitude of the resultant force FR is given by
FR = 2(FR)x 2 + (FR)y 2 = 2533.012 + 1002 = 542.31 N = 542 N
Ans.
The angle u of FR is
u = tan - 1 B
(FR)y
(FR)x
R = tan - 1 c
100
d = 10.63° = 10.6° b
533.01
Ans.
Location of the Resultant Force: Applying the principle of moments, Figs. a and b,
and summing the moments of the force components algebraically about point B ,
a +(MR)B = ©Mb;
3
-533.01(d) = -500 cos 30°(1) - 500 sin 30°(0.2) - 250 a b (0.5) - 300(2)
5
d = 2.17 m
Ans.
Ans:
FR = 542 N
u = 10.6° b
d = 2.17 m
352
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*4–124.
Replace the parallel force system acting on the plate by a
resultant force and specify its location on the x–z plane.
z
0.5 m
1m
SOLUTION
2 kN
5 kN
Resultant Force: Summing the forces acting on the plate,
(FR)y = ©Fy;
1m
FR = -5 kN - 2 kN - 3 kN
= - 10 kN
Ans.
1m
The negative sign indicates that FR acts along the negative y axis.
y
3 kN
0.5 m
Resultant Moment: Using the right-hand rule, and equating the moment of FR to
the sum of the moments of the force system about the x and z axes,
x
Ans:
FR = - 10 kN
353
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4–125.
Replace the force and couple system acting on the frame by
an equivalent resultant force and specify where the
resultant’s line of action intersects member AB, measured
from A.
A
2 ft
5
3
4
150 lb
4 ft
SOLUTION
+ F = ©F ;
:
Rx
x
4
FRx = 150 a b + 50 sin 30° = 145 lb
5
+ c FRy = ©Fy ;
3
FRy = 50 cos 30° + 150 a b = 133.3 lb
5
FR = 2(145)2 + (133.3)2 = 197 lb
u = tan - 1 a
a + MRA = ©MA ;
500 lb ft
3 ft
Ans.
133.3
b = 42.6°
145
B
C
30
50 lb
Ans.
4
145 d = 150 a b (2) - 50 cos 30° (3) + 50 sin 30° (6) + 500
5
d = 5.24 ft
Ans.
Ans:
FR = 197 lb
u = 42.6°a
d = 5.24 ft
354
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4–126.
Replace the force and couple system acting on the frame by
an equivalent resultant force and specify where the
resultant’s line of action intersects member BC, measured
from B.
A
2 ft
5
3
4
150 lb
4 ft
SOLUTION
+ F = ©F ;
:
Rx
x
4
FRx = 150 a b + 50 sin 30° = 145 lb
5
+ c FRy = ©Fy ;
3
FRy = 50 cos 30° + 150 a b = 133.3 lb
5
500 lb ft
3 ft
30
50 lb
FR = 2(145) + (133.3) = 197 lb
Ans.
u = tan - 1 a
Ans.
2
2
133.3
b = 42.6°
145
a + MRA = ©MA ;
B
C
4
145 (6) - 133.3 (d) = 150 a b (2) - 50 cos 30° (3) + 50 sin 30° (6) + 500
5
d = 0.824 ft
Ans.
Ans:
FR = 197 lb
u = 42.6°a
d = 0.824 ft
355
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4–127.
If FA = 7 kN and FB = 5 kN, represent the force system
acting on the corbels by a resultant force, and specify its
location on the x–y plane.
z
FB
150 mm
6 kN
750 mm
100 mm
FA
650 mm
SOLUTION
x
8kN
700 mm
O
100 mm
600 mm
150 mm
y
Equivalent Resultant Force: By equating the sum of the forces in Fig. a along the z
axis to the resultant force FR, Fig. b,
+ c FR = ©Fz;
- FR = - 6 - 5 - 7 - 8
FR = 26 kN
Ans.
Point of Application: By equating the moment of the forces shown in Fig. a and FR,
Fig. b, about the x and y axes,
(MR)x = ©Mx;
- 26(y) = 6(650) + 5(750) - 7(600) - 8(700)
y = 82.7 mm
(MR)y = ©My;
Ans.
26(x) = 6(100) + 7(150) - 5(150) - 8(100)
x = 3.85 mm
Ans.
Ans:
FR = 26 kN
y = 82.7 mm
x = 3.85 mm
356
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*4–128.
Determine the magnitudes of FA and FB so that the
resultant force passes through point O of the column.
z
FB
150 mm
6 kN
750 mm
100 mm
FA
650 mm
8kN
700 mm
O
100 mm
SOLUTION
x
600 mm
150 mm
y
Equivalent Resultant Force: By equating the sum of the forces in Fig. a along the z
axis to the resultant force FR, Fig. b,
+ c FR = ©Fz;
- FR = - FA - FB - 8 - 6
FR = FA + FB + 14
(1)
Point of Application: Since FR is required to pass through point O, the moment of
FR about the x and y axes are equal to zero. Thus,
(MR)x = ©Mx;
0 = FB (750) + 6(650) - FA (600) - 8(700)
750FB - 600FA - 1700 = 0
(MR)y = ©My;
(2)
0 = FA (150) + 6(100) - FB (150) - 8(100)
159FA - 150FB + 200 = 0
(3)
Solving Eqs. (1) through (3) yields
FA = 18.0 kN
FB = 16.7 kN
FR = 48.7 kN
Ans.
