Classical Mechanics (Kibble & Berkshire) · Problem 5.15

Energy conservation — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · nonlinear-oscillations-piecewise-linear

Problem

A partially damped oscillator satisfies the equation x¨+2κx˙+Ω2x=0,\ddot{x} + 2\kappa\dot{x} + \Omega^2 x = 0, where Ω\Omega is a positive constant and κ\kappa is given by κ={0x<0Kx>0\kappa = \begin{cases} 0 & x < 0 \\ K & x > 0 \end{cases} where KK is a positive constant such that K<ΩK < \Omega. Find the period of the oscillator and the ratio of successive maximum values of xx.
Intuitive explanation

This problem combines oscillation with position-dependent damping—the system only experiences friction when x>0x > 0. The key insight is that the motion splits into two regimes: underdamped oscillation in the positive half (where damping acts) and undamped oscillation in the negative half (where it doesn't). You'll need to track how energy dissipates during each positive excursion, then use that to find how successive peaks decay and how long a full cycle takes.

Intuitive explanation

This problem combines oscillation with position-dependent damping—the system only experiences friction when x>0x > 0. The key insight is that the motion splits into two regimes: underdamped oscillation in the positive half (where damping acts) and undamped oscillation in the negative half (where it doesn't). You'll need to track how energy dissipates during each positive excursion, then use that to find how successive peaks decay and how long a full cycle takes.

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