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cables. Cable AC is equipped with a turnbuckle so that
its tension can be adjusted and a strain gauge that allows
its tension to be measured. If the tension in cable AC is
40 N, what are the tensions in cables AB and AD?
BC
A
D
0.4 m0.4 m 0.48 m
0.64 m
Solution:
TAB
TAC
5
Problem 3.34 The structural joint is in equilibrium. If
FAD1000 lb and FDD5000 lb, what are FBand FC?
FB
FC
FAFD
80⬚
65⬚
35⬚
Solution: The equilibrium equations are
104
Problem 3.35 The collar Aslides on the smooth
vertical bar. The masses mAD20 kg and mBD10 kg.
When hD0.1 m, the spring is unstretched. When the
system is in equilibrium, hD0.3 m. Determine the
spring constant k.
B
A
h
k
0.25 m
Solution: The triangles formed by the rope segments and the hori-
zontal line level with Acan be used to determine the lengths Luand
Ls. The equations are
163.5 N and TD255.4 N. Now we go to the free body diagram for B,
Lu
Lu
Ls
0.1 m
0.3 m
0.25 m
0.25 m
mBg
B
K
δ
Problem 3.36* Suppose that you want to design a
cable system to suspend an object of weight Wfrom
the ceiling. The two wires must be identical, and the
dimension bis fixed. The ratio of the tension Tin each
wire to its cross-sectional area Amust equal a specified
value T/A D. The “cost” of your design is the total
volume of material in the two wires, VD2Apb2Ch2.
Determine the value of hthat minimizes the cost.
W
b
h
b
Solution: From the equation
TT
θθ
106
Problem 3.37 The system of cables suspends a
1000-lb bank of lights above a movie set. Determine
the tensions in cables AB,CD, and CE.
D
C
E
20 ft
B
18 ft
Solution: Isolate juncture A, and solve the equilibrium equations.
Repeat for the cable juncture C.
0.9659 D732.05 lb
Isolate juncture C. The angle between the positive xaxis and the cable
CA is ⊲180°˛⊳. The tension is
TCA DjTCAj⊲icos⊲180°C˛⊳ Cjsin⊲180°C˛⊳⊳,
W
y
x
D
Problem 3.38 Consider the 1000-lb bank of lights in
Problem 3.37. A technician changes the position of the
lights by removing the cable CE. What is the tension in
cable AB after the change?
Solution: The original configuration in Problem 3.35 is used to
solve for the dimensions and the angles. Isolate the juncture A, and
The new angles are given by the cosine law
Reduce and solve:
Isolate the juncture A. The angle between the cable AD and the positive
xaxis is ˛. The tension is:
The equilibrium conditions are
20 ft 18 ft
B
y
D
Solve: jTABjDcos ˛
0.989 D621.03 lb,
108
Problem 3.39 While working on another exhibit, a
curator at the Smithsonian Institution pulls the suspended
Voyager aircraft to one side by attaching three horizontal
cables as shown. The mass of the aircraft is 1250 kg.
Determine the tensions in the cable segments AB,BC,
and CD.
B
C
D
50°
30°
Solution: Isolate each cable juncture, beginning with Aand solve
Isolate juncture B. The angles are ˛D50°,ˇD70°, and the tension
cable BC is TBC DjTBCj⊲icos ˛Cjsin ˛⊳. The angle between the
cable BA and the positive xaxis is ⊲180 Cˇ⊳; the tension is
y
x
C
Problem 3.40 A truck dealer wants to suspend a 4000-
kg truck as shown for advertising. The distance bD
15 m, and the sum of the lengths of the cables AB and
BC is 42 m. Points A and C are at the same height. What
are the tensions in the cables?
C
A
b
40 m
B
Solution: Determine the dimensions and angles of the cables. Iso-
late the cable juncture B, and solve the equilibrium conditions. The
dimensions of the triangles formed by the cables:
15 m 25 m
b
L
110
Problem 3.41 The distance hD12 in, and the tension
in cable AD is 200 lb. What are the tensions in cables
AB and AC?
12 in.
12 in.
12 in.
8 in.
8 in. h
D
B
A
C
Solution: Isolated the cable juncture. From the sketch, the angles
are found from
y
B
α
12 in
Problem 3.42 You are designing a cable system to
support a suspended object of weight W. Because your
design requires points Aand Bto be placed as shown,
you have no control over the angle ˛, but you can choose
the angle ˇby placing point Cwherever you wish. Show
that to minimize the tensions in cables AB and BC, you
must choose ˇD˛if the angle ˛½45°.
Strategy: Draw a diagram of the sum of the forces
exerted by the three cables at A.W
C
B
A
β
α
Solution: Draw the free body diagram of the knot at point A. Then
draw the force triangle involving the three forces. Remember that ˛is
In this case, we solved the problem without writing the equations of
equilibrium. For reference, these equations are:
FxDTAB cos ˛CTAC cos ˇD0
y
TAB
TAC
Possible locations
for C lie on line
B
α
C?C?
Problem 3.43* The length of the cable ABC is 1.4 m.
The 2-kN force is applied to a small pulley. The system
is stationary. What is the tension in the cable?
C
B
A1 m
0.75 m
15⬚2 kN
0.75 m ,tan ˇDh
0.25 m
)hD0.458 m,˛D31.39°,ˇD61.35°
Now draw a FBD and solve for the tension. We can use either of the
equilibrium equations
Fx:Tcos ˛CTcos ˇC⊲2kN⊳sin 15°D0
TD1.38 kN
β
α
T
T
112
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 3.44 The masses m1D12 kg and m2D6kg
are suspended by the cable system shown. The cable BC
is horizontal. Determine the angle ˛and the tensions in
the cables AB,BC, and CD.
