int: 7 th session: 7-3- 1 engineers mechanics engineers mechanics statics statics chapter7chapter7...

Int: 7th Session: 7-3- 1

ENGINEERS MECHANICS STATICS

CHAPTER

7Lecture Notes:Professor A. Salam Al-AmmriSuhad Ibraheem MohammedAl-Khwarizmi College of EngineeringUniversity of [email protected]

Theorems of Pappus-Guldinus

Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad2/17/2012


Introduction

Theorms of Pappus-Guldinus

Sample Problem 7.3.1







2/17/2012 Professor Dr. A.Salam Al-Ammri & Suhad Ibraheem Mohammad

Contents


Introduction• Theorems of Pappus-Guldinus are useful to find a surface area or volume of

revolution.

Volume (V) of revolutionSurface area (As) of revolution




• 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




• 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




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



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




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



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



Calculate the mass m of concrete required to constructthe arched dam shown. Concrete has a density of 2.40 Mg/m3.


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



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.


mmr

rV

V

M

85.77

)10()2/15200(22690

10 6



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


int: 7 th session: 7-3- 1 engineers mechanics engineers mechanics statics statics chapter7chapter7...

Documents