Problem 3.80 The cable AB keeps the 8-kg collar Ain
place on the smooth bar CD. The yaxis points upward.
What is the tension in the cable?
0.4 m
0.15 m
y
BC
Solution: We develop the following position vectors and unit
vectors
Problem 3.81 Determine the magnitude of the normal
force exerted on the collar Aby the smooth bar.
0.4 m
0.15 m
z
y
BC
Solution: From Problem 3.81 we have
ND⊲78.5 N⊳j⊲357 N⊳⊲0.674iC0.735jC0.079k⊳
142
Problem 3.82* The 10-kg collar Aand 20-kg collar B
are held in place on the smooth bars by the 3-m cable
from Ato Band the force Facting on A. The force F
is parallel to the bar. Determine F.
y
F
(0, 3, 0) m
(0, 5, 0) m
Using the coordinates of points C,D,E, and Gfrom the picture, the
unit vectors are
two ways to determine the location of B. First, Bis 3 m from point A
along the line AB (which we do not know). Also, Blies on the line
distance EB. Some of the information in the equations is redundant.
However, we can solve for EB (and the coordinates of B). We get that
y
D (0, 5, 0)
m
We now have two fewer equation than unknowns. Fortunately, there
are two conditions we have not yet invoked. The bars at Aand B
are smooth. This means that the normal force on each bar can have
FD36.6N.
Other values obtained in the solution are EB D2.56 m,
Problem 3.83 The 100-lb crate is held in place on the
smooth surface by the rope AB. Determine the tension in
the rope and the magnitude of the normal force exerted
on the crate by the surface.
45⬚
A
B
30⬚
Solution: The free-body diagram is sketched. The equilibrium
equations are
Problem 3.84 The system shown is called Russell’s
traction. If the sum of the downward forces exerted at
Aand Bby the patient’s leg is 32.2 lb, what is the weight
W?
y
W
A
B
20⬚
25⬚
60⬚
x
Solution: The force in the cable is Weverywhere. The free-body
diagram of the leg is shown. The downward force is given, but the
horizontal force FHis unknown.
144
Problem 3.85 The 400-lb engine block is suspended
by the cables AB and AC. If you don’t want either TAB
or TAC to exceed 400 lb, what is the smallest acceptable
value of the angle ˛?
B
C
AAx
y
aa
TAB TAC
Problem 3.86 The cable AB is horizontal, and the box
on the right weighs 100 lb. The surface are smooth.
(a) What is the tension in the cable?
(b) What is the weight of the box on the left?
AB
20⬚
40⬚
Solution: We have the following equilibrium equations
Solving these equations sequentially, we find
Problem 3.87 Assume that the forces exerted on the
170-lb climber by the slanted walls of the “chimney”
are perpendicular to the walls. If he is in equilibrium
and is exerting a 160-lb force on the rope, what are the
magnitudes of the forces exerted on him by the left and
right walls?
4⬚
10⬚
3⬚
Solution: The forces in the free-body diagram are in the directions
shown on the figure. The equilibrium equations are:
146
Problem 3.88 The mass of the suspended object Ais
mAand the masses of the pulleys are negligible. Deter-
mine the force Tnecessary for the system to be in
equilibrium.
TA
Solution: Break the system into four free body diagrams as shown.
Carefully label the forces to ensure that the tension in any single cord
is uniform. The equations of equilibrium for the four objects, starting
with the leftmost pulley and moving clockwise, are:
terms of mAand g. From the first two equations, we get SD3T,
and RD3SD9T. Substituting these into the last equilibrium equation
results in 2TC2⊲3T⊳ C2⊲9T⊳ DmAg.
R
RR
F
S
S
S
S
S
mAg
Problem 3.89 The assembly A, including the pulley,
weighs 60 lb. What force Fis necessary for the system
to be in equilibrium?
F
A
148
Problem 3.90 The mass of block Ais 42 kg, and the
mass of block Bis 50 kg. The surfaces are smooth. If
the blocks are in equilibrium, what is the force F?
B
45⬚
A
20⬚
F
Solution: Isolate the top block. Solve the equilibrium equations.
The weight is. The angle between the normal force N1and the posi-
tive xaxis is. The normal force is. The force N2is. The equilibrium
conditions are
y
B
Problem 3.91 The climber Ais being helped up an icy
slope by two friends. His mass is 80 kg, and the direction
cosines of the force exerted on him by the slope are
cos xD0.286, cos yD0.429, cos zD0.857. The y
axis is vertical. If the climber is in equilibrium in the
position shown, what are the tensions in the ropes AB
and AC and the magnitude of the force exerted on him
by the slope?
