978-0073380292 Chapter 7 Part 5

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

0x18 in:

and then solve for a2and b2for 18 in:x36 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.