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Problem 6.89 The woman using the exercise machine
is holding the 80-lb weight stationary in the position
shown. What are the reactions at the built-in support E
and the pin support F?(Aand Care pinned connections.)
EF
80 lb
A
B
D
2 ft 2 in
60⬚
2 ft
C
1 ft 6 in
9 in
466
Problem 6.90 Determine the reactions on member
ABC at Aand B.
E
BC
D
A
9 in
8 in
80 lb
Problem 6.91 The mass of the suspended object is
mD50 kg. Determine the reactions on member ABC.
DE0.2 m
0.6 m
0.8 m
0.2 m
C
0.6 m
AB
m
Solution: Begin with an examination of the pulley at B.
Finally look at member ABC
468
Problem 6.92 The unstretched length of the string is
LO. Show that when the system is in equilibrium the
angle ˛satisfies the relation sin ˛D2⊲LO2F/k⊳L.
k
F
1
–
2L
1
–
4L
1
–
4L
αα
The sum of the moments about A
TDkL
2sin ˛LO
Element BE: The strategy is to determine Cy, which is the spring force
on BE. The moment about Eis
MEDL
4Cycos ˛L
2Bycos ˛L
2Bxcos ˛D0,
from which Cy
2CByDBx. The sum of forces:
FxDBxD0,
from which BxD0.
FyDCyCByCED0,
from which CyCByDEDF. The two simultaneous equations
are solved: CyD2F, and ByDF.
from which kL
2sin ˛LOD2F.
2LO2F
α
E
L
4L
4
Problem 6.93 The pin support Bwill safely support a
force of 24-kN magnitude. Based on this criterion, what
is the largest mass mthat the frame will safely support?
C
B
500 mm
100 mm
300 mm
m
300 mm 400 mm 400 mm
D
E
AF
Solution: The weight is given by WDmg D9.81 g
CyCy
470
Problem 6.94 Determine the reactions at Aand C.
3 ft
A
72 ft-lb
36 lb
C
B
3 ft
18 lb
Problem 6.95 Determine the forces on member AD.
400 N
200 N
C
D
130 mm
400 mm
B
A
FyDCCRy400 200 D0,
472
Problem 6.96 The frame shown is used to support
high-tension wires. If bD3 ft, ˛D30°, and WD
200 lb, what is the axial force in member HJ?B
A
C
D
E
F
G
J
H
IW
W
W
bbbb
αα
α
α
Solution: Joints Band Eare sliding joints, so that the reactions
are normal to AC and BF, respectively. Member HJ is supported by
pins at each end, so that the reaction is an axial force. The distance
hDbtan ˛D1.732 ft
Member ABC. The sum of the forces:
Ay
Dy
Gy
Gx
Dx
Ax
B
B
E
b
W
h
Problem 6.97 Determine the force exerted on the ball
by the bolt cutters and the magnitude of the axial force
in the two-force member AB.
20 in
A
20 lb
B
474
Problem 6.98 The woman exerts 20-N forces to the
pliers as shown.
(a) What is the magnitude of the forces the pliers exert
on the bolt at B?
(b) Determine the magnitude of the force the members
of the pliers exert on each other at the pinned
connection C.
25 mm 80 mm
Problem 6.99 Figure a is a diagram of the bones and
biceps muscle of a person’s arm supporting a mass.
Tension in the biceps muscle holds the forearm in the
horizontal position, as illustrated in the simple mechan-
ical model in Fig. b. The weight of the forearm is 9 N,
and the mass mD2 kg.
(a) Determine the tension in the biceps muscle AB.
(b) Determine the magnitude of the force exerted on
the upper arm by the forearm at the elbow joint C.
(a)
A
B
C
290
mm
50
mm
150 mm
9 N
200 mm
m
(b)
Solution: Make a cut through AB and BC just above the elbow
joint C. The angle formed by the biceps muscle with respect to the
forearm is ˛Dtan1290
50 D80.2°. The weight of the mass is WD
2⊲9.81⊳D19.62 N.
W
T
200
mm
150
mm
50
mm
9N
Cy
Cx
α
476
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 6.100 The bones and the tendons in a horse’s
rear leg are shown in Fig. a. A biomechanical model of
the leg is shown in Fig. b. If the horse is stationary and
the normal force everted on its leg by the ground is
ND1200 N, determine the tension s in the superficial
digital flexor BC and the patellar ligament DF.
40 cm
72 cm
8
cm
10
cm
8
cm
A
N
B
C
D
E
F
6 cm
6 cm
6 cm
3 cm
(a) (b)
Solution: The free-body diagrams for AB and AE are shown. The
Problem 6.101 The pressure force exerted on the
piston is 2 kN toward the left. Determine the couple
Mnecessary to keep the system in equilibrium.
