Mechanical Engineering Chapter 5 Problem The Hydraulic Piston Exerts Hori Zontal Force Support The Weight The

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subject Pages 13
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subject Authors Anthony M. Bedford, Wallace Fowler

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page-pf1
Problem 5.129 The hydraulic piston exerts a hori-
zontal force at Bto support the weight WD1500 lb
of the bucket of the excavator. Determine the magnitude
of the force the hydraulic piston must exert. (The vector
sum of the forces exerted at Bby the hydraulic piston,
the two-forces member AB, and the two-force member
BD must equal zero.)
14 in
CD
A
16 in
4 in
B
Solution: See the solution to Problem 5.23.
Problem 5.130 The member ACG of the front-end
loader is subjected to a load WD2 kN and is supported
by a pin support at Aand the hydraulic cylinder BC.
Treat the hydraulic cylinder as a two-force member.
(a) Draw a free-body diagrams of the hydraulic
cylinder and the member ACG.
(b) Determine the reactions on the member ACG.
0.5 m
1 m
0.75 m
1.5 m 1.5 m
W
G
C
A
B
Solution: This is a very simple Problem. The free body diagrams
Using the given value for Wand solving these equations, we get
BX
W
CX
1.5 m 1.5 m
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Problem 5.131 In Problem 5.130, determine the reac-
tions of the member ACG by using the fact that it is a
three-force member.
Solution: The easiest way to do this is take advantage of the fact
that for a three force member, the three forces must be concurrent. The
fact that the force at Cis horizontal and the weight is vertical make
it very easy to nd the point of concurrency. We then use this point
to determine the direction of the force through A. We can even know
geometry, we can determine the angle between the force Aand the
A
A
y
x
G
0.75 m
1.5 m 1.5 m
θ
368
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Problem 5.132 A rectangular plate is subjected to
two forces Aand B(Fig. a). In Fig. b, the two forces
are resolved into components. By writing equilibrium
equations in terms of the components Ax,Ay,Bx, and By,
show that the two forces Aand Bare equal in magnitude,
opposite in direction, and directed along the line between
their points of application. A
(a)
h
b
B
B
A
A
(b)
h
b
B
By
Bx
Axx
y
Ay
Solution: The sum of forces:
from which AXDBX
FYDAYCBYD0,
A
b
B
ByBx
y
Ay
Fig a
Fig b
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Problem 5.133 An object in equilibrium is subjected
to three forces whose points of application lie on a
straight line. Prove that the forces are coplanar.
F1
F2
F3
Solution: The strategy is to show that for a system in equilibrium
forces. The sum of the moments about the point P:
where the vectors are the position vectors of the points of the applica-
tion of the forces relative to the point P. (The position vectors lie in
the plane.) Dene
d12,d13 are parallel to the line L.) The component of the moment
parallel to the line Lis
But by denition, F2lies in the same plane as the line L, hence it is
normal to the cross product d12 ðe6D 0, and the term
F2Ðd12 ðeD0.
But this means that
F3Ðd13 ðeeD0,
which implies that F3also lies in the same plane as F2, since
d13 ðe6D 0.
Thus the two forces lie in the same plane. Since the choice of the point
about which to sum the moments was arbitrary, this process can be
repeated to show that F1lies in the same plane as F2.Thus all forces
lie in the same plane.
PL
370
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.
page-pf5
Problem 5.134 The suspended cable weighs 12 lb.
(a) Draw the free-body diagram of the cable. (The
tensions in the cable at Aand Bare not equal.)
(b) Determine the tensions in the cable at Aand B.
(c) What is the tension in the cable at its lowest point?
A
B
32
50
Solution:
(a) The FBD
(b) The equilibrium equations
50°
TA
TB
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Problem 5.135 Determine the reactions at the xed
support.
3m
A
4 kN
20 kN-m
Solving
AxD4kN,A
yD5kN,M
AD26 kN-m
3 kN
Problem 5.136 (a) Draw the free-body diagram of the
50-lb plate, and explain why it is statically indeterminate.
(b) Determine as many of the reactions at Aand Bas
possible.
12 in
A
20 in
y
Solution:
(a) The pin supports at Aand Bare two-force supports, thus there
for the moment. Thus there are four unknowns and only three
372
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Problem 5.137 The mass of the truck is 4 Mg. Its
wheels are locked, and the tension in its cable is TD
10 kN.
(a) Draw the free-body diagram of the truck.
(b) Determine the normal forces exerted on the truck’s
wheels by the road.
AL's
Towing
(003) 676-5942
mg
2 m 2.5 m 2.2 m
30°3 m
T
Solution: The weight is 40009.81D39.24 kN. The sum of the
moments about B
from which
The sum of the forces:
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Problem 5.138 Assume that the force exerted on the
head of the nail by the hammer is vertical, and neglect
the hammer’s weight.
(a) Draw the free-body diagram of the hammer.
(b) If FD10 lb, what are the magnitudes of the forces
exerted on the nail by the hammer and the normal
and friction forces exerted on the oor by the
hammer?
F
11 in.
65°
Solution: Denote the point of contact with the oor by B. The
NHD5.077F
NH
FN
HXB
374
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.
