Pappus Theorem -Machine Design
There
are two Theorem associated with Pappus Surface of Revolution and Body of Revolution. Both the theorem has numerous applications
in real life situations. This research paper was conducted to find out what are
these theorems by means of lieterature review and the how can these theorems be
used in daily life applications. A fine knowledge of
these theorem along with calculus can
be used in solving many daily life problems.
Contents
There are two Theorem associated
with Pappus (Al-Dhahir, M. W. 1957).
2.
Theorem 2: Body of Revolution
Suppose a surface
is formed by rotating of a plane curve. This plane rotates about a
non-traversing axis. Then according to the Surface of Revolution theorem, the
surface area (say A) of the surface will be equal to the product of the length of the curve (say L)and the distance
travelled (Say d ) of the centroid throughout the generation of the surface by
revolution. Suppose now we body of revolution which is formed by the revolution
of a plane area about a non-crossing axis then the volume of this body ( say V)
will be equal to the product of its area ( Say A) and the distance travelled
(say d) of the centroid all through the formation of the body.
Literature Review
There are two Theorem associated
with Pappus:
4.
Theorem 2: Body of Revolution
Suppose
a surface is formed by rotating of a plane curve. This plane rotates about a
non-traversing axis. Then according to the Surface of Revolution theorem, the
surface area (say A) of the surface will be equal to the product of the length of the curve (say L)and the distance travelled
(Say d ) of the centroid throughout the generation of the surface by
revolution. (Grattan-Guinness, I., & Bos, H. J. (Eds.).2000).
A= L*d = L*( 2*π*y)
dA=
( 2*π*y) dL
A = ƪ dA = ƪ ( 2*π*y) dL =2*π ƪ y dL
Theorem 2: Body of Revolution
Suppose now we body
of revolution which is formed by the revolution of a plane area about a non-crossing
axis then the volume of this body ( say V) will be equal to the product of its
area ( Say A) and the distance travelled (say d) of the centroid all through
the formation of the body.
V
=A*d = A*(2*π*y)
dV=
(2*π*y)dA
V=
ƪdV= ƪ(2*π*y)dA = 2*π ƪ ydA
Applications
of Theorems of Pappus-Guldinus
Application1:
Let us find the
1)
internal
surface area and
2)
volume of the bowl (Beer, F. P. 2010).
Application
2:
Find the centroid of the volume
which can be made by revolving the marked area about the horizontal axis. (Awrejcewicz,
J. 2012).
Application 3:
Determine the area of the
following figure.
We know that the axis of rotation
is the horizontal axis. Also, the forming curve is a plain circle.
The length will be ,L = 2*π*1m =
6.283 m
Area,A will be: A= = π(4 m)(6.283
m) = 79.0 m2
Results and conclusion
The
two theorems associated with Pappus: Surface of Revolution and Body of Revolution were studied in this report. Both the theorem has numerous applications in
real life situations.
Suppose a surface is formed by rotating of a plane curve. This plane rotates
about a non-traversing axis. Then according to the Surface of Revolution
theorem, the surface area (say A) of the surface will be equal to the product
of the length of the curve (say L)and
the distance travelled (Say d ) of the centroid throughout the generation of
the surface by revolution. Suppose now we body of revolution
which is formed by the revolution of a plane area about a non-crossing axis
then the volume of this body ( say V) will be equal to the product of its area
( Say A) and the distance travelled (say d) of the centroid all through the
formation of the body.
Bibliography:
1)
Al-Dhahir,
M. W. (1957). A simplified proof of the Pappus-Leisenring theorem. The
Michigan Mathematical Journal, 4(3), 225-226.
2)
Grattan-Guinness,
I., & Bos, H. J. (Eds.). (2000). From the calculus to set theory,
1630-1910: An introductory history. Princeton University Press.
3)
Beer,
F. P. (2010). Vector mechanics for engineers: statics and dynamics. Tata
McGraw-Hill Education.
4)
Awrejcewicz,
J. (2012). Classical mechanics: kinematics and statics (Vol. 28).
Springer.
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