1. Engineering Mechanics : STATICS
Lecture #07
By,
Noraniah Kassim
Hairul Mubarak b Hassim
Universiti Tun Hussein Onn Malaysia (UTHM),

2. DAJ 21003 ( Statics & Dynamics )

3. THEOREMS OF PAPPUS AND GULDINUS
Today’s Objective:
Students will be able to :
use the theorems of Pappus and Guldinus for finding the area and volume for a surface of revolution
Learning Topics:
Theorem of Pappus and
Guldinus
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4. THEOREM OF PAPPUS AND GULDINUS
Theorems of Pappus and Guldinus are used to find the surface area
and volume of any object of revolution
Surface area of revolution is generated
by revolving a planar curve about a
nonintersecting fixed axis in the plane
of the curve.
Volume of revolution is generated
by revolving a plane area about a
nonintersecting fixed axis in the plane
of the area.
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5. DAJ 21003 ( Statics & Dynamics )
SURFACE AREA
The area of a surface of revolution equals the product of the length of the
generating curve and the distance traveled by the centroid of the curve in
generating the surface area.
where,
A = surface area of revolution
θ = angle of revolution measured in
radians, θ ≤ 2π
r = perpendicular distance from the
axis of revolution to the centroid of
the generating curve
L = length of the generating curve
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6. DAJ 21003 ( Statics & Dynamics )
VOLUME
The volume of a body of revolution equals the product of the generating
area and the distance traveled by the centroid of the area in generating the
volume
where,
V = volume of revolution
θ = angle of revolution measured in
radians, θ ≤ 2π
r = perpendicular distance from the
axis of revolution to the centroid of
the generating area
A = generating area
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7. DAJ 21003 ( Statics & Dynamics )
EXAMPLE
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.
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8. DAJ 21003 ( Statics & Dynamics )
EXAMPLE
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.
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9. DAJ 21003 ( Statics & Dynamics )
EXAMPLE
Given: Homogeneous thin plate as shown
Find: Volume of the solid obtained by
rotating the area about (a) the x axis
Plan: Find the centroid of the plate using
the method of composite
ANSWERS:
2. D
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10. IN CLASS TUTORIAL (Continued)
Solution :
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11. IN CLASS TUTORIAL (Continued)
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12. DAJ 21003 ( Statics & Dynamics )
EXAMPLE (9-86)
Using integration, determine both the area and the distance yc to the centroid of the shaded area.
Then using the second theorem of PappusGuldinus, determine the volume of the solid generated by revolving the shaded area about the x axis.
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