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Statics 2e 213
Problem 2.153
For the triangular-shaped window of Prob. 2.109 on p. 95, use the cross product to determine the outward
normal unit vector (i.e., pointing away from the origin) and the area of the window.
Solution
Problem 2.154
A flat quadrilateral plate finite element is shown (see Example 2.21 on
p. 110 for a description of what a finite element is).
(a)
Describe and perform a test, using two cross products that will
verify if all four nodes (corners) lie in the same plane.
(b)
Determine the unit outward normal direction to the surface (i.e.,
pointing away from the origin).
(c) Determine the surface area of the element.
(11)
The unit vectors corresponding to En1and En2are
Ou1DEn1
n1
D.60 O{C60 O|C300 O
k/ cm2
311:8 cm2D0:1925 O{C0:1925 O|C0:9623 O
k; (12)
Problem 2.155
A contractor is required to prepare a soil surface so that it is planar and the
normal direction to the surface is within 0.5
ı
of vertical. If the elevations at
points
A
and
B
, relative to the elevation at point
C
, are
3:3 ft
and
4:2 ft
,
respectively, determine the angle between the soil surface’s normal direction
and the vertical, and if the surface is sufficiently level.
Problem 2.156
Repeat Prob. 2.155 if the elevations at points
A
and
B
, relative to the elevation
at point C, are 2:6 ft and 3:1 ft, respectively.
Problem 2.157
An ergonomically designed key for a computer keyboard has an approximately
flat surface defined by points
A
,
B
, and
C
and is subjected to a
1
N force in the
0.985, respectively. Determine the components of the force in directions
normal and tangent to the key’s motion. Comment on why this design
might be more effective than that in Part (a).
Problem 2.158
Impact of debris, both natural and artificial, is a significant hazard for spacecraft.
Space Shuttle windows are routinely replaced because of damage due to impact
with small objects, and recent flights have employed evasive maneuvers to
avoid impact with larger objects, whose orbits NASA constantly monitors. For
the triangular-shaped window and the relative velocity of approach
Ev
shown,
determine the components and vector components of the velocity in directions
normal and tangent to the window. Note that this information is needed before
an analysis of damage due to impact can be performed.
(4)
D.1300 O{C3500 O|C2800 O
k/ cm2;(5)
Problem 2.159
The velocity of air approaching the rudder of an aircraft has magnitude
900 ft/s
in the ydirection. The rudder rotates about line OA.
(a)
The position vector from
B
to
C
has the
x
,
y
, and
´
direction cosines
sin ˛
,
.cos ˛/.cos 20ı/
, and
.cos ˛/.sin 20ı/
, respectively. If the rud-
der is rotated so that
˛D10ı
, determine the components, and vector
components, of the air velocity in directions normal and tangent to the
surface of the rudder.
(b)
Using the geometry shown in Fig. P2.159, verify the direction cosines
stated in Part (a).
Problem 2.160
Beam
AB
has rectangular cross section where point
A
is at the origin of the
coordinate system and point
B
has the coordinates
B .1:2; 0:3; 2:4/ m
. The
vector
Er1D.1 O{C12 O|C1O
k/ m
is perpendicular to the axis
AB
of the beam
(ErAB ), and is parallel to the thin dimension of the cross section.
(a) Verify that Er1is indeed perpendicular to ErAB .
(b)
Determine the unit vector
Or2
that is perpendicular to both
Er1
and
ErAB
and
has the orientation shown in the figure.
(c)
The force
E
P
applied to point
B
of the beam has
1000
N magnitude and
direction angles
xD144ı
,
yD72ı
, and
´D60ı
. Determine
the components of the
1000
N force, namely
PAB
,
P1
, and
P2
, in the
directions
ErAB
,
Er1
, and
Or2
, respectively, as shown in the figure. Note: For
future work in mechanics of materials, it will be necessary to know these
forces.
Statics 2e 225
Problem 2.161
Repeat Prob. 2.160 if the coordinates of point
B
are
B .130; 60; 180/ in.
,
Er1D.6 O{C31 O|C6O
k/ in.
, and the force
E
P
has
500 lb
magnitude and direction
angles xD108ı,yD60ı, and ´D36ı.
Problem 2.162
Structure
ABCD
is rigid and is built-in at point
A
. Point
A
is at the origin of the
coordinate system, points
B
and
D
have the coordinates
B .16; 24; 48/ in.
and
D .26; 52; 2/ in
., and point
C
is at the midpoint of the straight bar
AB
. Por-
tion CD of the structure is perpendicular to bar AB.
(a)
Verify that portion
CD
of the structure is indeed perpendicular to bar
AB.
(b)
Determine the unit vector
Or
that is perpendicular to both
AB
and
CD
such that this vector has a positive xcomponent.
(c)
The force
E
P
applied to point
D
of the structure has
600 lb
magnitude
and direction angles
xD108ı
,
yD36ı
, and
´D60ı
. Determine
the components of the
600 lb
force, namely
PAB
,
PCD
, and
Pr
, in the
directions AB,CD, and Or, respectively.
NOTE:
The first printing of this book has an error in the last sentence of the statement for Part (c). The
force referenced in the last sentence should be
600 lb
(not
50
N). This error is corrected in the second and
subsequent printings of the book.
Problem 2.163
Repeat Prob. 2.162 if the coordinates of points
B
and
D
are
B .20; 60; 90/ mm
and
D .40; 110; 15/ mm
, and the force
E
P
has
50
N magnitude and direction
angles xD135ı,yD60ı, and ´D60ı.
Problem 2.164
Determine the smallest distance between point
O
and the infinite plane contain-
ing points A,B, and C.