Ans:
FA = 18.0 kN
FB = 16.7 kN
FR = 48.7 kN
357
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4–129.
The tube supports the four parallel forces. Determine the
magnitudes of forces FC and FD acting at C and D so that
the equivalent resultant force of the force system acts
through the midpoint O of the tube.
z
FD
600 N
D
FC
A
400 mm
SOLUTION
Since the resultant force passes through point O, the resultant moment components
about x and y axes are both zero.
©Mx = 0;
500 N
C
400 mm
x
z
B
200 mm
200 mm y
FD(0.4) + 600(0.4) - FC(0.4) - 500(0.4) = 0
FC - FD = 100
©My = 0;
O
(1)
500(0.2) + 600(0.2) - FC(0.2) - FD(0.2) = 0
FC + FD = 1100
(2)
Solving Eqs. (1) and (2) yields:
FC = 600 N
FD = 500 N
Ans.
Ans:
FC = 600 N
FD = 500 N
358
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4–130.
z
The building slab is subjected to four parallel column
loadings. Determine the equivalent resultant force and
specify its location (x, y) on the slab. Take F1 = 8 kN and
F2 = 9 kN.
12 kN
F1
F2
6 kN
x
8m
6m
SOLUTION
y
16 m
12 m
4m
Equivalent Resultant Force. Sum the forces along z axis by referring to
Fig. a
+ c (FR)z = ΣFz;
-FR = - 8 - 6 - 12 - 9
FR = 35 kN
Ans.
Location of the Resultant Force. Sum the moments about the x and y axes by
referring to Fig. a,
(MR)x = ΣMx;
-35 y = -12(8) - 6(20) - 9(20)
y = 11.31 m = 11.3 m
(MR)y = ΣMy;
Ans.
35 x = 12(6) + 8(22) + 6(26)
x = 11.54 m = 11.5 m
Ans.
Ans:
FR = 35 kN
y = 11.3 m
x = 11.5 m
359
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4–131.
z
The building slab is subjected to four parallel column
loadings. Determine F1 and F2 if the resultant force acts
through point (12 m, 10 m).
12 kN
F1
F2
6 kN
x
8m
6m
SOLUTION
y
16 m
12 m
4m
Equivalent Resultant Force. Sum the forces along z axis by referring to Fig. a,
+ c (FR)z = ΣFz;
- FR = - F1 - F2 - 12 - 6
FR = F1 + F2 + 18
Location of the Resultant Force. Sum the moments about the x and y axes by
referring to Fig. a,
(MR)x = ΣMx;
- (F1 + F2 + 18)(10) = - 12(8) - 6(20) - F2(20)
10F1 - 10F2 = 36
(MR)y = ΣMy;
(1)
(F1 + F2 + 18)(12) = 12(6) + 6(26) + F1(22)
12F2 - 10F1 = 12
(2)
Solving Eqs (1) and (2),
F1 = 27.6 kN
F2 = 24.0 kN
Ans.
Ans:
F1 = 27.6 kN
F2 = 24.0 kN
360
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*4–132.
If FA = 40 kN and FB = 35 kN, determine the magnitude
of the resultant force and specify the location of its point of
application (x,y) on the slab.
z
30 kN
0.75 m
FB 2.5 m
90 kN
20 kN
2.5 m
0.75 m
FA
0.75 m
x
SOLUTION
Equivalent Resultant Force: By equating the sum of the forces along the z axis to
the resultant force FR, Fig. b,
+ c FR = ©Fz;
y
3m
3m
0.75 m
- FR = - 30 - 20 - 90 - 35 - 40
FR = 215 kN
Ans.
Point of Application: By equating the moment of the forces and FR, about the x and
y axes,
(MR)x = ©Mx;
- 215(y) = - 35(0.75) - 30(0.75) - 90(3.75) - 20(6.75) - 40(6.75)
y = 3.68 m
(MR)y = ©My;
Ans.
215(x) = 30(0.75) + 20(0.75) + 90(3.25) + 35(5.75) + 40(5.75)
x = 3.54 m
Ans.
Ans:
FR = 215 kN
y = 3.68 m
x = 3.54 m
361
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4–133.
