Post on 14-Dec-2015
Int: 7th Session: 7-3- 1
ENGINEERS MECHANICS STATICS
CHAPTER
7Lecture Notes:Professor A. Salam Al-AmmriSuhad Ibraheem MohammedAl-Khwarizmi College of EngineeringUniversity of Baghdadasalaam.kecbu@uobaghdad.edu.iq
Theorems of Pappus-Guldinus
Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad2/17/2012
Int: 7th Session: 7-3- 2
Introduction
Theorms of Pappus-Guldinus
Sample Problem 7.3.1
Sample Problem 7.3.2
Sample Problem 7.3.3
Sample Problem 7.3.4
Sample Problem 7.3.5
Sample Problem 7.3.6
Sample Problem 7.3.7
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Contents
Int: 7th Session: 7-3- 3
Introduction• Theorems of Pappus-Guldinus are useful to find a surface area or volume of
revolution.
Volume (V) of revolutionSurface area (As) of revolution
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 4
Theorems of Pappus-Guldinus
• Surface of revolution is generated by rotating a plane curve about a fixed axis.
• Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation.
LyAFor
LyA
2,2
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 5
Theorems of Pappus-Guldinus
• Body of revolution is generated by rotating a plane area about a fixed axis.
• Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation.
AyV 2
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 6
Sample Problem 7.3.1
The circular arc is rotated through 360o about the y-axis.Determine the outer surface area of the resulting body, which is a portion of a sphere.
222
4
22222
22
4/4
sin
a
aaLxA
aa
x
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 7
Sample Problem 7.3.2The body shown in cross section is a complete circular ring formed by revolving the octagonal area about the z-axis. The entire surface is to be covered with a special coating . Determine this surface area.
2177100
220(4)40(2)60(22
6020402
2
mm
LrA
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 8
Sample Problem 7.3.3
The two circular arcs AB and BC are revolved about the vertical axis to obtain the surface of revolution shown. Compute the area A of the outside of this surface.
29.157
)2()()4(22
in
LrA
SymmetrybyCentroideG
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 9
Sample Problem 7.3.4The outside diameter of a pulley is 0.8 m, and the cross section of its rim is as shown. Knowing that the pulley is made of steel and that the density of steel is ρ=7.85x103 kg/m3,determine the mass and weight of the rim.
kgVm 0.60101065.71085.7 963
NmgW 58981.90.60
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 10
Calculate the mass m of concrete required to constructthe arched dam shown. Concrete has a density of 2.40 Mg/m3.
Sample Problem 7.3.5
kgVVm
AyV
my
y
A
Ayy
i
ii
9
2
2
2
)10(126.1)(2400
)70(4
)80(80)52.175(3
52.175
)70(4
)80(80
1203
)70(4)70(
412040)80(80
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 11
A hand-operated control wheel made of aluminum has the proportions shown in the cross-sectional view. The area of the total section shown is 15,200 mm2, and the wheel has a mass of 10 kg. Calculate the distance to the centroid of the half-section. The aluminum has a density of 2.69 Mg/m3.
Sample Problem 7.3.6
mmr
rV
V
M
85.77
)10()2/15200(22690
10 6
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad
Int: 7th Session: 7-3- 12
Sample Problem 7.3.7A steel die, shown in section ,has the form of a solid generated by revolving the shaded area around the z-axis.Calculate the mass of the die.
kgVm
mmm
ArV
5.84)01079.0(7830
01079.0)10(79.10
3
)60(460)60(
2
16060)100()200(2
2
336
2
2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad