78 In Fig. 12-82, a uniform beam of
length 12.0 m is supported by a horizontal cable and a hinge at angle u ϭ
50.0°. The tension in the cable is 400
N. In unit-vector notation, what are
(a) the gravitational force on the
beam and (b) the force on the beam
from the hinge?
79
Four bricks of length L,
identical and uniform, are stacked
on a table in two ways, as shown in
Fig. 12-83 (compare with Problem
63). We seek to maximize the overhang distance h in both arrangements. Find the optimum distances
a 1 , a 2 , b 1 , and b 2 , and calculate h for the two arrangements.
353
PROB LE M S
the loop with the rope hanging vertically when the child’s father
pulls on the child with a horizontal force and displaces the child
to one side. Just before the child is released from rest, the rope
makes an angle of 15Њ with the vertical and the tension in the
rope is 280 N. (a) How much does the child weigh? (b) What is
the magnitude of the (horizontal) force of the father on the child
just before the child is released? (c) If the maximum horizontal
force the father can exert on the child is 93 N, what is the maximum angle with the vertical the rope can make while the father is
pulling horizontally?
85
Figure 12-85a shows details of a finger in the crimp hold
of the climber in Fig. 12-50. A tendon that runs from muscles in
the forearm is attached to the far bone in the finger. Along the
way, the tendon runs through several guiding sheaths called pulleys. The A2 pulley is attached to the first finger bone; the A4 pulley is attached to the second finger bone. To pull the finger toward
the palm, the forearm muscles pull the tendon through the pulleys, much like strings on a marionette can be pulled to move parts
of the marionette. Figure 12-85b is a simplified diagram of the second finger bone, which has length d. The tendon’s pull
on the
bone acts at the point where the tendon enters the A4 pulley, at
distance d/3 along the bone. If the force components on each of
the four crimped fingers in Fig. 12-50 are F h ϭ 13.4 N and F v ϭ
162.4 N, what is the magnitude of ? The result is probably tolerable, but if the climber hangs by only one or two fingers, the A2 and
A4 pulleys can be ruptured, a common ailment among rock
climbers.
F
:
t
F
:
t
x
y
θ
Cable
Figure 12-82 Problem 78.
h
a 2
a 1
(a)
L
h
b 2
b 1
b 1
b 1
(b)
L
Figure 12-83 Problem 79.
y
x
A
B
θ
Figure 12-84 Problem 83.
86 A trap door in a ceiling is 0.91 m square, has a mass of 11 kg,
and is hinged along one side, with a catch at the opposite side. If
the center of gravity of the door is 10 cm toward the hinged side
from the door’s center, what are the magnitudes of the forces exerted by the door on (a) the catch and (b) the hinge?
87 A particle is acted on by forces given, in newtons, by
ϭ
F
:
1
80 A cylindrical aluminum rod, with an initial length of 0.8000
m and radius 1000.0 mm, is clamped in place at one end and then
stretched by a machine pulling parallel to its length at its other
end. Assuming that the rod’s density (mass per unit volume)
does not change, find the force magnitude that is required of the
machine to decrease the radius to 999.9 mm. (The yield strength
is not exceeded.)
81 A beam of length L is carried by three men, one man at one
end and the other two supporting the beam between them on a
crosspiece placed so that the load of the beam is equally divided
among the three men. How far from the beam’s free end is the
crosspiece placed? (Neglect the mass of the crosspiece.)
82 If the (square) beam in Fig. 12-6a and the associated sample
problem is of Douglas fir, what must be its thickness to keep the
compressive stress on it to of its ultimate strength?
83 Figure 12-84 shows a stationary arrangement of two crayon
boxes and three cords. Box A has a mass of 11.0 kg and is on a
ramp at angle u ϭ 30.0°; box B has a mass of 7.00 kg and hangs on a
cord. The cord connected to box A is parallel to the ramp, which is
frictionless. (a) What is the tension in the upper cord, and (b) what
angle does that cord make with the horizontal?
1
6
45°
80°
10°
F t
F h
F v
F c
d/3
(b)
Far
bone
Second
finger
bone
First
finger bone
A2 A4
Attached
Tendon
(a)
Figure 12-85 Problem 85.
84 A makeshift swing is constructed by making a loop in one end
of a rope and tying the other end to a tree limb. A child is sitting in
8.40 Ϫ 5.70 and ϭ 16.0 ϩ 4.10 . (a) What are the x component
and (b) y component of the force
that balances the sum of these
F
:
3
j ˆ
i ˆ
F
:
2
j ˆ
i ˆ
forces? (c) What angle does
have relative to the ϩx axis?
88 The leaning Tower of Pisa is 59.1 m high and 7.44 m in diameter. The top of the tower is displaced 4.01 m from the vertical. Treat
the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge
of toppling? (b) What angle would the tower then make with the
vertical?