page-pf9
Problem 5.139 The spring constant is kD9600 N/m
and the unstretched length of the spring is 30 mm. Treat
the bolt at Aas a pin support and assume that the surface
at Cis smooth. Determine the reactions at Aand the
normal force at C.
15 mm C
B
A
k
30 mm
24 mm
30°
Solution: The length of the spring is
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Problem 5.140 The engineer designing the release
mechanism shown in Problem 5.139 wants the normal
force exerted at Cto be 120 N. If the unstretched length
of the spring is 30 mm, what is the necessary value of
the spring constant k?
376
page-pfb
Problem 5.141 The truss supports a 90-kg suspended
object. What are the reactions at the supports Aand B?
300 mm
400 mm 700 mm
A
B
Solution: Treat the truss as a single element. The pin support at
The sum of the forces:
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Problem 5.142 The trailer is parked on a 15°slope.
Its wheels are free to turn. The hitch Hbehaves like a
pin support. Determine the reactions at Aand H.
x
y
15°
H
2.8 ft
1.4 ft
870 lb
Solution: The coordinate system has the xaxis parallel to the road.
The wheels are a one force reaction normal to the road, the pin His
a two force reaction. The position vectors of the points of the center
of mass and Hare:
The angle of the weight vector realtive to the positive xaxis is
378
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.
page-pfd
Problem 5.143 To determine the location of the point
where the weight of a car acts (the center of mass),
an engineer places the car on scales and measures the
normal reactions at the wheels for two values of ˛,
obtaining the following results:
˛A
y(kN) B(kN)
10°10.134 4.357
20°10.150 3.677
What are the distances band h?
α
Ay
Ax
W
B
h
b
2.7 m
x
y
The angle between the weight and the positive xaxis is ˇD270 ˛.
The weight vector at each of the two angles is
D013.8272bC5.0327h27.4054 D0
hD0.50 m
page-pfe
Problem 5.144 The bar is attached by pin supports to
collars that slide on the two xed bars. Its mass is 10 kg,
it is 1 m in length, and its weight acts at its midpoint.
Neglect friction and the masses of the collars. The spring
is unstretched when the bar is vertical (˛D0), and the
spring constant is kD100 N/m. Determine the values
of ˛in the range 0 ˛60°at which the bar is in
equilibrium.
α
k
Solution: The force exerted by the spring is given by FSDk⊲L
Lcos ˛⊳. The equations of equilibrium, from the free body diagram, are
and MBDLsin ˛⊳NACL
FS
NB
B
Spring Constant (K) in N/m vs Alpha (deg)
90000
30000
m
Spring Constant (K) in N/m vs Alpha (deg)
116
114
112
110
K
380
page-pff
Problem 5.145 With each of the devices shown you
can support a load Rby applying a force F. They are
called levers of the rst, second, and third class.
(a) The ration R/F is called the mechanical advantage.
Determine the mechanical advantage of each lever.
(b) Determine the magnitude of the reaction at Afor
each lever. (Express you answers in terms of F)
R
L
F
L
A
L
F
L
R
A
F
A
R
First-class lever Second-class lever
Solution:
Problem 5.146 The force exerted by the weight of the
horizontal rectangular plate is 800 N. The weight of the
plate acts at its midpoint. If you represent the reactions
exerted on the plate by the three cables by a single
equivalent force, what is the force, and where does its
line of action intersect the plate?
A
B
C
1 m
0.5 m
2 m
Solution: The equivalent force must equal the sum of the
reactions: FEQ DTACTBCTC.FEQ D300 C100 C400 D800 N.
y
TA
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Problem 5.147 The 20-kg mass is suspended by cables
attached to three vertical 2-m posts. Point Ais at
(1, 1.2, 0) m. Determine the reactions at the built-in
support at E.B
A
C
D
y
E
1 m
1 m
Solution: All distances will be in meters, all forces in Newtons,
and all moments in Newton-meters. To solve the three dimensional
point equilibrium problem at A, we will need unit vectors eAB,eAC,
after making the substitutions related to the force components yields
the tensions in the cables. They are
jTABjD150 N,
Also,jMGjD68.4 N-m
CD
TAD
382
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Problem 5.148 In Problem 5.147, the built-in support
of each vertical post will safely support a couple of
800 N-m magnitude. Based on this criterion, what is the
maximum safe value of the suspended mass?
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Problem 5.149 The 80-lb bar is supported by a ball
and socket support at A, the smooth wall it leans against,
and the cable BC. The weight of the bar acts at its
midpoint.
(a) Draw the free-body diagram of the bar.
(b) Determine the tension in cable BC and the reactions
at A.
x
A
C
B
3 ft
3 ft 5 ft
3 ft
z
4 ft
y
Solution: (a) The ball and socket is a three reaction force support;
the cable and the smooth wall are each one force reaction supports.
rAB DrBrAD2iC4j3k.
Solve:
The reactions at Aare found from the sums of forces:
384
page-pf13
Problem 5.150 The horizontal bar of weight Wis
supported by a roller support at Aand the cable BC. Use
the fact that the bar is a three-force member to determine
the angle ˛, the tension in the cable, and the magnitude
of the reaction at A.
L/2
AB
C
α
L/2
W
Solution: The sum of the moments about Bis
2. This equation cannot be satised

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