Problem 7.140
A scoop for handling animal food is shown. The scoop’s shape is one-half
of a truncated circular cone. Use the Pappus-Guldinus theorem to determine
the volume of food the scoop will hold, assuming the food is “level.” Also,
disregarding the handle, determine the area of sheet metal, in
cm2
, required to
fabricate the scoop.
Solution
Lines
➀
and
➁
shown at the right are used to produce the surface of the scoop. The
lengths and the positions of the centroids of these lines are
Problem 7.141
Determine the support reactions for the loading shown.
Solution
16 ft
The FBD for the beam is shown at the right. The distributed load is subdivided
into two triangular load distributions, and the resultant force for each of these
and the positions of these forces is shown in the FBD. The equilibrium equations
are
Problem 7.142
Determine the support reactions for the loading shown.
Solution
The FBD for the beam is shown at the right. The distributed load is subdivided
into two composite shapes shown, and the resultant force for each of these is
The equilibrium equations are
Problem 7.143
Determine the support reactions for the loading shown.
Problem 7.144
In Fig. P7.141, reposition the pin support 6ft to the right of point A, and determine the support reactions.
Solution
The FBD for the beam is shown at the right where point
C
is the location of the
repositioned pin support. The resultant forces for the distributed loads are
The equilibrium equations are
These equations are readily solved yielding
Problem 7.145
In Fig. P7.142, replace the pin and roller supports with a built-in support at A, and determine the support
reactions.
Solution
The FBD for the beam is shown at the right. The resultant forces
The equilibrium equations are:
Problem 7.146
In Fig. P7.143, reposition the roller support
2
m to the left of point
B
, and determine the support reactions.
Solution
The FBD for the beam is shown at the right where point
C
is the location of
the repositioned roller support. The resultant force for the distributed load is
Problem 7.147
A circular plate with
21 in:
radius is subjected to the pressure distribution shown.
By treating the pressure distribution as a solid of revolution, use the theorems
of Pappus and Guldinus to determine the total force applied to the plate.
Solution
The pressure distribution is rotated 360
ı
about the
y
axis, and its shape constitutes
an object of revolution. Using the composite regions shown at the right, and
applying the Pappus-Guldinus theorem
VDPQriAi
, where
Ai
is the area for
each distributed force with units of (
lb=in:2
)(
in:
) and
V
becomes the total force
P, we obtain
Problem 7.148
Water in a channel is retained by a gate with
0:5 ft
width (into the plane of the figure).
The gate is supported by a pin at
A
and a roller at
C
. The vertical wall
AD
is built
into the bottom of the channel. If the gate has negligible weight, determine the support
reactions.
Solution
Problem 7.149
Water in a channel is retained by a cylindrical gate with
2
m width. The gate is
supported by a pin at
B
and a cable between
A
and
C
. If the gate has negligible
weight, determine the force supported by the cable and the reactions at B.
Solution
Problem 7.150
A uniform right circular cone
C
with
40 mm
radius at its base and
0:1
N
weight is attached to a beam
AB
with negligible weight. The cone is
partially submerged in water. A block
D
with
0:2
N weight is placed
a distance
d
from the support at
A
. Determine the value of
d
so that
the system is in equilibrium in the position shown. Report
d
such that
a positive value means block
D
is to the left of
A
, and a negative value
means block
D
is to the right of
A
.Hint: The theorems of Pappus and
Guldinus may be useful.
Solution
The pressure at the tip of the cone is
Problem 7.151
A model for the gate of a dump truck is shown. Gate
ABC
is hinged at
A
, is
actuated by a horizontal hydraulic cylinder
BD
, and it has
48 in:
width into the
plane of the figure. The pins at
A
and
D
are fixed in space. The gate retains a
72 in:
depth of sand that may be modeled as a fluid with
0:06 lb=in:3
specific
weight. If gate
ABC
is uniform with
400 lb
weight that acts midway between
points
A
and
C
, determine the force in the hydraulic cylinder that will allow
the gate to begin to open.
NOTE:
The first printing of this book has the following error. The weight of the gate referenced in the last
sentence of the problem statement should be
400 lb
(not
200 lb
). This error is corrected in the second and
subsequent printings of the book.