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Problem 2.165
The vector from point
O
to point
P
has magnitude
40 mm
and has equal
direction angles with the
x
,
y
, and
´
axes. Determine the smallest distance
from point Pto the infinite plane containing points A,B, and C.
Problem 2.166
The product
Er1E
F
produces a vector
E
M
. The product
E
MEr2=jEr2j
produces a
scalar Mk, which is the component of E
Min the direction of Er2.
(a) Evaluate Mkby finding E
Mfirst, followed by the dot product.
(b) Evaluate Mkusing the scalar triple product.
Problem 2.167
As described in connection with Fig. 2.34 on p. 105, the scalar triple product
.E
AE
B/ E
C
provides the
volume of the parallelepiped formed by
E
A
,
E
B
, and
E
C
. Comment on how the results of the following triple
products compare to the value provided by .E
AE
B/ E
C:
(a) .E
AE
C / E
B.
(b) .E
BE
C / E
A.
(c) .E
CE
B/ E
A.
(d) .E
CE
A/ E
B.
Note: Concept problems are about explanations, not computations.
Problem 2.168
The manufacturer of a welded steel bracket specifies the work-
ing loads depicted in the figure of
Rn
versus
Rt
. Values of
Rn
and
Rt
that lie within the shaded region are allowable,
while values that lie outside of the region are unsafe. Such
a diagram is often called an interaction diagram because it
characterizes the combined effect that multiple loads have on
the strength of a component. For the loading and geometry
shown, determine the range of values load
P
may have and
still satisfy the manufacturer’s combined loading criterion.
Problem 2.169
For the loading and geometry shown, use the interaction diagram of Prob. 2.168
to determine if the manufacturer’s combined loading criterion is satisfied.
Problem 2.170
(a) Determine the resultant E
Rof the three forces E
FCE
PCE
Q.
(b)
If an additional force
E
T
in the
˙x
direction is to be added, determine
the magnitude it should have so that the magnitude of the resultant is as
small as possible.
Problem 2.171
(a) Determine the resultant E
Rof the three forces E
FCE
PCE
Q.
(b)
If an additional force
E
T
in the
˙x
direction is to be added, determine
the magnitude it should have so that the magnitude of the resultant is as
small as possible.
Problem 2.172
An electrical power transmission line runs between points
A
,
B
, and
C
, as
shown, and a new power transmission line between points
B
and
D
is to
be constructed. Using the coordinates of points
A
,
C
, and
D
given below,
determine the distance from point
A
to point
B
where the new power line
should connect to the existing power line such that the length of the new power
line is as small as possible. Also determine the length of the new power line.
Assume that power lines ABC and BD are straight.
The coordinates of points
A
,
C
, and
D
are
A.6000; 1000; 500/ ft
,
C.700; 4500; 900/ ft, and D.2000; 400; 1400/ ft.
Problem 2.173
An electrical power transmission line runs between points
A
,
B
, and
C
, as
shown, and a new power transmission line between points
B
and
D
is to
be constructed. Using the coordinates of points
A
,
C
, and
D
given below,
determine the distance from point
A
to point
B
where the new power line
should connect to the existing power line such that the length of the new power
line is as small as possible. Also determine the length of the new power line.
Assume that power lines ABC and BD are straight.
The coordinates of points
A
,
C
, and
D
are
A.3000; 600; 300/ m
,
C.600; 2200; 500/ m, and D.1000; 300; 800/ m.
Problem 2.174
An architect specifies the roof geometry shown for a building. Each of lines
EAB
,
BD
,
DC G
,
EG
, and
AC
are straight. Two forces of magnitude
P
and
Qacting in the ydirection are applied at the positions shown.
Determine the components of the force
P
in directions normal and parallel
to the roof at point A. Express your answer in terms of P.
Problem 2.175
An architect specifies the roof geometry shown for a building. Each of lines
EAB
,
BD
,
DC G
,
EG
, and
AC
are straight. Two forces of magnitude
P
and
Qacting in the ydirection are applied at the positions shown.
Determine the components of the force
Q
in directions normal and parallel
to the roof at point C. Express your answer in terms of Q.
Problem 2.176
A concrete surface for a parking lot is to be prepared such that it is planar and the
normal direction to the surface is
2ı˙0:5ı
of vertical so that rainwater will have
adequate drainage. If the coordinates of points
A
,
B
, and
C
are
A .300; 50; 2/ ft
,
B .20; 150; 5/ ft
, and
C .280; 450; 8/ ft
, determine the angle between the
concrete surface’s normal direction and the vertical, and if the concrete surface
meets the drainage specification cited above.
Problem 2.177
Repeat Prob. 2.176 if the coordinates of points
A
,
B
, and
C
are
A .100; 20; 1/ m
,
B .10; 50; 3/ m, and C .110; 160; 2/ m.
Problem 2.178
A specimen of composite material consisting of ceramic matrix and unidirec-
tional ceramic fiber reinforcing is tested in a laboratory under compressive load-
ing. If a
10 kN
force is applied in the
´
direction, determine the components,
and vector components, of this force in directions parallel and perpendicular
to the fiber direction f, where this direction has direction angle ´D40ıand
remaining direction angles that are equal (i.e., xDy).
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