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Volume 14: Pages 309-313, 2001
Exciting a Mass Revolving on a String at Constant Tension
4632 North Lane, Del Mar, California 92014‐4134 U.S.A.
The two‐dimensional problem of a mass revolving in a plane on the end of a string always held at constant tension is discussed in terms of a general analytic procedure developed recently, but the history of the problem goes back to Newton. As found here, the governing equation for the orbiting mass is a nonlinear form of the harmonic oscillator equation that does not appear to have a name in the literature. The analytic solution to this nonlinear equation is a circle with the central force always directed to the center of the circle. Newton found this solution using a geometrical method. Before and after a disturbance, within the plane of motion, the orbit will still be a circle but its size will be different in general. For any reasonable disturbance the solution is stable. A perturbation that increases the tangential speed of the particle will cause the radius to expand and one that slows down the mass will cause the radius to shrink such that the centrifugal force remains constant. Energy, angular momentum, and action are not conserved before and after the disturbance. It is believed that the results of studying this simple mechanical problem will increase our understanding of the far more complex three‐body problem, as outlined below.
Keywords: orbital problem, Newtonian solution, three‐body problem
Received: February 15, 2001; Published online: December 15, 2008