The Center of Gravity of Classes of Cylindrical Solids via a Computer Algebra System

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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