Mechanical Engineering Chapter 7 Problem Determine The Reactions Member Abcd And Knm Knmde Maf Msolution First

Problem 7.124 Determine the reactions on member

ABCD at Aand D.

AF

B

C

DE

2 kN/m

1 m

2 kN/m

Solution: First, replace the distributed forces with equivalent

concentrated forces, then solve for the loads. Note that BF and CE

are two force members.

Distributed Load on ABCD,F1

By the area analogy,

2 kN/m

y

586

7.124 (Continued)

Solving, we get

AxD7kN

AyD6kN

Problem 7.125 Estimate the centroid of the volume of

the Apollo lunar return conﬁguration (not including its

rocket nozzle) by treating it as a cone and a cylinder.

14 ft

10 ft

Nozzle

x

12.8 ft

y

Solution: The volume of the cone is

hCL

3.

VD15.2ft.

588

Problem 7.126 The shape of the rocket nozzle of the

Apollo lunar return conﬁguration is approximated by

revolving the curve shown around the xaxis. In terms

of the coordinate system shown, determine the centroid

of the volume of the nozzle.

y

x

y = 0.350 + 0.435x – 0.035x2

Solution:

Let dV be a disk of radius y and thickness dx

0

xD



a2x2

C2bc x5

2.83



a2xC2ab x2

C2bc x4

2.83

xD4.43

Problem 7.127 Determine the coordinates of the cen-

troid of the volume.

y

x

z

30 mm

40 mm

20 mm

120 mm

100

mm

Solution: From symmetry yDzD0

Problem 7.128 Determine the surface area of the

volume of revolution in Problem 7.127.

6 in

9 in

5 in

590

Problem 7.129 Determine the ycoordinate of the

center of mass of the homogeneous steel plate.

y

80 mm

20 mm

10 mm

20 mm

1

The centroid:

3



1

AixiA4x4

Problem 7.130 Determine the xcoordinate of the

center of mass of the homogeneous steel plate.

y

220 mm

Solution: The quarter circle: The equation of the circle is x2C

y2DR2. Take the elemental area to be a vertical strip of height

4D3.8013 ð104mm.

The area of the rectangle is ARD50⊲150⊳D0.75 ð104mm2. The

Problem 7.131 The area of the homogeneous plate is

10 ft2. The vertical reactions on the plate at Aand Bare

80 lb and 84 lb, respectively. Suppose that you want to

equalize the reactions at Aand Bby drilling a

1-ft-diameter hole in the plate. What horizontal distance

from Ashould the center of the hole be? What are the

resulting reactions at Aand B?

AB

WD2.56 ft.

The weight density is

X

AB

592

Problem 7.132 The plate is of uniform thickness and

is made of homogeneous material whose mass per unit

area of the plate is 2 kg/m2. The vertical reactions at

Aand Bare 6 N and 10 N, respectively. What is the x

coordinate of the centroid of the hole?

1 m

AB

Problem 7.133 Determine the center of mass of the

homogeneous sheet of metal.

x

y

Solution: Divide the object into four parts: (1) The lower plate,

(3) A3D4⊲12⊳D150.8in

x3D0,yD2⊲4⊳

AD

1

AiD354.796 in2.

The centroid for the composite:

3



Aixi

594

Problem 7.134 Determine the center of mass of the

homogeneous object.

y

60 mm

10 mm

20 mm

30

mm

z

Solution: Divide the object into three parts and the composite:

(1) A triangular solid 30 mm altitude, 60 mm base, and 10 mm thick.

(2) A rectangle 60 by 70 mm by 10 mm. (3) A semicircle with radius

(1) V1D1

(2) V2D60⊲70⊳⊲10⊳D42000 mm3,

x2D35 mm,

VD8.019 mm

zDV1z1CV2z2V3z3

Problem 7.135 Determine the center of mass of the

homogeneous object. y

z

x

1.5 in

z

y

5 in

Top View

Solution: Divide the object into ﬁve parts plus the composite.

(1) A solid cylinder with 1.5 in radius, 3 in long. (2) A rectangle 3

V3 14.137 5 0.5 0

V5 3.534 4.363 0 0

Composite 43.27 2.09 0.3267 0

The composite is

3



5



596

Problem 7.136 The arrangement shown can be used to

determine the location of the center of mass of a person.

A horizontal board has a pin support at Aand rests on

a scale that measures weight at B. The distance from A

to Bis 2.3 m. When the person is not on the board, the

scale at Bmeasures 90 N.

(a) When a 63-kg person is in position (1), the scale

at Bmeasures 496 N. What is the xcoordinate of

the person’s center of mass?

(b) When the same person is in position (2), the scale

measures 523 N. What is the xcoordinate of his

center of mass?

y

x

(1)

(2)

B

A

x

y

Solution:

Fy:AyCByWBD0

xD1.511 m

1.15 m WB

2.3 m

Problem 7.137 If a string is tied to the slender bar at

Aand the bar is allowed to hang freely, what will be the

angle between AB and the vertical?

A

B

8 in

4 in

Solution: When the bar hangs freely, the action line of the weight

will pass through the mass center. With a homogenous, slender bar,

the mass center corresponds to the centroid of the lines making up

the bar. Choose the origin at A, with the xaxis parallel to the lower

(3) L3D4D12.566 in,x3D8C2⊲4⊳

LD6.58 in,

598

Problem 7.138 When the truck is unloaded, the total

reactions at the front and rear wheels are AD54 kN

and BD36 kN. The density of the load of gravel is

D1600 kg/m3. The dimension of the load in the z

direction is 3 m, and its surface proﬁle, given by the

function shown, does not depend on z. What are the

total reactions at the front and rear wheels of the loaded

truck?

y

2.8 m 3.6 m

5.2 m

AB

y = 1.5 – 0.45x + 0.062x2

x

Solution: First, ﬁnd the location of the center of mass of the

unloaded truck (and its mass). Then ﬁnd the center of mass and mass

of the load. Combine to ﬁnd the wheel loads on the loaded truck.

Now we can ﬁnd the wheel loads on the loaded truck

Fx: no forces

Problem 7.139 The mass of the moon is 0.0123 times

the mass of the earth. If the moon’s center of mass is

383,000 km from the center of mass of the earth, what

is the distance from the center of mass of the earth to

the center of mass of the earth–moon system?

1C0.0123 ⊲383,000⊳

600