THE RESULTING SHAPE OF A HANGING CHAIN WITH A FUNCTION BOUNDARY CONDITION
Open Access
- Author:
- Heyd, Emma
- Area of Honors:
- Mathematical Sciences
- Degree:
- Bachelor of Science
- Document Type:
- Thesis
- Thesis Supervisors:
- Anita Mareno, Thesis Supervisor
David Scott Witwer, Thesis Honors Advisor
Eugene Boman, Faculty Reader - Keywords:
- Hanging Chain
Catenary
Calculus of Variations - Abstract:
- Mathematics has built the foundation for many areas of study. The creation of some of the concepts of mathematics are rooted in the need to solve physics problems. A common physics problem that is used as an example in calculus is the catenary problem. The catenary is the shape formed by a chain connected to two poles. This problem has been heavily studied and analyzed using the Newtonian approach of forces and a system being in equilibrium, we consider the calculus of variations approach to the hanging chain problem. The calculus of variations was created in parallel to calculus; however, they differ in how one would approach a problem such as the catenary. The calculus of variations focuses on the fact that a chain connected between two poles tends toward a shape that minimizes the system’s potential energy, thus forming a catenary. We derive the equation for the catenary then explore the consequences of two hanging problems using methods of the calculus of variations. The first problem having boundary conditions related to fixing the left endpoint to a specific point while constraining the right endpoint to freely move along the line, mx_1+b. The second problem allowing the left endpoint to move freely up and down a pole say, y = x_0 with the right endpoint the same as the first problem. Both problems yield a general solution of, y(x) = C_1*cosh ((x+C_2)/C_1 ), with C_1 and C_2 being constants dependent on the initial boundary conditions which we analyze. Specifically, the results of this research will need to be explored further in other research projects to have greater worldly applications. However, this research will offer assistance to other young mathematicians in learning and understanding a problem in the calculus of variations.