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Problem 3.1 In Active Example 3.1, suppose that the
angle between the ramp supporting the car is increased
from 20°to 30°. Draw the free-body diagram of the car
showing the new geometry. Suppose that the cable from
Ato Bmust exert a 1900-lb horizontal force on the car
to hold it in place. Determine the car’s weight in pounds.
AB
20⬚
Solution: The free-body diagram is shown to the right.
Problem 3.2 The ring weighs 5 lb and is in equilib-
rium. The force F1D4.5 lb. Determine the force F2and
the angle ˛.
x
y
30⬚
F2
F1
a
Solution: The free-body diagram is shown below the drawing. The
equilibrium equations are
84
Problem 3.3 In Example 3.2, suppose that the attach-
ment point Cis moved to the right and cable AC is
extended so that the angle between cable AC and the
ceiling decreases from 45°to 35°. The angle between
cable AB and the ceiling remains 60°. What are the
tensions in cables AB and AC?
B
C
A
60⬚
45⬚
Solution: The free-body diagram is shown below the picture.
The equilibrium equations are:
Problem 3.4 The 200-kg engine block is suspended
by the cables AB and AC. The angle ˛D40°. The free-
body diagram obtained by isolating the part of the system
within the dashed line is shown. Determine the forces
TAB and TAC.
B
C
AAx
y
aa
TAB TAC
(200 kg) (9.81 m/s2)
Solution:
TAB TAC
Problem 3.5 A heavy rope used as a mooring line for
a cruise ship sags as shown. If the mass of the rope is
90 kg, what are the tensions in the rope at Aand B?B40⬚
55⬚A
Problem 3.6 A physiologist estimates that the
masseter muscle of a predator, Martes, is capable of
exerting a force Mas large as 900 N. Assume that
the jaw is in equilibrium and determine the necessary
force Tthat the temporalis muscle exerts and the force
Pexerted on the object being bitten.
T
22⬚
P
M
Solution: The equilibrium equations are
86
Problem 3.7 The two springs are identical, with un-
stretched lengths 250 mm and spring constants kD
1200 N/m.
(a) Draw the free-body diagram of block A.
(b) Draw the free-body diagram of block B.
(c) What are the masses of the two blocks?
B
A
300 mm
280 mm
Solution: The tension in the upper spring acts on block Ain the
positive Ydirection, Solve the spring force-deflection equation for
the tension in the upper spring. Apply the equilibrium conditions to
block A. Repeat the steps for block B.
Similarly, the tension in the lower spring acts on block Ain the nega-
300 mm
Problem 3.8 The two springs in Problem 3.7 are iden-
tical, with unstretched lengths of 250 mm. Suppose that
their spring constant kis unknown and the sum of the
masses of blocks Aand Bis 10 kg. Determine the value
of kand the masses of the two blocks.
Solution: All of the forces are in the vertical direction so we will
Problem 3.9 The inclined surface is smooth (Remem-
ber that “smooth” means that friction is negligble). The
two springs are identical, with unstretched lengths of
250 mm and spring constants kD1200 N/m. What are
the masses of blocks Aand B?
B
A
300 mm
280 mm
30⬚
Solution:
mAg
NB
88
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 3.10 The mass of the crane is 20,000 kg. The
crane’s cable is attached to a caisson whose mass is
400 kg. The tension in the cable is 1 kN.
(a) Determine the magnitudes of the normal and
friction forces exerted on the crane by the
level ground.
(b) Determine the magnitudes of the normal and
friction forces exerted on the caisson by the
level ground.
Strategy: To do part (a), draw the free-body diagram
of the crane and the part of its cable within the
dashed line.
45°
Solution:
Ncrane D196.9kN,F
crane D0.707 kN
(b) Fy:Ncaisson 3.924 kN C1 kN sin 45°D0
Fx:1 kN cos 45°CFcaisson D0
Ncaisson D3.22 kN,F
caisson D0.707 kN
Ncrane
Fcrane
x
45°
45°
1 kN
3.924 kN
Ncaisson
Fcaisson
Problem 3.11 The inclined surface is smooth. The
100-kg crate is held stationary by a force Tapplied to
Solution:
(a) The FBD
Ν
Problem 3.12 The 1200-kg car is stationary on the
sloping road.
(a) If ˛D20°, what are the magnitudes of the total
normal and friction forces exerted on the car’s tires
by the road?
(b) The car can remain stationary only if the total
friction force necessary for equilibrium is not
greater than 0.6 times the total normal force.
What is the largest angle ˛for which the car can
remain stationary?
a
Solution:
F-:F11.772 kN sin ˛D0
N
11.772 kN
Problem 3.13 The 100-lb crate is in equilibrium on the
smooth surface. The spring constant is kD400 lb/ft. Let
Sbe the stretch of the spring. Obtain an equation for S
(in feet) as a function of the angle ˛.
a
90
Problem 3.14 A 600-lb box is held in place on the
smooth bed of the dump truck by the rope AB.
(a) If ˛D25°, what is the tension in the rope?
(b) If the rope will safely support a tension of 400 lb,
what is the maximum allowable value of ˛?
α
A
B
Problem 3.15 The 80-lb box is held in place on
the smooth inclined surface by the rope AB. Determine
the tension in the rope and the normal force exerted
on the box by the inclined surface.
A
B
30⬚
50⬚
Solution: The equilibrium equations (in terms of a coordinate
Problem 3.16 The 1360-kg car and the 2100-kg tow
truck are stationary. The muddy surface on which the
car’s tires rest exerts negligible friction forces on them.
What is the tension in the tow cable?