F
:
3
length 12.0 m is supported by a horizontal cable and a hinge at angle u ϭ
50.0°. The tension in the cable is 400
N. In unit-vector notation, what are
(a) the gravitational force on the
beam and (b) the force on the beam
from the hinge?
79
Four bricks of length L,
identical and uniform, are stacked
on a table in two ways, as shown in
Fig. 12-83 (compare with Problem
63). We seek to maximize the overhang distance h in both arrangements. Find the optimum distances
a 1 , a 2 , b 1 , and b 2 , and calculate h for the two arrangements.
353
PROB LE M S
the loop with the rope hanging vertically when the child’s father
pulls on the child with a horizontal force and displaces the child
to one side. Just before the child is released from rest, the rope
makes an angle of 15Њ with the vertical and the tension in the
rope is 280 N. (a) How much does the child weigh? (b) What is
the magnitude of the (horizontal) force of the father on the child
just before the child is released? (c) If the maximum horizontal
force the father can exert on the child is 93 N, what is the maximum angle with the vertical the rope can make while the father is
pulling horizontally?
85
Figure 12-85a shows details of a finger in the crimp hold
of the climber in Fig. 12-50. A tendon that runs from muscles in
the forearm is attached to the far bone in the finger. Along the
way, the tendon runs through several guiding sheaths called pulleys. The A2 pulley is attached to the first finger bone; the A4 pulley is attached to the second finger bone. To pull the finger toward
the palm, the forearm muscles pull the tendon through the pulleys, much like strings on a marionette can be pulled to move parts
of the marionette. Figure 12-85b is a simplified diagram of the second finger bone, which has length d. The tendon’s pull
on the
bone acts at the point where the tendon enters the A4 pulley, at
distance d/3 along the bone. If the force components on each of
the four crimped fingers in Fig. 12-50 are F h ϭ 13.4 N and F v ϭ
162.4 N, what is the magnitude of ? The result is probably tolerable, but if the climber hangs by only one or two fingers, the A2 and
A4 pulleys can be ruptured, a common ailment among rock
climbers.
F
:
t
F
:
t
x
y
θ
Cable
Figure 12-82 Problem 78.
h
a 2
a 1
(a)
L
h
b 2
b 1
b 1
b 1
(b)
L
Figure 12-83 Problem 79.
y
x
A
B
θ
Figure 12-84 Problem 83.
86 A trap door in a ceiling is 0.91 m square, has a mass of 11 kg,
and is hinged along one side, with a catch at the opposite side. If
the center of gravity of the door is 10 cm toward the hinged side
from the door’s center, what are the magnitudes of the forces exerted by the door on (a) the catch and (b) the hinge?
87 A particle is acted on by forces given, in newtons, by
ϭ
F
:
1
80 A cylindrical aluminum rod, with an initial length of 0.8000
m and radius 1000.0 mm, is clamped in place at one end and then
stretched by a machine pulling parallel to its length at its other
end. Assuming that the rod’s density (mass per unit volume)
does not change, find the force magnitude that is required of the
machine to decrease the radius to 999.9 mm. (The yield strength
is not exceeded.)
81 A beam of length L is carried by three men, one man at one
end and the other two supporting the beam between them on a
crosspiece placed so that the load of the beam is equally divided
among the three men. How far from the beam’s free end is the
crosspiece placed? (Neglect the mass of the crosspiece.)
82 If the (square) beam in Fig. 12-6a and the associated sample
problem is of Douglas fir, what must be its thickness to keep the
compressive stress on it to of its ultimate strength?
83 Figure 12-84 shows a stationary arrangement of two crayon
boxes and three cords. Box A has a mass of 11.0 kg and is on a
ramp at angle u ϭ 30.0°; box B has a mass of 7.00 kg and hangs on a
cord. The cord connected to box A is parallel to the ramp, which is
frictionless. (a) What is the tension in the upper cord, and (b) what
angle does that cord make with the horizontal?
1
6
45°
80°
10°
F t
F h
F v
F c
d/3
(b)
Far
bone
Second
finger
bone
First
finger bone
A2 A4
Attached
Tendon
(a)
Figure 12-85 Problem 85.
84 A makeshift swing is constructed by making a loop in one end
of a rope and tying the other end to a tree limb. A child is sitting in
8.40 Ϫ 5.70 and ϭ 16.0 ϩ 4.10 . (a) What are the x component
and (b) y component of the force
that balances the sum of these
F
:
3
j ˆ
i ˆ
F
:
2
j ˆ
i ˆ
forces? (c) What angle does
have relative to the ϩx axis?
88 The leaning Tower of Pisa is 59.1 m high and 7.44 m in diameter. The top of the tower is displaced 4.01 m from the vertical. Treat
the tower as a uniform, circular cylinder. (a) What additional displacement, measured at the top, would bring the tower to the verge
of toppling? (b) What angle would the tower then make with the
vertical?
F
:
3