If the resultant force is required to act at the center of the
slab, determine the magnitude of the column loadings FA
and FB and the magnitude of the resultant force.
z
30 kN
0.75 m
FB 2.5 m
90 kN
20 kN
2.5 m
0.75 m
FA
0.75 m
x
SOLUTION
3m
Equivalent Resultant Force: By equating the sum of the forces along the z axis to
the resultant force FR,
+ c FR = ©Fz;
y
3m
0.75 m
- FR = - 30 - 20 - 90 - FA - FB
FR = 140 + FA + FB
(1)
Point of Application: By equating the moment of the forces and FR, about the x and
y axes,
(MR)x = ©Mx;
- FR(3.75) = -FB(0.75) - 30(0.75) - 90(3.75) - 20(6.75) - FA(6.75)
FR = 0.2FB + 1.8FA + 132
(MR)y = ©My;
(2)
FR(3.25) = 30(0.75) + 20(0.75) + 90(3.25) + FA(5.75) + FB(5.75)
FR = 1.769FA + 1.769FB + 101.54
(3)
Solving Eqs.(1) through (3) yields
FA = 30 kN
FB = 20 kN
FR = 190 kN
Ans.
Ans:
FA = 30 kN
FB = 20 kN
FR = 190 kN
362
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4–134.
Replace the two wrenches and the force, acting on the pipe
assembly, by an equivalent resultant force and couple
moment at point O.
100 N · m
300 N
z
C
SOLUTION
O
0.5 m
Force And Moment Vectors:
A
0.6 m
B
0.8 m
100 N
x
F1 = 5300k6 N
F3 = 5100j6 N
45°
F2 = 2005cos 45°i - sin 45°k6 N
200 N
180 N · m
= 5141.42i - 141.42k6 N
M 1 = 5100k6 N # m
M 2 = 1805cos 45°i - sin 45°k6 N # m
= 5127.28i - 127.28k6 N # m
Equivalent Force and Couple Moment At Point O:
FR = ©F;
FR = F1 + F2 + F3
= 141.42i + 100.0j + 1300 - 141.422k
= 5141i + 100j + 159k6 N
Ans.
The position vectors are r1 = 50.5j6 m and r2 = 51.1j6 m.
M RO = ©M O ;
M RO = r1 * F1 + r2 * F2 + M 1 + M 2
i
= 30
0
+
j
0.5
0
i
0
141.42
k
0 3
300
j
1.1
0
k
0
- 141.42
+ 100k + 127.28i - 127.28k
=
122i - 183k N # m
Ans.
Ans:
FR = 5141i + 100j + 159k6 N
MRO = 5122i - 183k6 N # m
363
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4–135.
z
Replace the force system by a wrench and specify the
magnitude of the force and couple moment of the wrench
and the point where the wrench intersects the x–z plane.
200 N
400 N
200 N
4
x
3
4
FR = e c 200 a b - 400 d i - 200j + 200 a bk f
5
5
= { -280i - 200j + 160k} N
The magnitude of FR is
Ans.
The direction of FR is deined by
uFR =
3m
2m
SOLUTION
FR = 2 ( - 280 ) 2 + ( - 200 ) 2 + 1602 = 379.47 N = 379 N
y
5
3
Resultant Force. Referring to Fig. a
0.5 m
O
- 280i - 200j + 160k
FR
=
= - 0.7379i - 0.5270j + 0.4216k
FR
379.47
Resultant Moment. The line of action of MR of the wrench is parallel to that of FR.
Also, assume that MR and FR have the same sense. Then
uMR = - 0.7379i - 0.5270j + 0.4216k
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4–135. Continued
Referring to Fig. a, where the origin of the x′, y′, z′ axes is the point where the
wrench intersects the xz plane,
(MR)x' = ΣMx'; - 0.7379 MR = - 200(z - 0.5)
(1)
2
4
(MR)y' = ΣMy'; -0.5270 MR = - 200 a b(z - 0.5) - 200 a b(3 - x) + 400(z - 0.5) (2)
5
5
(MR)z' = ΣMz'; 0.4216 MR = 200 x + 400(2)
(3)
Solving Eqs (1), (2) and (3)
MR = 590.29 N # m = 590 N # m
Ans.
z = 2.6778 m = 2.68 m
Ans.
x = -2.7556 m = - 2.76 m
Ans.
Ans:
FR = 379 N
MR = 590 N # m
z = 2.68 m
x = -2.76 m
365
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*4–136.
z
Replace the five forces acting on the plate by a wrench.
Specify the magnitude of the force and couple moment for
the wrench and the point P(x, z) where the wrench intersects
the x–z plane.
800 N
4m
2m
4m
2m
200 N
400 N
SOLUTION
x
Resultant Force. Referring to Fig. a
FR = { -600i - (300 + 200 + 400)j - 800k} N
= 5-600i - 900j - 800k6 N
Then the magnitude of FR is
FR = 2( - 600)2 + ( - 900)2 + ( - 800)2 = 1345.36 N = 1.35 kN
Ans.
The direction of FR is deined by
uFR =
- 600i - 900j - 800k
FR
=
= -0.4460i - 0.6690j - 0.5946k
FR
1345.36
Resultant Moment.
