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Problem 7.65
A solid is generated by rotating the shaded area shown 360
ı
about the
y
axis.
Determine the volume and surface area of this solid in terms of the dimension h.
Solution
To determine the volume of the solid of revolution, the Pappus-Guldinus theorem will be used with the
composite areas shown, where
To determine the surface area of the object, the Pappus-Guldinus theorem will be
used with the composite lines shown at the right, where
Problem 7.66
Repeat Prob. 7.65 if the solid is generated by rotating the shaded area about the
x
axis.
2.2h/ .2h/D2h2; A3D h2
4:(1)
The distance from the axis of revolution to the centroid of each region is
Employing the Pappus-Guldinus theorem, the volume of the object is
Problem 7.67
Determine the volume and surface area for the solid of revolution in Example 7.2 on p. 439.
Solution
The object from Example 7.2 is a solid of revolution that is produced
by the generating area shown at the right where the ´axis is the axis of
revolution. To determine the volume of the solid, the Pappus-Guldinus
theorem will be used with the composite areas shown, where
The distance from the axis of revolution to the centroid of each region is
Employing the Pappus-Guldinus theorem, the volume of the object is
Problem 7.68
A metal Sierra cup is used by campers as a multipurpose utensil for drinking,
holding food, and so on. It is an object of revolution that has the generating curve
shown. If the cup is made of titanium sheet that weighs
0:004 lb=in:2
, determine
the volume of fluid the cup is capable of holding and the total weight of the cup.
Solution
Employing the Pappus-Guldinus theorem, the volume of the Sierra cup is
Problem 7.69
A funnel is to be made of thin sheet metal using the generating curve shown. If the
funnel is to be plated with
0:005 mm
thick chrome on all surfaces, determine the
volume of chrome required.
Solution
The volume of chrome required to plate the funnel is the product of the thickness of the
chrome plating (
0:005 mm
) and the total surface area of the funnel (inside plus outside
surface area). To determine the surface area of the funnel, the Pappus-Guldinus theorem
will be used with the composite lines shown, where
Employing the Pappus-Guldinus theorem, the area of one surface of the funnel (i.e., the inside surface area or
the outside surface area) is
Problem 7.70
A pressure vessel is to be constructed as a solid of revolution using the generat-
ing area shown.
(a)
Determine the volume of material needed to construct the pressure vessel.
(b) Determine the outside surface area.
(c) Determine the inside surface area.
Solution
Part (a)
To determine the volume of the pressure vessel, the Pappus-Guldinus theorem will be used with
the composite areas shown, where
The distance from the axis of revolution to the centroid of each area is
Qr1D4
3 .45 mm/D19:10 mm;Qr2D45 mm
2D22:5 mm;(3)
Part (b)
To determine the outside surface area of the pressure vessel, the
Pappus-Guldinus theorem will be used with the composite lines shown at
the right, where
1054 Solutions Manual
Employing the Pappus-Guldinus theorem, the outside surface area of the pressure vessel is
Part (c)
To determine the inside surface area of the pressure vessel, the
Pappus-Guldinus theorem will be used with the composite lines shown at
the right, where
Problem 7.71
The penstock shown is retrofitted to an existing dam to deliver wa-
ter to a turbine generator so that electricity may be produced. The
penstock has circular cross section with
24 in:
diameter throughout
its length, including section
BC
, which is a
90ı
elbow. Determine
the total weight of portion
ABCD
of the penstock, including the
water that fills it. The penstock is made of thin-walled steel with a
weight of 0:07 lb=in:2, and the water weighs 0:036 lb=in:3.
2.24 in:/ .12 in:/2D17;055 in:3:(3)
2.24 in:/2 .12 in:/D2842 in:2:(4)
Using these results, the total weight of section BC is
Problem 7.72
Determine the volume and surface area for the hemispherical solid with a conical cavity shown in Fig. P7.9
on p. 446.
Problem 7.73
The area of Prob. 7.12 on p. 446 is revolved 360
ı
about the line
xD 1
m to create a solid of revolution.
Determine the volume and surface area of the solid. Hint: This problem is straightforward if you first solve
Prob. 7.12 on p. 446 and Prob. 7.29 on p. 447.
Solution
The generating area
A
for the solid of revolution is shown at the right. From
the solution to Problem 7.12, the area Aand the xlocation of its centroid are
Hence, the length of each line, and the distance from the axis of revolution to the centroid of each line, are
Problem 7.74
A shelf in a grocery store supports 100 bags of rice, each bag
weighing
1lb
. Consider the arrangements shown: (a) The
bags are stacked at a uniform height, (b) the bags are stacked
twice as high on the right-hand side as on the left-hand side,
(c) the bags are stacked twice as high on the right-hand side
as on the left-hand side with a linear variation, and (d) the
bags are stacked twice as high in the middle as at the two
ends with linear variations. For each of the arrangements,
develop an expression (or multiple expressions if needed)
for the distributed force was a function of position x.
Solution
Loading (c) Use composite shapes:
wcan be written in the form wDaCbx. Solve for aand bto obtain
Statics 2e 1059
Loading (d) Use composite shapes:
w
can be written in the form
wDaCbx
. Solve for
a1
and
b1
for
0x18 in:
and then solve for a2and b2for 18 in:x36 in:, to obtain
Problem 7.75
A cantilever beam supports a wall built of 1000 bricks, each
brick having 5kg mass. Consider the arrangements shown:
(a) The wall has uniform height, (b) the wall is twice as high
on the left-hand side as on the right-hand side, (c) the wall
is twice as high on the left-hand side as on the right-hand
side with a linear variation, and (d) the wall is twice as
high in the middle as at the two ends with linear variations.
For each of the arrangements, develop an expression (or
multiple expressions if needed) for the distributed force
w
as a function of position x.
(3)
wcan be written in the form wDaCbx. Solve for aand bto obtain
Statics 2e 1061
Loading (d) Use composite shapes:
Problem 7.76
Determine the support reactions, using composite shapes to represent the distributed load.
Loading (a), shown above, in Prob. 7.74.
Problem 7.77
Determine the support reactions, using composite shapes to represent the distributed load.
Loading (d), shown above, in Prob. 7.74.
Problem 7.78
Determine the support reactions, using composite shapes to represent the distributed load.
Loading (a), shown above, in Prob. 7.75.
Problem 7.79
Determine the support reactions, using composite shapes to represent the distributed load.
Loading (d), shown above, in Prob. 7.75.