18⬚
10⬚
26⬚
Solution: FBD of the car being towed
13.34 kN
N
26°
92
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 3.17 Each box weighs 40 lb. The angles are
measured relative to the horizontal. The surfaces are
smooth. Determine the tension in the rope Aand the
normal force exerted on box Bby the inclined surface.
D
C
B
A
70⬚
20⬚
45⬚
Solution: The free-body diagrams are shown.
The equilibrium equations for box Dare
Problem 3.18 A 10-kg painting is hung with a wire
supported by a nail. The length of the wire is 1.3 m.
(a) What is the tension in the wire?
(b) What is the magnitude of the force exerted on the
nail by the wire?
1.2 m
Solution:
(a) Fy:98.1N25
98.1 N
Problem 3.19 A 10-kg painting is hung with a wire
supported by two nails. The length of the wire is 1.3 m.
(a) What is the tension in the wire?
(b) What is the magnitude of the force exerted on each
nail by the wire? (Assume that the tension is the
same in each part of the wire.)
Compare your answers to the answers to Problem 3.18.
0.4 m0.4 m 0.4 m
Solution:
Fy:Tsin 27.3°49.05 N D0
TD107 N
(b) Examine one of the nails
Fx:RxTcos 27.3°CTD0
Fy:RyTsin 27.3°D0
RDRx2CRy2
RD50.5N
T
49.05 N
Rx
T
T
Problem 3.20 Assume that the 150-lb climber is in
equilibrium. What are the tensions in the rope on the
left and right sides?
14⬚15⬚
Solution:
y
94
Problem 3.21 If the mass of the climber shown in
Problem 3.20 is 80 kg, what are the tensions in the rope
on the left and right sides?
FxDTRcos⊲15°⊳TLcos⊲14°⊳D0
FyDTRsin⊲15°⊳CTLsin⊲14°⊳mg D0
Solving, we get
TLD1.56 kN,T
RD1.57 kN
14°15°
TL
TR
x
Problem 3.22 The construction worker exerts a 20-lb
force on the rope to hold the crate in equilibrium in the
position shown. What is the weight of the crate?
5⬚
30⬚
Solution: The free-body diagram is shown.
Problem 3.23 A construction worker on the moon,
where the acceleration due to gravity is 1.62 m/s2, holds
the same crate described in Problem 3.22 in the position
shown. What force must she exert on the cable to hold
the crate ub equilibrium (a) in newtons; (b) in pounds?
5⬚
30⬚
Solution: The free-body diagram is shown.
From Problem 3.22 we know that the weight is WD188 lb. Therefore
its mass is
The equilibrium equations for the part of the rope system where the
96
Problem 3.24 The person wants to cause the 200-lb
crate to start sliding toward the right. To achieve this,
the horizontal component of the force exerted on the
crate by the rope must equal 0.35 times the normal force
exerted on the crate by the floor. In Fig.a, the person
pulls on the rope in the direction shown. In Fig.b, the
person attaches the rope to a support as shown and pulls
upward on the rope. What is the magnitude of the force
he must exert on the rope in each case?
20⬚
(a)
Ffr D0.35N
(a) For equilibrium we have
(b) The person exerts the force F. Using the free-body diagram of
the crate and of the point on the rope where the person grabs the
rope, we find
Problem 3.25 A traffic engineer wants to suspend a
200-lb traffic light above the center of the two right
lanes of a four-lane thoroughfare as shown. Determine
the tensions in the cables AB and BC.
Solution:
Fx:6
p37 TAB C2
p5TBC D0
Problem 3.26 Cable AB is 3 m long and cable BC is
4 m long. The mass of the suspended object is 350 kg.
Determine the tensions in cables AB and BC.C
B
A
5m
Solution:
TAB
98
Problem 3.27 In Problem 3.26, the length of cable AB
is adjustable. If you don’t want the tension in either cable
AB or cable BC to exceed 3 kN, what is the minimum
acceptable length of cable AB?
2.11 kN <3kN
3.43 kN
Problem 3.28 What are the tensions in the upper and
lower cables? (Your answers will be in terms of W.
Neglect the weight of the pulley.)
45° 30°
W
Solution: Isolate the weight. The frictionless pulley changes the
direction but not the magnitude of the tension. The angle between the
right hand upper cable and the xaxis is ˛, hence
TU
TU
βα
100
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 3.29 Two tow trucks lift a 660-lb motorcycle
out of a ravine following an accident. If the motorcycle
is in equilibrium in the position shown, what are the
tensions in cables AB and AC?B
A
C
(12, 32) ft
(36, 36) ft
(26, 16) ft
x
y
Solution: The angles are
Problem 3.30 An astronaut candidate conducts exper-
iments on an airbearing platform. While she carries out
calibrations, the platform is held in place by the hori-
zontal tethers AB,AC, and AD. The forces exerted by
the tethers are the only horizontal forces acting on the
platform. If the tension in tether AC is 2 N, what are the
tensions in the other two tethers?
3.0 m 1.5 m
BC
TOP VIEW
D
4.0 m
3.5 m
A
Solution: Isolate the platform. The angles ˛and ˇare
3.0 m
B
102
Problem 3.31 The bucket contains concrete and
weighs 5800 lb. What are the tensions in the cables AB
and AC?
BC
A
(20, 34) ft
(5, 34) ft
(12, 16) ft
y
x
Solution: The angles are
Problem 3.32 The slider Ais in equilibrium and the
bar is smooth. What is the mass of the slider? 20⬚
45⬚
200 N
A
Solution: The pulley does not change the tension in the rope that
passes over it. There is no friction between the slider and the bar.
Eqns. of Equilibrium:
y
20°
T = 200 N
mg = (9.81) g