The line of action of MR of the wrench is parallel to that of FR. Also, assume that
both MR and FR have the same sense. Then
uMR = - 0.4460i - 0.6690j - 0.5946k
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600 N
y
300 N
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*4–136. Continued
Referring to Fig. a,
(MR)x′ = ΣMx′;
- 0.4460 MR = - 300z - 200(z - 2) - 400z
(1)
(MR)y′ = ΣMy′;
- 0.6690 MR = 800(4 - x) + 600z
(2)
(MR)z′ = ΣMz′;
- 0.5946 MR = 200(x - 2) + 400x - 300(4 - x)
(3)
Solving Eqs (1), (2) and (3)
MR = - 1367.66 N # m = - 1.37 kN # m
Ans.
x = 2.681 m = 2.68 m
Ans.
z = -0.2333 m = - 0.233 m
Ans.
The negative sign indicates that the line of action of MR is directed in the opposite
sense to that of FR.
Ans:
MR = - 1.37 kN # m
x = 2.68 m
z = - 0.233 m
367
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4–137.
z
Replace the three forces acting on the plate by a wrench.
Specify the magnitude of the force and couple moment for
the wrench and the point P(x, y) where the wrench intersects
the plate.
FB { 300k} N
FC {200j} N
C
y
B
x
P
x
SOLUTION
5m
3m
Resultant Force. Referring to Fig. a,
A
FR = {400i + 200j - 300k} N
FA {400i} N
Then, the magnitude of FR is
FR = 24002 + 2002 + ( - 300)2 = 538.52 N = 539 N
Ans.
The direction of FR is deined by
uFR =
400i + 200j - 300k
FR
=
= 0.7428i + 0.3714j - 0.5571k
FR
538.52
Resultant Moment. The line of action of MR of the wrench is parallel to that of FR.
Also, assume that both MR and FR have the same sense. Then
uMR = 0.7428i + 0.3714j - 0.5571k
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4–137. Continued
Referring to Fig. a,
(MR)x′ = ΣMx′; 0.7428 MR = 300y
(1)
(MR)y′ = ΣMy′; 0.3714 MR = 300 (3 - x)
(2)
(MR)z′ = ΣMz; -0.5571 MR = - 200x - 400 (5 - y)
(3)
Solving Eqs (1), (2) and (3)
MR = 1448.42 N # m = 1.45 kN # m
Ans.
x = 1.2069 m = 1.21 m
Ans.
y = 3.5862 m = 3.59 m
Ans.
Ans:
FR = 539 N
MR = 1.45 kN # m
x = 1.21 m
y = 3.59 m
369
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4–138.
Replace the loading by an equivalent resultant force and
couple moment acting at point O.
50 lb/ft
9 ft
O
9 ft
50 lb/ft
SOLUTION
+ c FR = ΣF ;
FR = 0
Ans.
a+ MRO = ΣMO ;
MRO = 225 (6) = 1350 lb # ft = 1.35 kip # ft
Ans.
Ans:
FR = 0
MRO = 1.35 kip # ft
370
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4–139.
Replace the distributed loading with an equivalent resultant
force, and specify its location on the beam measured from
point O.
3 kN/m
O
3m
SOLUTION
1.5 m
Loading: The distributed loading can be divided into two parts as shown in Fig. a.
Equations of Equilibrium: Equating the forces along the y axis of Figs. a and b,
we have
+ T FR = ©F;
FR =
1
1
(3)(3) + (3)(1.5) = 6.75 kN T
2
2
Ans.
If we equate the moment of FR, Fig. b, to the sum of the moment of the forces in
Fig. a about point O, we have
a + (MR)O = ©MO;
1
1
(3)(3)(2) - (3)(1.5)(3.5)
2
2
x = 2.5 m
- 6.75(x) = -
Ans.
Ans:
FR = 6.75 kN
x = 2.5 m
371
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*4–140.
w
Replace the loading by an equivalent resultant force and
specify its location on the beam, measured from point A.
5 kN/m
2 kN/m
A
x
B
4m
2m
SOLUTION
Equivalent Resultant Force. Summing the forces along the y axis by referring to
Fig. a
+c (FR)y = ΣFy;
-FR = -2(6) -
1
(3)(6)
2
Ans.
FR = 21.0 kNT
Ans.
Location of the Resultant Force. Summing the moments about point A,
1
a + (MR)A = ΣMA; - 21.0(d) = - 2(6)(3) - (3)(6)(4)
2
d = 3.429 m = 3.43 m
Ans.
Ans:
FR = 21.0 kN
d = 3.43 m
372
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4–141.
Currently eighty-five percent of all neck injuries are caused
by rear-end car collisions. To alleviate this problem, an
automobile seat restraint has been developed that provides
additional pressure contact with the cranium. During
dynamic tests the distribution of load on the cranium has
been plotted and shown to be parabolic. Determine the
equivalent resultant force and its location, measured from
point A.
