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

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subject Pages 9
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subject Authors Anthony M. Bedford, Wallace Fowler

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Problem 7.124 Determine the reactions on member
ABCD at Aand D.
AF
B
C
DE
2 kN/m
1 m
1 m
1 m
1 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
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7.124 (Continued)
Solving, we get
AxD7kN
AyD6kN
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Problem 7.125 Estimate the centroid of the volume of
the Apollo lunar return conguration (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
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Problem 7.126 The shape of the rocket nozzle of the
Apollo lunar return conguration 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
a2x2
C2bc x5
2.83
a2xC2ab x2
C2bc x4
2.83
xD4.43
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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
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Problem 7.129 Determine the ycoordinate of the
center of mass of the homogeneous steel plate.
y
80 mm
20 mm
20 mm
10 mm
20 mm
1
The centroid:
3
1
AixiA4x4
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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 ARD50150D0.75 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
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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
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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) A3D412D150.8in
x3D0,yD24
AD
1
AiD354.796 in2.
The centroid for the composite:
3
Aixi
594
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Problem 7.134 Determine the center of mass of the
homogeneous object.
y
60 mm
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) V1D1
(2) V2D6070⊳⊲10D42000 mm3,
x2D35 mm,
VD8.019 mm
zDV1z1CV2z2V3z3
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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 ve 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
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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
B
A
A
x
y
Solution:
Fy:AyCByWBD0
xD1.511 m
1.15 m WB
2.3 m
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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) L3D4D12.566 in,x3D8C24
LD6.58 in,
598
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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 prole, 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, nd the location of the center of mass of the
unloaded truck (and its mass). Then nd the center of mass and mass
of the load. Combine to nd the wheel loads on the loaded truck.
Now we can nd the wheel loads on the loaded truck
Fx: no forces
page-pff
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 earthmoon system?
1C0.0123 383,000
600

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