Statics 2e 153
Problem 2.98
A wall-mounted jib crane consists of an I beam that is supported by a pin at
point
A
and a cable at point
C
, where
A
and
C
lie in the
xy
plane. A crate
at
E
is supported by a cable that is attached to a trolley at point
B
where the
trolley may move along the length of the I beam. The forces supported by
cables
CD
and
BE
are
3kN
and
5kN
, respectively. For the value of angle
˛
given below, determine expressions for the forces the two cables apply to the I
beam. ˛D70ı.
Problem 2.99
A wall-mounted jib crane consists of an I beam that is supported by a pin at
point
A
and a cable at point
C
, where
A
and
C
lie in the
xy
plane. A crate at
E
is supported by a cable that is attached to a trolley at point
B
where the trolley
may move along the length of the I beam. The forces supported by cables
CD
and
BE
are
3kN
and
5kN
, respectively. For the value of angle
˛
given below,
determine expressions for the forces the two cables apply to the I beam. General
values of ˛where 90ı˛90ı.
Dq64.cos2˛Csin2˛/ C36 m:(3)
Since
cos2˛Csin2˛D1
for any value of
˛
, the magnitude of
ErCD
, which is the same as the length of cable
CD, is always
rCD D10 m:(4)
Problem 2.100
A coordinate system that is often used for problems in mechanics is the spherical
coordinate system, as shown. With this coordinate system, the location of a
point
P
in three dimensions is specified using a radial distance
r
where
r0
,
an angle
(sometimes called the azimuthal angle) where
0ı360ı
, and
an angle
(sometimes called the polar angle) where
0ı180ı
. If values
for
r
,
, and
are known, determine the direction angles for the position vector
from the origin of the coordinate system to point P.
Problem 2.101
For the spherical coordinate system described in Prob. 2.100, if the direction
angles for the position vector from the origin of the coordinate system to point
P
are known, determine the values for
r
,
, and
.Hint: Several approaches
may be used to solve this problem, but a straightforward solution is to first solve
Prob. 2.100, and then use those results to solve this problem.
Problem 2.102
The rear wheel of a multispeed bicycle is shown. The wheel has 32 spokes, with one-half being on either
side (spokes on sides
A
and
B
are shown in the figure in black and red, respectively). For the tire to be
properly centered on the frame of the bicycle, points
A
and
B
of the hub must be positioned at the same
distance
d
from the centerline of the tire. To make room for the sprocket cluster, bicycle manufacturers
give spokes on side
B
of the wheel a different orientation than spokes on side
A
. For the following
questions, assume the tire is in the process of being manufactured, so that all spokes on side
A
have the
same force FAand all spokes on side Bhave the same force FB.
(a)
Determine the ratio of spoke forces
FB=FA
so that the resultant force in the
x
direction applied to
the hub by all 32 spokes is zero. Hint: Although each spoke has a different orientation, all spokes on
side
A
have the same length, and similarly all spokes on side
B
have the same length. Furthermore,
all spokes on side
A
have the same
x
component of force, and all spokes on side
B
have the same
x
component of force.
(b) On which side of the wheel are the spokes most severely loaded?
(c)
Briefly describe a new design in which spokes on both sides of the wheel are equally loaded and
points Aand Bare at the same distance dfrom the centerline of the tire.
158 Solutions Manual
Problem 2.103
(a) Determine the angle between vectors E
Aand E
B.
(b) Determine the components of E
Aparallel and perpendicular to E
B.
(c) Determine the vector components of E
Aparallel and perpendicular to E
B.
Problem 2.104
(a) Determine the angle between vectors E
Aand E
B.
(b) Determine the components of E
Aparallel and perpendicular to E
B.
(c) Determine the vector components of E
Aparallel and perpendicular to E
B.
Problem 2.105
(a) Determine the angle between vectors E
Aand E
B.
(b) Determine the components of E
Aparallel and perpendicular to E
B.
(c) Determine the vector components of E
Aparallel and perpendicular to E
B.
Problem 2.106
(a) Determine the angle between vectors E
Aand E
B.