A
12 lb/ft
0.5 ft
w
w 12(1 2x2) lb/ft
B
18 lb/ft
x
SOLUTION
FR =
x =
L
L
w(x) dx =
x w(x) dx
L
=
w(x)dx
L0
12 A 1 + 2 x2 B dx = 12 cx +
2 3 0.5
x d = 7 lb
3
0
x(12) A 1 + 2 x2 B dx
x4 0.5
x2
+ (2) d
2
4 0
7
0.5
L0
0.5
7
12c
=
x = 0.268 ft
Ans.
Ans.
Ans:
FR = 7 lb
x = 0.268 ft
373
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4–142.
4 kN/m
Replace the distributed loading by an equivalent resultant
force, and specify its location on the beam, measured from
the pin at A.
2 kN/m
A
SOLUTION
B
3m
3m
Equivalent Resultant Force. Summing the forces along the y axis by referring to
Fig. a,
+c (FR)y = ΣFy;
-FR = -2(6) -
1
(2)(3)
2
FR = 15.0 kN T
Ans.
Location of the Resultant Force. Summing the Moments about point A,
a + (MR)A = ΣMA;
- 15.0(d) = - 2(6)(3) -
1
(2)(3)(5)
2
d = 3.40 m
Ans.
Ans:
FR = 15.0 kN
d = 3.40 m
374
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4–143.
Replace this loading by an equivalent resultant force and
specify its location, measured from point O.
6 kN/m
4 kN/m
O
2m
1.5 m
SOLUTION
Equivalent Resultant Force. Summing the forces along the y axis by referring to
Fig. a,
+c (FR)y = ΣFy;
- FR = - 4(2) -
1
(6)(1.5)
2
FR = 12.5 kN
Ans.
Location of the Resultant Force. Summing the Moment about point O,
a + (MR)O = ΣMO;
-12.5(d) = - 4(2)(1) -
1
(6)(1.5)(2.5)
2
d = 1.54 m
Ans.
Ans:
FR = 12.5 kN
d = 1.54 m
375
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*4–144.
The distribution of soil loading on the bottom of a
building slab is shown. Replace this loading by an
equivalent resultant force and specify its location, measured
from point O.
O
50 lb/ft
100 lb/ft
12 ft
300 lb/ft
9 ft
SOLUTION
+ c FR = ©Fy; FR = 50(12) + 21 (250)(12) + 21 (200)(9) + 100(9)
= 3900 lb = 3.90 kip c
a + MRo = ©MO;
Ans.
3900(d) = 50(12)(6) + 21 (250)(12)(8) + 12 (200)(9)(15) + 100(9)(16.5)
d = 11.3 ft
Ans.
Ans:
FR = 3.90 kip c
d = 11.3 ft
376
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4–145.
8 kN/m
Replace the loading by an equivalent resultant force and
couple moment acting at point O.
5 kN/m
O
1.5 m
0.75 m
0.75 m
SOLUTION
Equivalent Resultant Force And Couple Moment About Point O. Summing the
forces along the y axis by referring to Fig. a,
+c (FR)y = ΣFy;
1
1
FR = - (3)(1.5) - 5(2.25) - (5)(0.75)
2
2
= -15.375 kN = 15.4 kN T
Ans.
Summing the Moment about point O,
a + (MR)O = ΣMO;
1
(MR)O = - (3)(1.5)(0.5) - 5(2.25)(1.125)
2
1
- (5)(0.75)(2.5)
2
= -18.46875 kN # m = 18.5 kN # m (clockwise) Ans.
Ans:
FR = 15.4 kN
(MR)O = 18.5 kN # m (clockwise)
377
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4–146.
6 kN/m
Replace the distributed loading by an equivalent resultant
force and couple moment acting at point A.
6 kN/m
3 kN/m
A
B
3m
3m
SOLUTION
Equivalent Resultant Force And Couple Moment About Point A. Summing the
forces along the y axis by referring to Fig. a,
+ c (FR)y = ΣFy;
1
1
FR = - (3)(3) - 3(6) - (3)(3)
2
2
= - 27.0 kN = 27.0 kN T
Ans.
Summing the moments about point A,
a+ (MR)A = ΣMA;
1
1
(MR)A = - (3)(3)(1) - 3(6)(3) - (3)(3)(5)
2
2
= - 81.0 kN # m = 81.0 kN # m (clockwise)
Ans.
Ans:
FR = 27.0 kN
(MR)A = 81.0 kN # m (clockwise)
378
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4–147.
Determine the length b of the triangular load and its
position a on the beam such that the equivalent resultant
force is zero and the resultant couple moment is 8 kN # m
clockwise.
a
b
4 kN/m
A
2.5 kN/m
9m
SOLUTION
1
1
(2.5)(9) - (4)(b)
2
2
+ c FR = 0 = ΣFy ;
0 =
a+MRA = ΣMA;
1
1
2
- 8 = - (2.5)(9)(6) + (4)(5.625) aa + (5.625) b
2
2
3
b = 5.625 m
a = 1.54 m
Ans.
Ans.
Ans:
a = 1.54 m
379
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*4–148.
