Statics 2e 249
Problem 2.179
An automobile body panel is subjected to a force
E
F
from a stiffening strut.
Assuming that region
ABC
of the panel is planar, determine the components,
and vector components, of
E
F
in the directions normal and parallel to the panel.
Problem 2.180
The steering wheel and gearshift lever of an automobile are shown. Points
A
,
C
,
and
B
lie on the axis of the steering column, where point
A
is at the origin of
the coordinate system, point
B
has the coordinates
B .120; 0; 50/ mm
, and
point
C
is
60 mm
from point
A
. The gearshift lever from
C
to
D
has
240 mm
length and the direction angles given below.
(a)
Two of the direction angles for the position vector from point
C
to
D
are
yD50ı
and
´D60ı
. Knowing that this position vector has a
negative xcomponent, determine x.
(b)
Determine the unit vector
Or
that is perpendicular to directions
AB
and
CD such that this vector has a positive ´component.
(c)
If the force applied to the gearshift lever is
E
PD.6O{4O|C12 O
k/ N
,
determine the component of E
P, namely Pr, in the direction of Or.
Problem 2.181
The steering wheel and gearshift lever of an automobile are shown. Points
A
,
C
,
and
B
lie on the axis of the steering column, where point
A
is at the origin of
the coordinate system, point
B
has the coordinates
B .120; 0; 50/ mm
, and
point
C
is
60 mm
from point
A
. The gearshift lever from
C
to
D
has
240 mm
length and the direction angles given below.
(a)
Two of the direction angles for the position vector from point
C
to
D
are
yD45ı
and
´D70ı
. Knowing that this position vector has a
negative xcomponent, determine x.
(b)
Determine the unit vector
Or
that is perpendicular to directions
AB
and
CD such that this vector has a positive ´component.
(c)
If the force applied to the gear shift lever is
E
PD.9O{6O|C18 O
k/ N
,
determine the component of E
P, namely Pr, in the direction of Or.
Problem 2.182
An I beam is positioned from points
A
to
B
. Because its strength and deforma-
tion properties for bending about an axis through the web of the cross section
are different than those for bending about an axis parallel to the flanges, it is
necessary to also characterize these directions. This can be accomplished by
specifying just one of the direction angles for the direction of the web from
A
to
C
, which is perpendicular to line
AB
, plus the octant of the coordinate
system in which line AC lies.
(a)
If direction angle
´D30ı
for line
AC
, determine the remaining direc-
tion angles for this line.
(b)
Determine the unit vector in the direction perpendicular to the web of the
beam (i.e., perpendicular to lines AB and AC ).
Statics 2e 257
Problem 2.183
Determine the smallest distance between the infinite lines passing through bars
AB and CD.
Problem 2.184
A portion of a downhill ski run between points
C
and
D
is to be constructed
on a mountainside, as shown. Let the portion of the mountainside defined by
points
A
,
B
, and
C
be idealized to be planar. For the values of
xA
,
yB
, and
´C
given below, determine the distance from point
A
to point
D
where the
ski run should intersect line
AB
so that the run will be as steep as possible.
Hint: Consider a force acting on the slope in the
´
direction, such as perhaps
a skier’s weight – or better yet, a
1lb
weight – and follow the approach used
in Example 2.20 on p. 108 to resolve this weight into normal and tangential
components; the direction of the tangential component will give the direction
of steepest descent.
xAD1500 ft, yBD2000 ft, and ´CD800 ft.
Problem 2.185
A portion of a downhill ski run between points
C
and
D
is to be constructed
on a mountainside, as shown. Let the portion of the mountainside defined by
points
A
,
B
, and
C
be idealized to be planar. For the values of
xA
,
yB
, and
´C
given below, determine the distance from point
A
to point
D
where the
ski run should intersect line
AB
so that the run will be as steep as possible.
Hint: Consider a force acting on the slope in the
´
direction, such as perhaps
a skier’s weight – or better yet, a
1lb
weight – and follow the approach used
in Example 2.20 on p. 108 to resolve this weight into normal and tangential
components; the direction of the tangential component will give the direction
of steepest descent.
xAD1200 ft, yBD1600 ft, and ´CD900 ft.
Problem 2.186
The tetrahedron shown arises in advanced mechanics, and it is necessary to
relate the areas of the four surfaces. Show that the surface areas are related
by
AxDAcos x
,
AyDAcos y
, and
A´DAcos ´
where
A
is the area
of surface
ABC
, and
cos x
,
cos y
, and
cos ´
are the direction cosines for
normal direction
En
.Hint: Find
En
by taking the cross product of vectors along
edges
AB
,
AC
, and/or
BC
, and note the magnitude of this vector is
2A
. Then,
by inspection, write expressions for
Ax
,
Ay
, and
A´
(e.g.,
AxDy´=2
and so
on).
Solution