Problem: Oscillation without damping

Problem

Imagine a perfectly isolated pendulum, in total vacuum, with a frictionless pivot. It starts oscillating with $A_0 = 30\;\text{cm}$. What is the amplitude after $N$ oscillations? Discuss the implications: mean lifetime of the oscillation, dissipated energy, and why no real system is truly conservative.

Solution

Without dissipation, mechanical energy is conserved, so the amplitude stays constant at every oscillation:

$\ev{A(N) = A_0 = 30\;\text{cm} \text{ for every } N}$

The pendulum would oscillate forever: infinite mean lifetime, zero dissipated energy.

But this is an ideal limit. Every real system has some residual damping: friction at the pivot, gas residue in the “vacuum”, internal friction of the string, even radiative losses. Sooner or later the oscillation dies out.

Perfectly conservative simple harmonic motion does not exist in nature: it is a useful idealisation for understanding the limiting case, not a realisable system.

Collegamenti

Argomenti: Oscillations and simple harmonic motion Concetti: Damped oscillations · Simple harmonic motion · Simple pendulum Competenze: Analysis of limiting cases and thought experiments Oggetti: Simple pendulum