The form is used to cast a concrete wall having a width
of 5 m. Determine the equivalent resultant force the wet
concrete exerts on the form AB if the pressure distribution
due to the concrete can be approximated as shown. Specify
the location of the resultant force, measured from point B.
B
p
1
p (4z 2 ) kPa
4m
SOLUTION
L
dA =
L0
4
A
1
2
8 kPa
4z dz
4
3
2
= c (4)z2 d
3
0
z
= 21.33 kN>m
FR = 21.33(5) = 107 kN
L
zdA =
L0
4
Ans.
3
4z2 dz
4
5
2
= c (4)z2 d
5
0
= 51.2 kN
z =
51.2
= 2.40 m
21.33
Ans.
Also, from the back of the book,
A =
2
2
ab = (8)(4) = 21.33
3
3
FR = 21.33 (5) = 107 kN
Ans.
z = 4 - 1.6 = 2.40 m
Ans.
Ans:
FR = 107 kN
z = 2.40 m
380
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4–149.
If the soil exerts a trapezoidal distribution of load on the
bottom of the footing, determine the intensities w1 and w2 of
this distribution needed to support the column loadings.
80 kN
60 kN
1m
50 kN
2.5 m
SOLUTION
3.5 m
1m
w2
Loading: The trapezoidal reactive distributed load can be divided into two parts
as shown on the free-body diagram of the footing, Fig. a. The magnitude and location measured from point A of the resultant force of each part are also indicated in
Fig. a.
w1
Equations of Equilibrium: Writing the moment equation of equilibrium about
point B, we have
a + ©MB = 0; w2(8) ¢ 4 -
8
8
8
8
≤ + 60 ¢ - 1 ≤ - 80 ¢ 3.5 - ≤ - 50 ¢ 7 - ≤ = 0
3
3
3
3
w2 = 17.1875 kN>m = 17.2 kN>m
Ans.
Using the result of w2 and writing the force equation of equilibrium along the
y axis, we obtain
+ c ©Fy = 0;
1
(w - 17.1875)8 + 17.1875(8) - 60 - 80 - 50 = 0
2 1
w1 = 30.3125 kN>m = 30.3 kN>m
Ans.
Ans:
w2 = 17.2 kN>m
w1 = 30.3 kN>m
381
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4–150.
Replace the loading by an equivalent force and couple
moment acting at point O.
6 kN/m
15 kN
500 kN m
O
7.5 m
4.5 m
SOLUTION
+ c FR = ©Fy ;
FR = -22.5 - 13.5 - 15.0
= -51.0 kN = 51.0 kN T
a + MRo = ©Mo ;
Ans.
MRo = -500 - 22.5(5) - 13.5(9) - 15(12)
= - 914 kN # m
= 914 kN # m (Clockwise)
Ans.
Ans:
FR = 51.0 kN T
MRO = 914 kN # m b
382
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4–151.
Replace the loading by a single resultant force, and specify
the location of the force measured from point O.
6 kN/m
15 kN
500 kN m
O
7.5 m
4.5 m
SOLUTION
Equivalent Resultant Force:
+ c FR = ©Fy ;
- FR = - 22.5 - 13.5 - 15
FR = 51.0 kN T
Ans.
Location of Equivalent Resultant Force:
a + (MR)O = ©MO ;
- 51.0(d) = -500 - 22.5(5) - 13.5(9) - 15(12)
d = 17.9 m
Ans.
Ans:
FR = 51.0 kN T
d = 17.9 m
383
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*4–152.
Replace the loading by an equivalent resultant force and
couple moment acting at point A.
400 N/m
B
A
3m
3m
SOLUTION
Equivalent Resultant Force And Couple Moment At Point A. Summing the forces
along the y axis by referring to Fig. a,
+ c (FR)y = ΣFy;
FR = - 400(3) -
1
(400)(3)
2
= -1800 N = 1.80 kN T
Ans.
Summing the moment about point A,
a+ (MR)A = ΣMA;
(MR)A = - 400(3)(1.5) -
1
(400)(3)(4)
2
= -4200 N # m = 4.20 kN # m (clockwise) Ans.
Ans:
FR = 1.80 kN
(MR)A = 4.20 kN # m (clockwise)
384
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4–153.
400 N/m
Replace the loading by a single resultant force, and specify
its location on the beam measured from point A.
B
A
3m
3m
SOLUTION
Equivalent Resultant Force. Summing the forces along the y axis by referring
to Fig. a,
+ c (FR)y = ΣFy;
-FR = -400(3) -
1
(400)(3)
2
FR = 1800 N = 1.80 kN T
Ans.
Location of Resultant Force. Summing the moment about point A by referring
to Fig. a,
a+ (MR)A = ΣMA;
- 1800 d = -400(3)(1.5) -
1
(400)(3)(4)
2
d = 2.333 m = 2.33 m
Ans.
Ans:
FR = 1.80 kN
d = 2.33 m
385
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4–154.
