## The Center of Gravity of Classes of Cylindrical Solids via a Computer Algebra System
*Tilak de Alwis*
`talwis@selu.edu`
Mathematics
**Southeastern Louisiana University**
Department of Mathematics
USA
### Abstract
In this paper, we will show how to use a computer algebra system (CAS) to study the center of gravity of several classes of three dimensional
cylinders. Today many mathematics instructors and researchers are using CAS to enhance their teaching and research. Some of the popular CAS include,
Derive, Matlab, Maple, Axiom, and Mathematica. Even though this paper uses Mathematica as the CAS of choice, other systems can be used as well.
Mathematica is a general purpose CAS. It is capable of performing numerical or algebraic calculations, and two or three dimensional graphing.
The graphics capabilities of Mathematica combined with its powerful programming language makes it an ideal medium to visualize mathematical or
physical phenomena (see [3], [4], [5], [7], [9] and [12] ). Because of its animation and sound generation capabilities, Mathematica can also be
used as a multimedia studio (see [6], [10] and [11] ). The paper also illustrates this aspect, in the context of studying the center of gravity
of several classes of cylinders. For general references on Mathematica, the reader can refer to [2], [13], [16], and [17].
As our first example, consider the solid bounded by the -plane, the fixed elliptic cylinder x^2/a^2 +y^2/b^2 = 1 where a and b are positive
constants, and the plane through the point
(0, 0, c) where c is a positive constant, with variable normal vector <s, t, 1> where s and t are real parameters. Observe that the variable plane just described forms the "roof" of the solid. As the parameters s and t change, the solid changes, and hence its center of gravity G changes (see [1], [14] and [15] ). One can calculate the coordinates of G using the symbolic integration capabilities of Mathematica. Further, the "Eliminate" command of Mathematica can be used to find the locus of the center of gravity G, for changing s and t. Motivated by these computer experiments, an interesting theorem was discovered:
Theorem: Consider the solid bounded by the XY-plane, fixed
elliptic cylinder x^2/a^2 + y^2/b^2
= 1, and the plane through the fixed
point (0,0, c) c>0, with variable
normal vector <s, t, 1> where
s and t are real parameters. Then
the center of gravity G of the solid
is given by G = ( -a^2*s/(4c), -b^2*t/(4c),
(4c^2 + a^2*s^2 + b^2*t^2)/(8c)
). Furthermore, for changing s and
t, the locus of G is an elliptic
paraboloid, given by the equation
z = (c/2) + 2c(x^2/a^2 +y^2/b^2).
The paper also describes how to use the Mathematica programming language to create an animation of the center of gravity G. When the animation is run, one can observe the point G traversing along an elliptic paraboloid, reinforcing our theorem.
Our second example investigates another, more general class
of cylindrical solids. This time
we will consider the solid bounded
by the -plane, the fixed astroidal
cylinder x^(2/n)/a^2 +y^(2/n)/b^2
= 1 where a and b are positive constants,
n>1 is an odd integer, and the plane
through the point (0, 0, c) where
c is a positive constant, with variable
normal vector <s, t, 1> where
s and t are real parameters. A third
class of cylindrical solid one can
consider is bounded by the XY-plane,
the parabolic cylinder y^2 = 4a*x
where a is a positive constant,
the plane x = b where b is a constant,
and the plane through the point
(0, 0, c) where c is a positive
constant, with variable normal vector <s, t, 1>
where s and t are real parameters.
In each case, Mathematica can be
used to investigate the behavior
of the center of gravity of the
corresponding solid. Further details,
and the theorems one can obtain,
are omitted in this abstract.
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