Mechanical Engineering Chapter 7 Problem The Mass The Bar Per Unit Length Kgm Choose The Dimension

Problem 7.103 The mass of the bar per unit length is

2 kg/m. Choose the dimension bso that part BC of the

suspended bar is horizontal. What is the dimension b,

and what are the resulting reactions on the bar at A?

b

30⬚

1 m

A

B

C

Then

Fx:AxD0

Fy:Ayg⊲1.0m⊳g⊲b⊳ D0)

AxD0,A

yD61.6N,bD2.14 m

ρ

g(1.0 m)

ρ

gb

Problem 7.104 The semicircular part of the homoge-

neous slender bar lies in the x–zplane. Determine the

center of mass of the bar.

y

Solution: The bar is divided into three segments plus the com-

posite. The lengths and the centroids are given in the table: The

composite length is:

LD

3



Li.

Problem 7.105 The density of the cone is given by the

equation D0⊲1Cx/h⊳, where 0is a constant. Use

the procedure described in Example 7.17 to show that

the mass of the cone is given by mD⊲7/4⊳0V, where

Vis the volume of the cone, and that the xcoordinate

of the center of mass of the cone is xD⊲27/35⊳h.

x

z

y

R

h

Solution: Consider an element of volume dV of the cone in the

form of a “disk” of width dx. The radius of such a disk at position x

is (R/h)x,sodV D[⊲R/h⊳x]2dx.

The mass of the cone is

572

Problem 7.106 A horizontal cone with 800-mm length

and 200-mm radius has a built-in support at A. Its density

is D6000⊲1C0.4x2⊳kg/m3, where xis in meters.

What are the reactions at A?

y

A

800 mm

200 mm

x

Solution: The strategy is to determine the distance to the line of

action of the weight, from which to apply the equilibrium conditions.

y

200

Problem 7.107 In Active Example 7.18, suppose that

bar 1 is replaced by a bar with the same dimensions

that consists of aluminum alloy with a density of

2600 kg/m3. Determine the xcoordinate of the center

of mass of the machine part.

1

2

y

80 mm

240 mm

40 mm

The xcoordinate of the center of mass is

Problem 7.108 The cylindrical tube is made of

aluminum with mass density 2700 kg/m3. The cylin-

drical plug is made of steel with mass density

7800 kg/m3. Determine the coordinates of the center of

mass of the composite object.

y

x

z

y

z

A

20 mm

35 mm

Section A–A

x

y

100

mm

100

mm

Tube

Plug

Solution: The volume of the aluminum tube is

574

Problem 7.109 In Example 7.19, suppose that the

object is redesigned so that the radius of the hole in

the hollow cylinder is increased from 2 in to 3 in. What

is the xcoordinate of the center of mass of the object?

Front View

zx

y

Side View

10 in

2 in

4 in

12 in

5 in 5 in

Problem 7.110 A machine consists of three parts. The

Part Mass (kg) x(mm) y(mm) z(mm)

2 4.5 150 70 0

to position part 3 so that the center of mass location

Solution: The composite mass is mD2.0C4.5C2.5D9 kg.

2.5D82 mm

2.5D122 mm

2.5D16 mm

Problem 7.111 Two views of a machine element

are shown. Part 1 is aluminum alloy with density

2800 kg/m3, and part 2 is steel with density 7800 kg/m3.

Determine the coordinates of its center of mass.

60 mm

yy

18 mm

2

1

24 mm

8 mm

so their masses are

m1DS1V1D⊲2800⊳⊲17.92 ð105⊳D0.502 kg,

576

Problem 7.112 The loads F1DF2D25 kN. The

mass of the truss is 900 kg. The members of the truss are

homogeneous bars with the same uniform cross section.

(a) What is the xcoordinate of the center of mass of the

truss? (b) Determine the reactions at Aand G.

A

B

G

F2

F1

4 m

C

4 m

3 m

D

E

3 m

y

x

Solution:

(a) The center of mass of the truss is located at the centroid of

the composite line of the axes of the members. The lengths of

(b) The equilibrium equations for the truss are

Problem 7.113 With its engine removed, the mass of

the car is 1100 kg and its center of mass is at C. The

mass of the engine is 220 kg.

(a) Suppose that you want to place the center of mass

Eof the engine so that the center of mass of the

car is midway between the front wheels Aand the

rear wheels B. What is the distance b?

(b) If the car is parked on a 15°slope facing up the

slope, what total normal force is exerted by the

road on the rear wheels B?

1.14 m

A

b

2.60 m

B

0.6 m

0.45 m

C

E

xD2.6

2D1.3m,

mD0.475 m.

(b) Assume that the engine has been placed in the new position, as

given in Part (a). The sum of the moments about Bis

578

Problem 7.114 The airplane is parked with its landing

gear resting on scales. The weights measured at A,B, and

Care 30 kN, 140 kN, and 146 kN, respectively. After a

crate is loaded onto the plane, the weights measured at A,

B, and Care 31 kN, 142 kN, and 147 kN, respectively.