3 kN/m
Replace the distributed loading by an equivalent resultant
force and specify where its line of action intersects a
horizontal line along member AB, measured from A.
B
A
3m
2 kN/m
4m
SOLUTION
Equivalent Resultant Force. Summing the forces along the x and y axes by referring
to Fig. a,
+ (FR)x = ΣFx;
S
(FR)x = - 2(4) = - 8 kN = 8 kN d
+ c (FR)y = ΣFy;
(FR)y = - 3(3) = - 9 kN = 9 kN T
Then
FR = 2(FR)2x + (FR)2y = 282 + 92 = 12.04 kN = 12.0 kN
Ans.
u = tan-1 c
Ans.
And
(FR)y
(FR)x
9
d = tan-1 a b = 48.37° = 48.4° d
8
Location of the Resultant Force. Summing the moments about point A, by referring
to Fig. a,
a+ (MR)A = ΣMA ;
- 8x - 9y = -3(3)(1.5) - 2(4)(2)
8x + 9y = 29.5
(1)
386
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4–154. Continued
Along AB, x = 0. Then Eq (1) becomes
8(0) + 9y = 29.5
y = 3.278 m
Thus, the inter section point of line of action of FR on AB measured to the right from
point A is
d = y = 3.28 m
Ans.
Ans:
FR = 12.0 kN
u = 48.4° d
d = 3.28 m
387
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4–155.
3 kN/m
Replace the distributed loading by an equivalent resultant
force and specify where its line of action intersects a vertical
line along member BC, measured from C.
B
A
3m
2 kN/m
4m
SOLUTION
Equivalent Resultant Force. Summing the forces along the x and y axes by referring
to Fig. a,
+ (FR)x = ΣFx;
S
(FR)x = - 2(4) = - 8 kN = 8 kN d
+ c (FR)y = ΣFy;
(FR)y = - 3(3) = - 9 kN = 9 kN T
Then
FR = 2(FR)2x + (FR)2y = 282 + 92 = 12.04 kN = 12.0 kN
Ans.
u = tan-1 c
Ans.
And
(FR)y
(FR)x
9
d = tan-1 a b = 48.37° = 48.4° d
8
Location of the Resultant Force. Summing the moments about point A, by referring
to Fig. a,
a+ (MR)A = ΣMA;
- 8x - 9y = - 3(3)(1.5) - 2(4)(2)
8x + 9y = 29.5
(1)
388
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4–155. Continued
Along BC, y = 3 m. Then Eq (1) becomes
8x + 9(3) = 29.5
x = 0.3125 m
Thus, the intersection point of line of action of FR on BC measured upward from
point C is
d = 4 - x = 4 - 0.3125 = 3.6875 m = 3.69 m
Ans.
Ans:
FR = 12.0 kN
u = 48.4° d
d = 3.69 m
389
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*4–156.
Determine the length b of the triangular load and its
position a on the beam such that the equivalent resultant
force is zero and the resultant couple moment is 8 kN # m
clockwise.
a
b
6 kN/m
A
2 kN/m
4m
SOLUTION
Equivalent Resultant Force And Couple Moment At Point A. Summing the forces
along the y axis by referring to Fig. a, with the requirement that FR = 0,
+ c (FR)y = ΣFy;
0 = 2(a + b) -
1
(6)(b)
2
2a - b = 0
(1)
Summing the moments about point A, with the requirement that (MR)A = 8 kN # m,
a+ (MR)A = ΣMA; -8 = 2(a + b) c4 -
1
1
1
(a + b)d - (6)(b) a4 - bb
2
2
3
- 8 = 8a - 4b - 2ab - a2
(2)
Solving Eqs (1) and (2),
a = 1.264 m = 1.26 m
Ans.
b = 2.530 m = 2.53 m
Ans.
Ans:
a = 1.26 m
b = 2.53 m
390
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4–157.
w
Determine the equivalent resultant force and couple
moment at point O.
9 kN/m
w ( 13 x3 ) kN/m
O
x
3m
SOLUTION
Equivalent Resultant Force And Couple Moment About Point O. The differential
1
force indicated in Fig. a is dFR = w dx = x3dx. Thus, summing the forces along the
3
y axis,
3m
1 3
+ c (FR)y = ΣFy;
FR = - dFR = x dx
L
L0 3
= -
1 4 3m
x
12 L 0
= - 6.75 kN = 6.75 kNT
Ans.
Summing the moments about point O,
a+ (MR)O = ΣMO;
(MR)O =
=
=
L
(3 - x)dFR
1
(3 - x) a x3dxb
3
L0
3m
L0
3m
= a
ax3 -
1 4
x b dx
3
1 5 3m
x4
x b`
4
15
0
= 4.05 kN # m (counterclockwise)
Ans.
