chapter 3.d(theorem pg)

8/18/2019 Chapter 3.d(Theorem PG)

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Engineering Mechanics :STATICS

Lecture #07By,

Noraniah KassimUniversiti Tun Hussein Onn Malaysia(UTHM),



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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



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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%



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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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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.

chapter 3.d(theorem pg)

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