Determine the mass and the xand ycoordinates of the

center of mass of the crate.

x

10 m

6 m B

C

A

6 m

Solution: The weight of the airplane is WAD30 C140 C146 D

316 kN. The center of mass of the airplane:

Myaxis D30⊲10⊳xAWAD0,

Problem 7.115 A suitcase with a mass of 90 kg is

placed in the trunk of the car described in Example 7.20.

The position of the center of mass of the suitcase is

xsD0.533 m, ysD0.762 m, and zsD0.305 m. If

the suitcase is regarded as part of the car, what is the

new position of the car’s center of mass?

Solution: In Example 7.20, the following results were obtained

zcD0.769 m

The new center of mass is at

Problem 7.116 A group of engineering students

constructs a miniature device of the kind described in

Example 7.20 and uses it to determine the center of mass

of a miniature vehicle. The data they obtain are shown

in the following table:

Wheelbase =36 in

Track =30 in Measured Loads (lb)

˛D0˛D10°

Left front wheel, NLF 35 32

Right front wheel, NRF 36 33

Left rear wheel, NLR 27 34

Right rear wheel, NRR 29 30

Determine the center of mass of the vehicle. Use the

same coordinate system as in Example 7.20.

xD36⊲35 C36⊳

WD20.125 in.

Mxaxis DzW⊲Track⊳⊲NRF CNRR⊳D0,

from which

zD⊲30⊳⊲36 C29⊳

WD15.354 in.

from which

yDxWcos⊲10°⊳⊲36⊳⊲32 C33⊳

D8.034 in.

580

c

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Problem 7.117 Determine the centroid of the area by

letting dA be a vertical strip of width dx.

x

y

y = x2

(1, 1)

Solution: The area: The length of the vertical strip is ⊲1y⊳,so

Problem 7.118 Determine the centroid of the area in

Problem 7.117 by letting dA be a horizontal strip of

height dy .

Solution: The area: The length of the horizontal strip is x, hence

the element of area is

Divide by the area: xD3

8

Problem 7.119 Determine the centroid of the area.

x

60 cm

y

80 cm

0

The x-coordinate:

Divide by the area: xDw

2D40 cm

The y-coordinate:

1

h2dx D1

Dab

2CabDab

2D1800 cm2

Check: This is the familiar result. check.

The x-coordinate:

b

0a

bxCaxdx Dax3

3bCax2

2b

0

Dab2

6.

Divide by the area: xDb

3D20 cm

The y-coordinate:

A

ydAD1

2b

0a

bxCa2

dx

D b

6aa

bxCa3b

0

Dba2

6.

D56.36 cm

D27.27 cm

582

c

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Problem 7.120 Determine the centroid of the area.

160 mm

20 mm

40 mm

y

40 mm

Solution: Divide the object into ﬁve areas:

(2) The rectangle 120 mm by 80 mm,

(4) The circle of 20 mm radius, and

(2) A2D9600 mm2,

x4D120 mm, y4D120 mm.

AD90.3mm .

yDA1y1CA2y2CA3y3A4y4

Problem 7.121 The cantilever beam is subjected to a

triangular distributed load. What are the reactions at A?

y

x

200 N/m

A

Solution: The load distribution is a straight line with intercept

wD200 N/m at xD0, and slope 200

10 D20 N/m2. The sum

of the moments is

MDMA10

⊲20xC200⊳x dx D0,

Problem 7.122 What is the axial load in member BD

of the frame?

5 m

10 m

100 N/m

5 m

A

BD

C

E

Solution: The distributed load is two straight lines: Over the

interval 0 y5 the intercept is wD0atyD0 and the slope is

C100

5D20.

MED5

⊲20y⊳y dy C10

100ydy,

d1D⊲2

3⊳5D3.333 m.

The centroid distance of the rectangle is d2D7.5 m. The moment

about Eis

MEDd1F1Cd2F2D4583.33 Nm check.

The Complete Structure: The sum of the moments about Eis

MD10ARCMED0,

where ARis the reaction at A, from which ARD458.33 N.

The element ABC : Element BD is a two force member, hence ByD0.

The sum of the moments about C:

MCD5Bx10AyD0,

where Ayis equal and opposite to the reaction of the support, from

which

BxD2AyD2ARD916.67 N.

Since the reaction in element BD is equal and opposite, BxD

916.67 N, which is a tension in BD.

Bx

By

Dy

Cx

Ex

Cy

Cx

584

c

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Problem 7.123 An engineer estimates that the maxi-

mum wind load on the 40-m tower in Fig. a is described

by the distributed load in Fig. b. The tower is supported

by three cables A,B, and Cfrom the top of the tower to

equally spaced points 15 m from the bottom of the tower

(Fig. c). If the wind blows from the west and cables B

and Care slack, what is the tension in cable A? (Model

the base of the tower as a ball and socket support.)

N

15 m

A

B

C

(c)

40 m

(a)

200 N/m

400 N/m

(b)

Solution: The load distribution is a straight line with the intercept

wD400 N/m, and slope 5. The moment about the base of the tower

due to the wind load is

200 N/m

θ