978-0073380292 Chapter 7 Part 3

subject Type Homework Help
subject Pages 14
subject Words 4914
subject Authors Francesco Costanzo, Gary Gray, Michael Plesha

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Statics 2e 1007
Problem 7.30
For the line shown:
(a)
Set up the integrals for integration with respect to
x
, including the limits of
integration, that will yield the xand ypositions of the centroid.
(b) Repeat Part (a) for integrations with respect to y.
(c)
Evaluate the integrals in Parts (a) and/or (b) by using computer software such
as Mathematica or Maple.
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Problem 7.31
For the line shown:
(a)
Set up the integrals for integration with respect to
x
, including the limits
of integration, that will yield the xand ypositions of the centroid.
(b)
Evaluate the integrals in Part (a) by using computer software such as
Mathematica or Maple.
NyDRQy dL
RdL DR2mm
0.2x x2/q1C.22x/2dx
R2mm
:(5)
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Problem 7.32
For the line shown:
(a)
Set up the integrals for integration with respect to
x
, including the limits
of integration, that will yield the xand ypositions of the centroid.
(b)
Evaluate the integrals in Part (a) by using computer software such as
Mathematica or Maple.
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Problem 7.33
A solid cone is shown. Use integration to determine the position of the centroid.
Express your answers in terms of rand h.
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Problem 7.34
The solid shown has a cylindrical hole with
1in:
radius. Use integration to
determine the volume of the solid and the coordinates of the centroid.
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Problem 7.35
The punch shown is used for cutting holes in fabric. It has the shape of a
truncated cone with a conical hole. Use integration to determine the volume of
the punch and the coordinates of the centroid.
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Problem 7.36
For the hemisphere with a conical cavity shown in Fig. P7.9 on p. 446, use integration to determine the
x
location of the centroid. Express your answers in terms of R.
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Problem 7.37
A solid of revolution is produced by revolving the shaded area shown 360
ı
around
the yaxis. Use integration to determine the coordinates of the centroid.
2.y/D1
2x :(5)
The ylocation of the centroid of the area is given by
0:5 in:1
2x .2/dx
0:5 in:
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Problem 7.38
Two planets
A
and
B
have circular orbits, and each rotates in the same plane about
the center of mass
G
for the two-planet system. The distance between the planets’
centers of mass (points
A
and
B
) is always
4105km
. Planet
A
has spherical shape
with radius
rAD6000 km
and uniform density
AD5000 kg=m3
. Planet
B
has
spherical shape with radius
rBD1500 km
and uniform density
BD4000 kg=m3
.
Determine the distances
dA
and
dB
from each planet’s center of mass to the system’s
center of mass G.
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Problem 7.39
An instrument for measuring the direction of wind is shown. It consists of a
hemispherical aluminum cap, cylindrical plastic body, and a triangular sheet
metal fin. The instrument is symmetric about the
xy
plane, and the alu-
minum, plastic, and sheet metal have specific weights of
aD0:1 lb=in:3
,
pD0:04 lb=in:3
, and
sm D0:09 lb=in:2
, respectively. Determine the
x
,
y
,
and ´positions of the center of gravity.
in.2
2.6 in./.5 in./ 2 in. 2in. C5
3in.
D1:350 lb D3:667 in.
NxDPQxiwi
PwiD.15:75 in./.1:676 lb/C.7:5 in./.7:540 lb/C.2 in./.1:350 lb/
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Problem 7.40
The radio antenna shown is symmetric about its horizontal member. The
horizontal member weighs
0:6 lb=ft
, the vertical members weigh
0:1 lb=ft
, and
the circuit box weighs
8lb
with center of gravity at point
A
. Determine the
location of the antenna’s center of gravity, measured from the left-hand end.
1.0:6 lb=ft/ .180 in:/ft
12 in:D9:00 lb 180 in:
2
3.0:1 lb=ft/ .40 in:/ft
12 in:D0:333 lb 30 in:
5.0:1 lb=ft/ .40 in:/ft
12 in:D0:333 lb 60 in:
7.0:1 lb=ft/ .80 in:/ft
12 in:D0:667 lb 135 in:
8.0:1 lb=ft/ .100 in:/ft
12 in:D0:883 lb 150 in:
9 8 lb 195 in:
The xlocation of the center of gravity is given by
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Problem 7.41
In Example 7.2 on p. 439, let the cylindrical portion be steel with specific weight
490 lb=ft3
and the
truncated cone portion be aluminum alloy with specific weight
170 lb=ft3
. Determine the coordinates of
the center of gravity.
D.1:5 in:/ .10:69 lb/C.4in:/ .1:648 lb/C.5:5 in:/ .0:2060 lb/
(2)
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Problem 7.42
In Prob. 7.7 on p. 445, let the cylindrical portion be cast iron with specific weight
450 lb=ft3
and the
hemispherical portion be aluminum alloy with specific weight
170 lb=ft3
. Determine the coordinates of the
center of gravity.
D.2:5 in:/ .36:82 lb/C.6:125 in:/ .5:563 lb/
(2)
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Problem 7.43
A square plate having
1:25 lb=in:2
weight has a circular hole. Along the right-
hand edge of the plate, a circular cross section rod having
0:75 lb=in:
weight is
welded to it. Determine the xposition of the center of gravity.
3.0:75 lb=in./ .1in:/D0:75 lb 1:1 in:
D.0:5 in:/ .1:25 lb/C.0:5 in:/ .0:245 lb/C.1:1 in:/ .0:75 lb/
(1)
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Problem 7.44
An object consists of a rectangular solid with
sD5g=cm3
, a rectangular thin
plate with
pD10 g=cm2
, and a thin wire with
wD60 g=cm
. The object is
symmetric about the y´ plane. Determine the ´position of the center of mass.
cm2.20 cm/.10 cm/D2000 g0
3 .wire/ 60 g
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Problem 7.45
The frame shown consists of a semicircular bar
ABC
, and straight bars
ADC
and
DE
, all with
bD0:5 lb=in:
weight. A circular counterweight with
cD0:1 lb=in:2
is to be welded to the frame at point
E
. Determine the radius
r
of the counterweight so that the center of gravity for the assembly is located
at point D.
D
23:56 lb C15 lb C6lb C0:1 lb
in.2r2
(2)
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Problem 7.46
The assembly shown consists of a flywheel, a counterweight, and a bolt that
attaches the counterweight to the flywheel. The flywheel is uniform with mass
per area of
2g=mm2
and has a
40 mm
diameter hole as shown. Determine the
mass
mC
of the counterweight so that the center of mass of the assembly is
at point
O
. The counterweight is attached to the flywheel by a bolt with mass
mBD800
g that is screwed into a
10 mm
diameter hole that passes completely
through the flywheel.
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Problem 7.47
A floor lamp consists of a half-circular base with weight per area of
0:06 lb=in:2
, tubes
AB
and
BC
, each having weight
0:05 lb=in:
, and the lamp shade at
C
with weight
2lb
. Tube
BC
is parallel to the
x
axis. Determine the coordinates of the center of
gravity.
Solution
Region wiQxiQyi
10:06 lb=in.21

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