C
A
α
B
D
70⬚
m2
m1
117.7 N
TAB
TBC
α
B
Solution: We have 4 unknowns and 4 equations
Solving we find
˛D79.7°,T
AB D119.7N,T
BC D21.4N,T
CD D62.6N
58.86 N
Problem 3.45 The weights W1D50 lb and W2are
suspended by the cable system shown. Determine the
weight W2and the tensions in the cables AB,BC,
and CD.
AD
BC
W1
W2
30 in 30 in 30 in
20 in 16 in
Solution: We have 4 unknowns and 4 equilibrium equations to use
TAB
Problem 3.46 In the system shown in Problem 3.45,
assume that W2DW1/2. If you don’t want the tension
anywhere in the supporting cable to exceed 200 lb, what
is the largest acceptable value of W1?
Solution:
FCy :2
p229 TBC C8
TAB
114
Problem 3.47 The hydraulic cylinder is subjected to
three forces. An 8-kN force is exerted on the cylinder
at Bthat is parallel to the cylinder and points from B
toward C. The link AC exerts a force at Cthat is parallel
to the line from Ato C. The link CD exerts a force at
Cthat is parallel to the line from Cto D.
(a) Draw the free-body diagram of the cylinder. (The
cylinder’s weight is negligible).
(b) Determine the magnitudes of the forces exerted by
the links AC and CD.
1 m
0.6 m Scoop
AB
D
C
0.15 m
0.6 m
1 m
Hydraulic
cylinder
Solution: From the figure, if Cis at the origin, then points A,B,
and Dare located at
A⊲0.15,0.6⊳
B⊲0.75,0.6⊳
y
D
FCD
Problem 3.48 The 50-lb cylinder rests on two smooth
surfaces.
(a) Draw the free-body diagram of the cylinder.
(b) If ˛D30°, what are the magnitudes of the forces
exerted on the cylinder by the left and right
surfaces?
α
45°
FyD⊲jNRjcos ˇCjNLjcos ˛jWj⊳jD0.
Problem 3.49 For the 50-lb cylinder in Problem 3.48,
obtain an equation for the force exerted on the cylinder
by the left surface in terms of the angle ˛in two ways:
(a) using a coordinate system with the yaxis vertical,
(b) using a coordinate system with the yaxis parallel to
the right surface.
Solution: The solution for Part (a) is given in Problem 3.48 (see
free body diagram).
y
αβ
116
Problem 3.50 The two springs are identical, with
unstretched length 0.4 m. When the 50-kg mass is
suspended at B, the length of each spring increases to
0.6 m. What is the spring constant k?
C
B
A
k
k
0.6 m
Solution:
kD1416 N/m
F
F
490.5 N
Problem 3.51 The cable AB is 0.5 m in length. The
unstretched length of the spring is 0.4 m. When the
50-kg mass is suspended at B, the length of the spring
increases to 0.45 m. What is the spring constant k?
C
B
A
k
0.7 m
Solution: The Geometry
0.72D0.52C0.4522⊲0.5⊳⊲0.45⊳cos ˇ
0.45 m Dsin
0.5mDsin ˇ
0.7m
ˇD94.8°,D39.8°D45.4°
Now do the statics
FDk⊲0.45 m 0.4m⊳
Fy:TAB sin CFsin 490.5ND0
Solving: kD7560 N/m
θφ
β
0.7 m
490.5 N
F
TAB
Problem 3.52 The small sphere of mass mis attached
to a string of length Land rests on the smooth surface
of a fixed sphere of radius R. The center of the sphere
is directly below the point where the string is attached.
Obtain an equation for the tension in the string in terms
of m,L,h, and R.
R
hL
m
Solution: From the geometry we have
Problem 3.53 The inclined surface is smooth. Deter-
mine the force Tthat must be exerted on the cable to
hold the 100-kg crate in equilibrium and compare your
answer to the answer of Problem 3.11.
T
60⬚
Solution:
60°
N
3T
118
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 3.54 In Example 3.3, suppose that the mass
of the suspended object is mAand the masses of the
pulleys are mBD0.3mA,mCD0.2mA, and mDD0.2mA.
Show that the force Tnecessary for the system to be in
equilibrium is 0.275mAg.
T
A
B
B
B
DD
CC
Solution: From the free-body diagram of pulley C
TD
Problem 3.55 The mass of each pulley of the system
is mand the mass of the suspended object Ais mA.
Determine the force Tnecessary for the system to be in
equilibrium.
Solution: Draw free body diagrams of each pulley and the object
120
Problem 3.56 The suspended mass m1D50 kg. Neg-
lecting the masses of the pulleys, determine the value
of the mass m2necessary for the system to be in
equilibrium.
A
B
C
m1
m2
Solution:
TT1
T
Problem 3.57 The boy is lifting himself using the
block and tackle shown. If the weight of the block and
tackle is negligible, and the combined weight of the boy
and the beam he is sitting on is 120 lb, what force
does he have to exert on the rope to raise himself at
a constant rate? (Neglect the deviation of the ropes from
the vertical.)
Solution: A free-body diagram can be obtained by cutting the four
122
Problem 3.58 Pulley systems containing one, two, and
three pulleys are shown. Neglecting the weights of the
pulleys, determine the force Trequired to support the
weight Win each case.
T T T
W
W
W
(a) One pulley
(b) Two pulleys
(c) Three pulleys
Solution:
(b) For two pulleys
TT
W
