Ponytail Motion Essay

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SIAM J. APPL. MATH. Vol. 70, No. 7, pp. 2667–2672 c 2010 Society for Industrial and Applied Mathematics Downloaded 09/25/12 to 190.250.80.100. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PONYTAIL MOTION∗ JOSEPH B. KELLER† Abstract. A jogger’s ponytail sways from side to side as the jogger runs, although her head does not move from side to side. The jogger’s head just moves up and down, forcing the ponytail to do so also. We show in two ways that this vertical motion is unstable to lateral perturbations. First we treat the ponytail as a rigid pendulum, and then we treat it as a flexible string; in each case, it is hanging from a support which is moving up and down periodically, and we solve the linear equation for small lateral oscillation. The angular displacement of the pendulum and the amplitude of each mode of the string satisfy Hill’s equation. This equation has solutions which grow exponentially in time when the natural frequency of the pendulum, or that of a mode of the string, is close to an integer multiple of half the frequency of oscillation of the support. Then the vertical motion is unstable, and the ponytail sways. Key words. instability, parametric resonance, Hill’s equation AMS subject classifications. 34F15, 70J40 DOI. 10.1137/090760477 1. Introduction. The ponytail of a running jogger sways from side to side, but the jogger’s head generally does not move from side to side. The head just moves up and down, so the ponytail also moves up and down with it. But, as we shall show, this vertical motion of the hanging ponytail is unstable to lateral perturbations. The resulting lateral motion, the swaying, is an example of parametric excitation, a phenomenon which is common in oscillating mechanical and electrical systems. We shall demonstrate this instability, and analyze the resulting motion, in

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