Ans:
FR = 6.75 kNT
(MR)O = 4.05 kN # m (counterclockwise)
391
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4–158.
w
Determine the magnitude of the equivalent
resultant force and its location, measured from
point O.
w (4 2 x ) lb/ft
8.90 lb/ft
4 lb/ft
x
O
6 ft
SOLUTION
dA = wdx
FR =
L
dA =
L0
6
( 4 + 21x ) dx
= c 4x +
4 3 6
x2 d
3 0
FR = 43.6 lb
L
xdF =
L0
6
Ans.
3
( 4x + 2x2 ) dx
= c 2x2 +
4 5 6
x2 d
5 0
= 142.5 lb # ft
x =
142.5
= 3.27 ft
43.6
Ans.
Ans:
FR = 43.6 lb
x = 3.27 ft
392
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4–159.
w
The distributed load acts on the shaft as shown. Determine
the magnitude of the equivalent resultant force and specify
its location, measured from the support, A.
28 lb/ft
w (2x¤ 8x 18) lb/ft
18 lb/ft
10 lb/ft
x
A
B
1 ft
2 ft
2 ft
SOLUTION
4
FR =
L
L-1
2
3
4
x dF =
x =
L-1
4
8x
+ 18 x ` = 73.33 = 73.3 lb
2
-1
2
( 2 x2 - 8 x + 18 ) dx = x3 -
2
4
8
3
( 2 x3 - 8 x2 + 18 x ) dx = x4 - x3 +
Ans.
18 2 4
x ` = 89.166 lb # ft
2
-1
89.166
= 1.22 ft
73.3
d = 1 + 1.22 = 2.22 ft
Ans.
Ans:
d = 2.22 ft
393
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*4–160.
Replace the distributed loading with an equivalent resultant
force, and specify its location on the beam measured from
point A.
w
w (x2 3x 100) lb/ft
370 lb/ft
100 lb/ft
A
x
B
SOLUTION
15 ft
Resultant: The magnitude of the differential force dFR is equal to the area of the element shown shaded in Fig. a. Thus,
dFR = w dx = a x2 + 3x + 100 b dx
Integrating dFR over the entire length of the beam gives the resultant force FR.
+T
FR =
LL
dFR =
L0
L
a x2 + 3x + 100 b dx = ¢
= 2962.5 lb = 2.96 kip
15 ft
x3
3x2
+
+ 100x ≤ `
3
2
0
Ans.
Location: The location of dFR on the beam is xc = x measured from point A. Thus,
the location x of FR measured from point A is given by
x =
LL
xcdFR
LL
=
dFR
L0
15 ft
xax2 + 3x + 100bdx
2962.5
=
¢
15 ft
x4
+ x3 + 50x2 ≤ `
4
0
2962.5
= 9.21 ft Ans.
Ans:
FR = 2.96 kip
x = 9.21 ft
394
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4–161.
w
Replace the loading by an equivalent resultant force and
couple moment acting at point O.
p x
w w0 cos ( 2L
(
x
O
L
SOLUTION
Equivalent Resultant Force And Couple Moment About Point O. The differential
p
xbdx. Thus, summing the
force indicated in Fig. a is dFR = w dx = aw0 cos
2L
forces along the y axis,
+ c (FR)y = ΣFy;
FR = -
L
L
dFR = -
LO
¢w0 cos
= -
L
2Lw0
p
asin
xb `
p
2L
O
= -
2Lw0
2Lw0
=
T
p
p
p
x≤dx
2L
Ans.
Summing the moments about point O,
a+ (MR)O = ΣMO;
(MR)O = -
L
xdFR
x aw0 cos
L
= -
LO
= -w0 a
= -a
= a
p
xb dx
2L
L
p
2L
4L2
p
x + 2 cos
xb `
x sin
p
2L
2L
p
O
2p - 4
bw0L2
p2
2p - 4
bw0L2 (clockwise)
p2
Ans.
Ans:
FR =
2Lw0
p
(MR)O = a
395
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bw0L2 (clockwise)
p2
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4–162.
Wet concrete exerts a pressure distribution along the wall
of the form. Determine the resultant force of this
distribution and specify the height h where the bracing strut
should be placed so that it lies through the line of action of
the resultant force. The wall has a width of 5 m.
p
4m
p
1
(4 z /2) kPa
SOLUTION
Equivalent Resultant Force:
+ F = ©F ;
:
R
x
-FR = - LdA = L0
FR =
4m
L0
h
z
8 kPa
wdz
a 20z2 b A 103 B dz
1
z
= 106.67 A 103 B N = 107 kN ;
Ans.
Location of Equivalent Resultant Force:
z =
LA
zdA
LA
=
dA
=
=
L0
z
zwdz
L0
L0
z
wdz
4m
L0
L0
1
4m
4m
L0
z c A 20z2 B (103) d dz
A 20z2 B (103)dz
1
c A 20z2 B (10 3) d dz
4m
3
A 20z2 B (103)dz
1
= 2.40 m
Thus,
h = 4 - z = 4 - 2.40 = 1.60 m
Ans.
Ans:
FR = 107 kN
h = 1.60 m
396
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