8/18/2019 Chapter 3.d(Theorem PG)
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Engineering Mechanics :STATICS
Lecture #07By,
Noraniah KassimUniversiti Tun Hussein Onn Malaysia(UTHM),
8/18/2019 Chapter 3.d(Theorem PG)
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3
THEOREMS OF PAPPUS AND GULDINUS
Learning Topics:
• Theorem of Pappus and
Guldinus
Today’s Objective:
Students will be able to :
a use the theorems of Pappus and
Guldinus for finding the area and
!olume for a surface of re!olution
8/18/2019 Chapter 3.d(Theorem PG)
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4
THEOREM OF PAPPUS AND GULDINUS
Theorems of Pappus and Guldinus are used to find the surface area
and !olume of an" ob#ect of re!olution
Surface area of revolution is generated
b" re!ol!ing a planar cur!e about a
nonintersecting fi$ed a$is in the plane
of the cur!e%
Volume of revolution is generated
b" re!ol!ing a plane area about a
nonintersecting fi$ed a$is in the plane
of the area%
8/18/2019 Chapter 3.d(Theorem PG)
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SURFACE AREA
Lr A θ =
The area of a surface of re!olution e&uals the product of the length of the
generating cur!e and the distance tra!eled b" the centroid of the cur!e ingenerating the surface area%
where'
A ( surface area of re!olution
) ( angle of re!olution measured in
radians' ) * +,
r ( perpendicular distance from the
a$is of re!olution to the centroid of
the generating cur!e
- ( length of the generating cur!e
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VOLUME
The !olume of a bod" of re!olution e&uals the product of the generating
area and the distance tra!eled b" the centroid of the area in generating the!olume
Ar V θ =
where'
. ( !olume of re!olution
) ( angle of re!olution measured in
radians' ) * +,
r ( perpendicular distance from the
a$is of re!olution to the centroid of
the generating area
A ( generating area
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EXAMPLE
• Surface of re!olution is generated b" rotating a plane cur!e about a fi$ed a$is%
• Area of a surface of re!olution is
e&ual to the length of the generatingcur!e times the distance tra!eled b"
the centroid through the rotation%
L y A π +=
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8
EXAMPLE
• /od" of re!olution is generated b" rotating a planearea about a fi$ed a$is%
• .olume of a bod" of re!olution is
e&ual to the generating area times
the distance tra!eled b" the centroid
through the rotation%
A yV π +=
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EXAMPLE
Given: 0omogeneous thin plate as shown
Find: .olume of the solid obtained b"
rotating the area about 1a the $ a$is
Plan: 2ind the centroid of the plate using
the method of composite
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IN CLASS TUTORIAL (Continued)
Solution :
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IN CLASS TUTORIAL (Continued)
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E3AMP-E 14567
Using integration, determine both the area and the distance yc to the centroid of
the shaded area.
Then using the second theorem of a!!us"u#dinus, determine the $o#ume of the
so#id generated b% re$o#$ing the shaded area about the x a&is.
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