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Problem 5.87 The force Facting on the boom ABC
at Cpoints in the direction of the unit vector 0.512i
0.384jC0.768kand its magnitude is 8 kN. The boom
is supported by a ball and socket at Aand the cables BD
and BE. The collar at Bis fixed to the boom.
(a) Draw the free-body diagram of the boom.
(b) Determine the tensions in the cables and the reac-
tions at A.
y
z
x
A
C
D
E
B
2 m
2 m
1 m
1.5 m 2 m
2 m
Solution:
In component form we have
Moment equilibrium requires:
In components:
p10.25 TBD⊲2m⊳
F
328
Problem 5.88 The cables BD and BE in Problem 5.87
will each safely support a tension of 25 kN. Based on
this criterion, what is the largest acceptable magnitude
of the force F?
Solution: We have the force and distances:
The moment equations are
Problem 5.89 The suspended load exerts a force FD
600 lb at A, and the weight of the bar OA is negligible.
Determine the tensions in the cables and the reactions at
the ball and socket support O.
(0, 10, 4) ft
B
(0, 6, –10) ft
C
A(8, 6, 0) ft
x
y
–Fj
O
z
Solution: From the diagram, the important points in this problem
are A(8, 6, 0), B(0, 10, 4), C(0, 6, 10), and the origin O(0, 0, 0)
with all dimensions in ft. We need unit vectors in the directions Ato
Band Ato C. Both vectors are of the form
330
Problem 5.90 In Problem 5.89, suppose that the
suspended load exerts a force FD600 lb at Aand bar
OA weighs 200 lb. Assume that the bar’s weight acts at
its midpoint. Determine the tensions in the cables and
the reactions at the ball and socket support O.
Solution: Point Gis located at (4, 3, 0) and the position vector of
Gwith respect to the origin is
The mathematical representation for all other forces and moments from
Problem 5.89 remain the same (the numbers change!). Each equation
of equilibrium has a new term reflecting the addition of the weight of
the bar. The new force equilibrium equation is
The new moment equilibrium equation is
Problem 5.91 The 158,000-kg airplane is at rest on
the ground (zD0 is ground level). The landing gear
carriages are at A,B, and C. The coordinates of the point
Gat which the weight of the plane acts are (3, 0.5, 5) m.
What are the magnitudes of the normal reactions exerted
on the landing gear by the ground?
y
x
6 m
21 m
A
B
C
G
6 m
Solution:
FYDNRCNLCNFWD0
mg
W0.5 m
332
Problem 5.92 The horizontal triangular plate is
suspended by the three vertical cables A, B, and C.
The tension in each cable is 80 N. Determine the x
and zcoordinates of the point where the plate’s weight
effectively acts.
x
z
0.3 m
0.4 m
C
A
B
(x, 0, z)
y
Solution:
Solving
xD0.1m,zD0.1333 m
Z
80 N
240 N
Problem 5.93 The 800-kg horizontal wall section is
supported by the three vertical cables, A,B, and C. What
are the tensions in the cables?
7 m
7 m
A
B
C
Solution: All dimensions are in mand all forces are in N. Forces
A,B,C, and Wact on the wall at (0, 0, 0), (5, 14, 0), (12, 7, 0),
Problem 5.94 The bar AC is supported by the cable
BD and a bearing at Athat can rotate about the zaxis.
The person exerts a force FD10j(lb) at C. Determine
the tension in the cable and the reactions at A.
C
14 in8 in
B
A
z
x
y
334
Problem 5.95 The L-shaped bar is supported by a
bearing at Aand rests on a smooth horizontal surface
at B. The vertical force FD4 kN and the distance
bD0.15 m. Determine the reactions at Aand B.
y
A
Fb
x
z:MAZC0.2F0.2BD0
AZ
B
z
b = 0.15 m
F = 4 kN
Problem 5.96 In Problem 5.95, the vertical force FD
4 kN and the distance bD0.15 m. If you represent the
reactions at Aand Bby an equivalent system consisting
of a single force, what is the force and where does its
line of action intersect the xzplane?
MAD0.4k(kN-m)
We want a single equivalent force, Rthat has the same resultant force
and moment about Aas does the set A,B, and MA.
xRD0.1m
y
F
b
R
Problem 5.97 In Problem 5.95, the vertical force FD
4 kN. The bearing at Awill safely support a force of
2.5-kN magnitude and a couple of 0.5 kN-m magnitude.
Based on these criteria, what is the allowable range of
the distance b?
MAZC0.2F0.2BD0
MAYDC0.5 (kN-m), and solve we get AYD2.5atbD0.4875 m
However, 0.3 is the physical limit of the device.