A
B
MC
400 mm
350 mm300 mm
45°
Solution: From the diagram, the coordinates of point Bare ⊲d, d⊳
where dD0.3 cos⊲45°⊳. The distance bcan be determined from the
Pythagorean Theorem as bD⊲0.35⊳2d2. From the diagram, the
angle D37.3°. From these calculations, the coordinates of points B
and Care B(0.212, 0.212), and C(0.491, 0) with all distances being
measured in meters. All forces will be measured in Newtons.
The unit vector from Ctoward Bis uCB D0.795iC0.606j.
AC
B
d
d
b
45°
θ
y
478
Problem 6.102 In Problem 6.101, determine the forces
on member AB at Aand B.
Solution: In the solution of Problem 6.101, we found that the force
and that the moment acting on member BC about point Awas given by
MD747.6kN-m (clockwise). Member AB must be in equilibrium,
and we ensured moment equilibrium in solving Problem 6.101.
From the free body diagram, the equations for force equilibrium are
FXDAXCFBCuBCX DAX2000 N D0,
and FYDAYCFBCuBCY DAYC1524 N D0.
Thus, AXD2000 N, and AYD1524 N.
y
x
AX
AY
A
B
M
Problem 6.103 In Example 6.8, suppose that the
object being held by the plier’s is moved to the left
so that the horizontal distance from Dto the object at
Edecreases from 30 mm to 20 mm. Draw a sketch of
the pliers showing the new position of the object. What
forces are exerted on the object at Eas a result of the
150-N forces on the pliers?
150 N 70 mm 30 mm
30 mm 30 mm
30 mm
BD
150 N
A
C
E
Problem 6.104 The shovel of the excavator is
supported by a pin support at Eand the two-force
member BC. The 300-lb weight Wof the shovel acts at
the point shown. Determine the reactions on the shovel
at Eand the magnitude of the axial force in the two-force
member BC.
W
12 in
15 in
3 in
Shovel
Hydraulic
cylinder
12 in
C
B
A
D
E
7 in 20 in
Solution: The angle
480
Problem 6.105 The shovel of the excavator has a pin
support at E. The position of the shovel is controlled by
the horizontal hydraulic piston AB, which is attached to
the shovel through a linkage of the two-force members
BC and BD. The 300-lb weight Wof the shovel acts
at the point shown. What is the magnitude of the force
the hydraulic piston must exert to hold the shovel in
equilibrium?
W
12 in
15 in
3 in
Shovel
Hydraulic
cylinder
12 in
C
B
A
D
E
7 in 20 in
Solution: From the solution to Problem 6.104 we know that the
force in member BC is
Problem 6.106 The woman exerts 20-N forces on the
handles of the shears. Determine the magnitude of the
forces exerted on the branch at A.
B
AC
D
E
25 mm25 mm36 mm
65 mm 20 N
20 N
482
Problem 6.107 The person exerts 40-N forces on the
handles of the locking wrench. Determine the magnitude
of the forces the wrench exerts on the bolt at A.
AB
E
D
C
50 mm 30 mm 75 mm
8 mm 40 mm
40 N
40 N
Solution: Recognize that DE is a 2-force member. Examine
CxD1312.5N,C
yD100 N,F
DE D1320 N
Now examine ABC
MB:A⊲50 mm⊳Cx⊲40 mm⊳D0
Fx:BxCxD0
Fy:ByCyAD0
Solving: AD1050 N,B
xD1312.5N ByD1150 N
Answer: AD1050 N
FDE
75
A
Cx
Cy
Bx
Problem 6.108 In Problem 6.107, determine the
magnitude of the force the members of the wrench exert
on each other at Band the axial force in the two-force
member DE.
Solution: From the previous problem we have
Problem 6.109 The device is designed to exert a large
force on the horizontal bar at Afor a stamping operation.
If the hydraulic cylinder DE exerts an axial force of
800 N and ˛D80°, what horizontal force is exerted on
the horizontal bar at A?
A
90°
250 mm
250 mm
250 mm
B
D
CE
400 mm
α
Solution: Define the x-ycoordinate system with origin at C. The
projection of the point Don the coordinate system is
Py
Fy
Fx
B
D
484
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
Problem 6.110 This device raises a load Wby
extending the hydraulic actuator DE. The bars AD and
BC are 4 ft long, and the distances bD2.5 ft and hD
1.5 ft. If WD300 lb, what force must the actuator exert
to hold the load in equilibrium? A
C
B
DE
b
h
W
Solution: The angle ADC is ˛Dsin1h
4D22.02°. The
distance CD is dD4 cos ˛.
The complete structure as a free body: The sum of the forces:
FyDWCCyCDyD0.
CyD97.7lb,
DyD202.3lb,
and CxDDx.
Divide the system into three elements: the platform carrying the
FyDACBCWD0.
(1) dCxhCydB D0. The sum of the forces:
FxDCxExD0, from which
(2) ExCxD0,
A
E
Cx
DxDy
Cy
Ex
Ey
(3) CyEyCBD0
(4) dDyChDxdA D0.
These are four equations in the four unknowns: EX,EY,Dx,CXand
DX
