978-0073380292 Chapter 7 Part 8

Problem 7.112

A breakwater along an oceanfront is to be constructed of concrete, and the

formwork for retaining the concrete while it is poured is shown. For every 6ft

of width (into the plane of the figure), the formwork has a horizontal support

BC

and a support

DF

. The stakes at points

A

,

E

, and

F

may be modeled as

pins, and the weights of all members except the concrete may be neglected. If

the concrete is modeled as a ﬂuid with

150 lb=ft3

specific weight, determine the

force supported by members BC and DF .

Solution

The FBDs for members AB and CDE are shown at

the right, where the forces in members BC and DF

Problem 7.113

A breakwater along an oceanfront is to be constructed of concrete, and the

formwork for retaining the concrete while it is poured is shown. For every 6ft

of width (into the plane of the figure), the formwork has a horizontal support

BC

and a support

DF

. The stakes at points

A

,

E

, and

F

may be modeled as

pins, and the weights of all members except the concrete may be neglected. If

the concrete is modeled as a ﬂuid with

150 lb=ft3

specific weight, determine the

force supported by members BC and DF .

Solution

Problem 7.114

The cross section through a spherical tank is shown. The upper and lower

portions of the tank are attached using 72 bolts that are uniformly spaced

around the perimeter of the tank. The upper portion of the tank has a small

dome that contains a gas. The ﬂuid in the tank has specific weight of

0:06 lb=in:3

and approximately spherical shape. Assuming all the bolts support the same

force, determine the force each bolt supports due to the ﬂuid and gas pressures

if:

(a) The gas is not pressurized (i.e., it is at atmospheric pressure).

(b) The gas is pressurized to 5psi.

Solution

Problem 7.115

The tank shown has a cylindrical midsection with hemispherical ends. Each of

the hemispherical ends is attached to the cylindrical midsection using 60 bolts

that are uniformly spaced around the perimeter of the tank. At

D

, the tank has

a circular access plate that is attached using 12 bolts that are uniformly spaced.

The tank is fully filled with a ﬂuid having density

900 kg=m3

. Assume that each

of the bolts at

B

supports the same force, each of the bolts at

C

supports the

same force, and each of the bolts at

D

supports the same force. However, the

forces supported by the bolts at

B

,

C

, and

D

are probably different. Assume the

piping that enters the tank at

A

is ﬂexible and has negligible weight. Determine

the force each bolt supports due to the ﬂuid pressure if the ﬂuid at

A

is at

atmospheric pressure.

Solution

Problem 7.116

Repeat Prob. 7.115 if the ﬂuid at Ais at 10 kN=m2pressure.

Problem 7.117

The cross section through the valve of a fuel injector for an engine is shown,

where the tip of the valve has conical shape. If the fuel is at

500 kN=m2

pressure,

determine the force

F

that must be applied to keep the valve closed. Hint: The

pressure due to weight of the fuel is negligible compared to 500 kN=m2.

Problem 7.118

Grain is contained in a silo. The walls of the silo are fixed, and the door

ABCD

can be opened to allow the grain to pour out. Door

ABCD

is ﬂat, with

8in:

depth (into the plane of the figure). Idealize the grain to be a ﬂuid with

0:025 lb=in:3

specific weight. In the position shown, the hydraulic cylinder

EG

is horizontal. Neglect the weights of the individual members. Determine the

force the hydraulic cylinder

EG

must support to keep the door in equilibrium.

Report your answer, using a positive value for tension in the hydraulic cylinder

and a negative value for compression.

Solution

Using the FBD for the hydraulic cylinder

EG

, we sum forces in the

x

and

y

directions to obtain the

equilibrium equations

Problem 7.119

A tank with two compartments contains pressurized gas on the left and oil on

the right that are separated by a gate

AB

having

20 in:

width into the plane

of the figure. The gate is supported by a hinge at

A

and a stop at

B

. The oil

compartment is vented so that the surface of the oil is subjected to atmospheric

pressure. Determine the value of the gas pressure that will cause the gate to

begin to open. The specific weight of the oil is

0:02 lb=in:3

, and the specific

weight of the gas is negligible.

Solution

The FBD for the gate is shown at the right, where

BxD0

when the gate

begins to open.

Using Eq. (5), we obtain the gas pressure needed to begin opening the gate as

Problem 7.120

For the area shown, use composite shapes to determine the

x

and

y

positions of the

centroid.

Solution

Using the composite areas shown at the right, the area of

each region and the position of its centroid are

Area AiQxiQyi

1.8 cm/.10 cm/D80 cm25cm 10 cm

NxDPQxiAi

PAi

(1)

3.10 cm/.30 cm2/C.5 cm/.12:57 cm2/

NyDPQyiAi

PAi

(4)

3.6 cm/.30 cm2/C.10 cm/.12:57 cm2/

Problem 7.121

For the area shown, use composite shapes to determine the

x

and

y

positions of the

centroid.

Solution

Using the two composite shapes shown, the xand ypositions of the centroid are

2.0:5 in:/ .2:5 in:/

The denominator in the above expressions is the area of the shape, namely 4:25 in:2.

Problem 7.122

For the area shown, use composite shapes to determine the

x

and

y

positions of

the centroid.

Problem 7.123

For the area shown, use composite shapes to determine the

x

and

y

positions of

the centroid.

Problem 7.124

For the area shown, use integration to determine the

x

and

y

positions of the

centroid.

Solution

For the area of this problem, a vertical area element is more convenient than a

horizontal area element. Thus, with

ytDpx

and

ybDx=4

, the area element

dA, and its centroid Qxand Qyare

2.ytCyb/D1

The xlocation of the centroid is given by

0xpx1

4xdx

5x5=2 1

12 x3ˇ

ˇ

4m

0

The ylocation of the centroid is given by

0

1

2pxC1

4xpx1

4xdx

4x21

96 x3ˇ

ˇ

4m

0

The denominator in the above expressions is the area of the shape, namely 10/3 m2.

Problem 7.125

For the truncated circular cone shown, use composite shapes to determine the

location of the centroid.

Solution

The

x

position of the centroid will be determined using the method of compos-

ite shapes where the object is divided into the two regions shown in the figure.

For each region, the volume

Vi

and its centroid

Qxi

are summarized in the table

The denominator in the above expressions is the volume of the object, namely 6:367105mm3.

Problem 7.126

For the truncated circular cone shown, use integration to determine the location

of the centroid.

Problem 7.127

The truncated circular cone shown has a truncated conical hole.

(a)

Fully set up the integral, including the limits of integration, that will yield

the centroid of the object.

(b)

Evaluate the integral determined in Part (a) using computer software such

as Mathematica or Maple.

Problem 7.128

The truncated circular cone shown has a truncated conical hole and is made

of a material with density

0:002 g=mm3

. Let the conical hole be filled with a

material with density 0:003 g=mm3.

(a)

Fully set up the integral, including the limits of integration, that will yield

the center of mass of the object.

(b)

Evaluate the integral determined in Part (a) using computer software such

as Mathematica or Maple.

Problem 7.129

The bullet-shaped object is a solid of revolution that is composed of materials

with densities

1

and

2

. Set up the integral, including the limits of integration,

that will yield the xposition of the center of mass.