Re-tightening of a slack pendulum string: a spring model of the impulsive event

Abstract In a classic pendulum problem, a string survives the static tension test at the lowest point but goes slack during the subsequent swing; the stone follows a projectile path and re-enters the circular trajectory with a finite outward radial speed. The ideal inextensible-string model gives the required radial impulse but not its duration or peak force. We introduce a simple spring model that supplies the missing time scale, with a natural frequency that ties together every higher derivative of the radial motion in one general relation, yielding finite expressions for the peak tension, maximum jerk, and impulse duration, and showing that snap vanishes at maximum jerk as a direct consequence of the quarter-period phase shift between derivatives. A nonlinear bungee-cord force law changes the peak tension by only about six percent, so the linear model remains a good approximation for the peak tension. The model describes only the initial radial arrest of the stone, the loading quarter-cycle. Because the tension rises as a simple sine of time, the model also gives a closed-form instant at which a real string of known breaking strength fails --- within the first three percent of the event, long before the modeled tension approaches its peak.