A
(3, 0, 4) m
B
(2, 2, 0) m
C
(5, 2, ⫺1) m
y
x
z
Solution: Get the unit vectors parallel to the ropes using the coor-
dinates of the end points. Express the tensions in terms of these unit
vectors, and solve the equilibrium conditions. The rope tensions, the
normal force, and the weight act on the climber. The coordinates
C
B
150
Problem 3.92 Consider the climber Abeing helped by
his friends in Problem 3.91. To try to make the tensions
in the ropes more equal, the friend at Bmoves to the
position (4, 2, 0) m. What are the new tensions in the
ropes AB and AC and the magnitude of the force exerted
on the climber by the slope?
Solution: Get the unit vectors parallel to the ropes using the coor-
Problem 3.93 A climber helps his friend up an icy
slope. His friend is hauling a box of supplies. If the
mass of the friend is 90 kg and the mass of the supplies
is 22 kg, what are the tensions in the ropes AB and CD?
Assume that the slope is smooth. That is, only normal
forces are exerted on the man and the box by the slope.
75
40⬚
C
B
20⬚
A
Solution: Isolate the box. The weight vector is
152
Problem 3.94 The 2800-lb car is moving at constant
speed on a road with the slope shown. The aerodynamic
forces on the car the drag DD270 lb, which is parallel
to the road, and the lift LD120 lb, which is perpendic-
ular to the road. Determine the magnitudes of the total
normal and friction forces exerted on the car by the road.
DL
15⬚
Solution: The free-body diagram is shown. If we write the equi-
librium equations parallel and perpendicular to the road, we have:
Problem 3.95 An engineer doing preliminary design
studies for a new radio telescope envisions a triangular
receiving platform suspended by cables from three equ-
ally spaced 40-m towers. The receiving platform has
a mass of 20 Mg (megagrams) and is 10 m below the
tops of the towers. What tension would the cables be
subjected to?
TOP VIEW
rAD37.5iC0jC10k,rBD18.764iC32.5jC10k,
The distance from the origin to any cable connection on the platform is
rcD5.774iC10jC0k.
The unit vector parallel to the tension acting between the points A,a
in the direction of Ais by definition
ecC D0.4667iC0.8082jC0.3592k
The tensions in the cables are expressed in terms of the unit vectors,
TaA DjTaAjeaA ,TbB DjTbBjebB,TcC DjTcCjecC.
The equilibrium conditions are
C0.3592jTcC 196200j⊳kD0
The commercial package TK Solver Plus was used to solve these
equations. The results:
37.511.547 D21.07°,
154
Problem 3.96 To support the tent, the tension in the
rope AB must be 35 lb. What are the tensions in the
ropes AC,AD, and AE?
(3, 0, 3) ft
z
y
eAB D1.00i
eAC D0.885iC0.147j0.442k
eAD D0.857iC0.286jC0.429k
is
Problem 3.97 Cable AB is attached to the top of the
vertical 3-m post, and its tension is 50 kN. What are the
tensions in cables AO,AC, and AD?
y
D
C
5 m
5 m
Solution: Get the unit vectors parallel to the cables using the
coordinates of the end points. Express the tensions in terms of these
The tensions in the cables are expressed in terms of the unit vectors,
156
Problem 3.98* The 1350-kg car is at rest on a
plane surface with its brakes locked. The unit vector
enD0.231iC0.923jC0.308kis perpendicular to the
surface. The yaxis points upward. The direction cosines
of the cable from Ato Bare cos xD0.816, cos yD
0.408, cos zD0.408, and the tension in the cable is
1.2 kN. Determine the magnitudes of the normal and
friction forces the car’s wheels exert on the surface.
en
ep
y
z
x
B
Solution: Assume that all forces act at the center of mass of the
y
Problem 3.99* The brakes of the car in Problem 3.98
are released, and the car is held in place on the
plane surface by the cable AB. The car’s front wheels
are aligned so that the tires exert no friction forces
parallel to the car’s longitudinal axis. The unit vector
epD0.941iC0.131jC0.314kis parallel to the plane
surface and aligned with the car’s longitudinal axis.
What is the tension in the cable?
158