Problem — Damping and isochronism

Problem

A real pendulum, left to itself, oscillates with decreasing amplitude. Does the period also change? Explain in five lines why, and which type of damping preserves isochronism.

Solution

With viscous damping (friction force proportional to velocity) the angular frequency of the oscillations becomes $\omega_d = \omega_0\sqrt{1 - \zeta^2}$ where $\zeta$ is the damping factor and $\omega_0$ the undamped natural angular frequency. For light damping ($\zeta \ll 1$) we have $\omega_d \approx \omega_0$, so $\ev{T \text{ practically unchanged}}$ The amplitude decays exponentially over time, but the pace of the oscillations stays the same: isochronism is preserved. Only in the strong-damping regime ($\zeta \to 1$) does the period lengthen noticeably, until it disappears altogether: for $\zeta \geq 1$ the system no longer oscillates and returns to equilibrium without overshooting. Linear/viscous damping is therefore the type that best preserves isochronism.

Links

Topics: Oscillazioni e moto armonico Concepts: Oscillazioni smorzate · Pendolo semplice Skills: Analisi di casi limite e fantafisica Objects: Pendolo semplice