(b) Determine the components of E
Aparallel and perpendicular to E
B.
(c) Determine the vector components of E
Aparallel and perpendicular to E
B.
Problem 2.107
A slide on a child’s play structure is to be supported in part by strut
CD
(railings
are omitted from the sketch for clarity). End
C
of the strut is to be positioned
along the outside edge of the slide, halfway between ends
A
and
B
. End
D
of
the strut is to be positioned on the
y
axis so that the angle
†ACD
between the
slide and the strut is a right angle. Determine the distance
h
that point
D
should
be positioned.
Problem 2.108
A whistle is made of a square tube with a notch cut in its edge, into which a
baffle is brazed. Determine the dimensions dand for the baffle.
Solution
dD5:77 cm:(4)
Problem 2.109
A flat, triangular-shaped window for the cockpit of an airplane is to have the
corner coordinates shown. Specify the angles
A
,
B
, and
C
and dimensions
dAB ,dBC , and dAC for the window.
Problem 2.110
The corner of an infant’s bassinet is shown. Determine angles
˛
and
ˇ
and
dimensions
a
and
b
of the side and end pieces so the corners of the bassinet
will properly meet when assembled.
Problem 2.111
The roof of an
8ft
diameter grain silo is to be made using 12 identical triangular
panels. Determine the value of angle
needed and the smallest value of
d
that
can be used.
Problem 2.112
For the description and figure indicated below, determine the components of the cord force in directions
parallel and perpendicular to rod
CD
. If released from rest, will the cord force tend to make bead
E
slide
toward Cor D?
Use the description and figure for Prob. 2.94 on p 82.
D.26:3 O{C68:5 O|C8:35 O
k/ N.
(6)
As a partial check of the accuracy of our answers, we evaluate the magnitudes of
E
Fk
and
E
F?
in Eqs. (5) and
(6) to obtain
FkDq.20:75 N/2C.15:56 N/2C.62:23 N/2D67:4 N;(7)
Problem 2.113
For the description and figure indicated below, determine the components of the cord force in directions
parallel and perpendicular to rod
CD
. If released from rest, will the cord force tend to make bead
E
slide
toward Cor D?
Use the description and figure for Prob. 2.95 on p 82.
(3)
Using this result, the perpendicular component is then
F2
EG DF2
?CF2
k)F?DqF2
EG F2
kDq.100 N/2.48:72 N/2D87:3 N:(4)
The vector form of the parallel component, E
Fk, is
ErCD
k/ mm
Problem 2.114
A cantilever I beam has a cable at end
B
that supports a force
E
F
, and
ErAB
is the
position vector from end
A
of the beam to end
B
. Position vectors
Er1
and
Er2
are
parallel to the flanges and web of the I beam, respectively. For determination
of the internal forces in the beam (discussed in Chapter 8), and for mechanics
of materials analysis, it is necessary to know the components of the force in
the axial direction of the beam (
AB
) and in directions parallel to the web and
flanges.
Using the dot product, show that
Er1
,
Er2
, and
ErAB
are orthogonal to one
another.
Problem 2.115
A cantilever I beam has a cable at end
B
that supports a force
E
F
, and
ErAB
is the
position vector from end
A
of the beam to end
B
. Position vectors
Er1
and
Er2
are
parallel to the flanges and web of the I beam, respectively. For determination
of the internal forces in the beam (discussed in Chapter 8), and for mechanics
of materials analysis, it is necessary to know the components of the force in
the axial direction of the beam (
AB
) and in directions parallel to the web and
flanges.
Determine the scalar and vector components of E
Fin direction ErAB .
Problem 2.116
A cantilever I beam has a cable at end
B
that supports a force
E
F
, and
ErAB
is the
position vector from end
A
of the beam to end
B
. Position vectors
Er1
and
Er2
are
parallel to the flanges and web of the I beam, respectively. For determination
of the internal forces in the beam (discussed in Chapter 8), and for mechanics
of materials analysis, it is necessary to know the components of the force in
the axial direction of the beam (
AB
) and in directions parallel to the web and
flanges.
Determine the scalar and vector components of E
Fin direction Er1.