336
Problem 5.98 The 1.1-m bar is supported by a ball
and socket support at Aand the two smooth walls. The
tension in the vertical cable CD is 1 kN.
(a) Draw the free-body diagram of the bar.
(b) Determine the reactions at Aand B.
B
C
D
A
400 mm
700 mm
x
y
z
600 mm
Solution:
(a) The ball and socket cannot support a couple reaction, but can
support a three force reaction. The smooth surface supports one-
contain only the unknown reaction at B. This will require the
position vectors of Band Drelative to A, which in turn will
1.1D33.1°.
From which,
BZD0.3819
0.6D0.6365 kN,
The reactions at Aare determined from the sums of the forces:
FZD⊲BZCAZ⊳kD0,from which AZD0.6365 kN
Problem 5.99 The 8-ft bar is supported by a ball and
socket at A, the cable BD, and a roller support at C. The
collar at Bis fixed to the bar at its midpoint. The force
FD50k(lb). Determine the tension in the cable BD
y
4 ft
2 ft
D
C
The angle formed by the rod is
rDD4iC2k
and rCD⊲8 cos 22°⊳iD7.4162i.
C7.4162CYkD0,
1.192 D62.92 lb
7.4162 D17.27 lb.
338
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
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Problem 5.100 Consider the 8-ft bar in Problem 5.99.
The force FDFyj50k(lb). What is the largest value
of Fyfor which the roller support at Cwill remain on
the floor?
3.7081 1.50
0.1160 0.5960 0.7946
C
ijk
7.4162 30
1.192 D62.92 lb.
3.708 D34.54 lb
Problem 5.101 The tower is 70 m tall. The tension in
each cable is 2 kN. Treat the base of the tower Aas a
built-in support. What are the reactions at A?
y
B
eBD D0.2265i0.7926jC0.5661k,
eBC D0.5812i0.8137jC0k.
FYD⊲AY4.7682⊳jD0,
from which AYD4.768 kN,
FZD⊲AZC0.2434⊳kD0,
340
Problem 5.102 Consider the tower in Problem 5.101.
If the tension in cable BC is 2 kN, what must the ten-
sions in cables BD and BE be if you want the couple
exerted on the tower by the built-in support at Ato be
zero? What are the resulting reactions at A?
Solve:
Problem 5.103 The space truss has roller supports at
B,C, and Dand is subjected to a vertical force FD
20 kN at A. What are the reactions at the roller supports? F
A (4, 3, 4) m
y
342
Problem 5.104 In Example 5.8, suppose that the cable
BD is lengthened and the attachment point Bis moved to
the end of the bar at C. The positions of the attachment
point Dand the bar are unchanged. Draw a sketch of
the bar showing cable BD in its new position. Draw the
free-body diagram of the bar and apply equilibrium to
determine the tension in the cable and the reactions at A.
x
y
z
⫺100j (lb)
A
B
C
30⬚
(2, 2, ⫺1) ftD
Solution: The sketch and free-body diagram are shown.
We must express the force exerted on the bar by cable BD in terms of
its components. The bar AC is 4 ft long. The vector from Cto Dis
Problem 5.105 The 40-lb door is supported by hinges
at Aand B. The yaxis is vertical. The hinges do not
exert couples on the door, and the hinge at Bdoes not
exert a force parallel to the hinge axis. The weight of
the door acts at its midpoint. What are the reactions at
Aand B?
z
40
4 ft
x
y
1 ft
1 ft
A
B
D1.2856iC3.5jC1.532k.
The position vectors of the hinges:
rADj,rBD6j.
344
Problem 5.106 The vertical cable is attached at A.
Determine the tension in the cable and the reactions at
the bearing Bdue to the force FD10i30j10k(N).
A
y
x
200 mm
100 mm
100 mm
200 mm
B
Solution: The position vector of the point of application of the
force is
rBD0.1i.
0.1D60 N,
MBY DC1 N-m,
MBZ D0.1TC3D3 N-m.
Problem 5.107 In Problem 5.106, suppose that the z
component of the force Fis zero, but otherwise Fis
unknown. If the couple exerted on the shaft by the
bearing at Bis MBD6j6kN-m, what are the force
Fand the tension in the cable?
346
Problem 5.108 The device in Problem 5.106 is badly
designed because of the couples that must be supported
by the bearing at B, which would cause the bearing to
“bind”. (Imagine trying to open a door supported by
only one hinge.) In this improved design, the bearings
at Band Csupport no couples, and the bearing at C
does not exert a force in the xdirection. If the force
FD10i30j10k(N), what are the tension in the
vertical cable and the reactions at the bearings Band C?
A
y
z
C
x
200 mm
50 mm
50 mm
100 mm
200 mm
F
B
