Problem 2.82
A theodolite is an instrument that measures horizontal and vertical angular
orientations of a line of sight. The line of sight may be established optically,
or by laser, and many forms of theodolite are used in surveying, construction,
astronomy, and manufacturing. A simple optical theodolite is shown. After
the instrument is aligned so that its base is in a horizontal plane and a desired
reference direction is selected (such as perhaps north in the case of a surveying
instrument), the telescopic sight is used to establish a line of sight, and then
horizontal and vertical angles hand vare measured.
(a)
If
h
and
v
are known, derive an expression that gives the direction
angles x,y, and ´for the line of sight.
(b)
If
hD30ı
and
vD60ı
, use your answer to Part (a) to determine
x
,
y, and ´.
Problem 2.83
With direction angles, general practice is to use values of
x
,
y
, and
´
between
0ı
and
180ı
, and this
is sufficient to characterize the orientation of any vector in three dimensions. When direction angles are
measured in this way, the sum of any two direction angles is always
90ı
or greater. Offer an argument that
shows this statement is true.
Problem 2.84
If the direction cosines for the satellite antenna shown are to be
cos xD0:286
,
cos yD0:429
, and
cos ´D0:857
, determine the values of angles
h
and
v
.
Problem 2.85
Using a theodolite with a laser rangefinder, a researcher at point
A
locates a pair
of rare birds nesting at point
B
and determines the direction angles
xD53ı
,
yD38ı
, and
´D81ı
and distance 242 m for a position vector from point
A
to
B
. For subsequent observations, the researcher would like to use position
C
,
and thus while at point
A
, she also measures the direction angles
xD142ı
,
yD63ı
, and
´D65ı
and distance 309 m for a position vector from point
A
to
C
. For observation from point
C
, determine the direction angles and distance
to the nest at B.
Problem 2.86
A robot maneuvers itself by using a laser guidance system
where the positions of reflective targets, such as at points
B
and
C
, are known. By scanning the room with a laser that
emanates from point
A
, and sensing the reflections from targets
B
and
C
, the robot may compute the direction cosines of
position vectors as given below. The locations of other points,
such as point
D
at the base of the ramp, are also known, but
such locations do not have reflective targets. Determine the
position vector from points
A
to
D
so that the robot may move
from its current position to the base of the ramp at point D.
for ErABWcos xD 0:4212; cos yD 0:9025; cos ´D0:0902;
for ErAC Wcos xD0:2676; cos yD 0:9635; cos ´D0:
Problem 2.87
A mountain climbing team establishes base camps at points
A
and
B
, and the
next camp is to be at point
C
. Camps
A
and
B
have elevations of
11;500 ft
and
16;000 ft
above sea level, respectively. With an optical telescope, the direction
angles for position vectors from
A
to
B
, and
B
to
C
, and
A
to
C
are determined
as follows:
for ErABWxD34:3ı; yD63:2ı; ´D70:2ı;
for ErB C WxD104:7ı; yD27:3ı; ´D67:6ı;
for ErAC WxD59:1ı; yD42:0ı; ´D64:6ı:
Determine the elevation above sea level of camp C.
140 Solutions Manual
Problem 2.88
Bars
AC
and
DG
are straight and parallel to the
x
and
y
axes, respectively.
BE
is a cable whose tensile force is 100 lb. For the dimensions given, determine
expressions for the force the cable exerts on
B
and the force the cable exerts on
Eusing Cartesian vector representation. xD4in:and yD7in.
Problem 2.89
Bars
AC
and
DG
are straight and parallel to the
x
and
y
axes, respectively.
BE
is a cable whose tensile force is 100 lb. For the dimensions given, determine
expressions for the force the cable exerts on
B
and the force the cable exerts on
Eusing Cartesian vector representation. xD6in:and yD12 in.
Problem 2.90
A crane consists of a quarter-circular bar that lies in a plane parallel to the
x´
plane with a low-friction collar at point
B
that may slide on the bar. The forces
supported by cables
AB
and
BC
are
30 lb
and
1500 lb
, respectively, and the
coordinates of point
A
are given in the figure. For the value of angle
˛
given
below, determine expressions for the forces the two cables apply to the collar at
point B.˛D0ı.
Problem 2.91
A crane consists of a quarter-circular bar that lies in a plane parallel to the
x´
plane with a low-friction collar at point
B
that may slide on the bar. The forces
supported by cables
AB
and
BC
are
30 lb
and
1500 lb
, respectively, and the
coordinates of point
A
are given in the figure. For the value of angle
˛
given
below, determine expressions for the forces the two cables apply to the collar at
point B.˛D30ı.
Problem 2.92
A crane consists of a quarter-circular bar that lies in a plane parallel to the
x´
plane with a low-friction collar at point
B
that may slide on the bar. The forces
supported by cables
AB
and
BC
are
30 lb
and
1500 lb
, respectively, and the
coordinates of point
A
are given in the figure. For the value of angle
˛
given
below, determine expressions for the forces the two cables apply to the collar at
point B.˛D60ı.
Problem 2.93
A crane consists of a quarter-circular bar that lies in a plane parallel to the
x´
plane with a low-friction collar at point
B
that may slide on the bar. The forces
supported by cables
AB
and
BC
are
30 lb
and
1500 lb
, respectively, and the
coordinates of point
A
are given in the figure. For the value of angle
˛
given
below, determine expressions for the forces the two cables apply to the collar at
point B. General values of ˛where 0ı˛90ı.
Problem 2.94
The structure consists of a quarter-circular rod
AB
with radius 150 mm that
is fixed in the
xy
plane.
CD
is a straight rod where
D
may be positioned at
different locations on the circular rod.
GE
is an elastic cord whose support at
G
lies in the
y´
plane, and the bead at
E
may have different positions
d
. For
the values of
b
,
h
,
d
, and
´
provided, determine the coordinates of
E
, and if the
elastic cord supports a tensile force of 100 N, write a vector expression using
Cartesian vector representation for the force
E
FEG
the cord exerts on bead
E
.
bD4,hD3,dD260 mm;and ´D240 mm.
Problem 2.95
The structure consists of a quarter-circular rod
AB
with radius 150 mm that
is fixed in the
xy
plane.
CD
is a straight rod where
D
may be positioned at
different locations on the circular rod.
GE
is an elastic cord whose support at
G
lies in the
y´
plane, and the bead at
E
may have different positions
d
. For
the values of
b
,
h
,
d
, and
´
provided, determine the coordinates of
E
, and if the
elastic cord supports a tensile force of 100 N, write a vector expression using
Cartesian vector representation for the force
E
FEG
the cord exerts on bead
E
.
bD3,hD4,dD195 mm;and ´D270 mm.
Problem 2.96
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. ˛D0ı.
Problem 2.97
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. ˛D40ı.