Problem — Wet pendulum

Problem

A pendulum is partly immersed in a viscous liquid (for example dense oil) and set oscillating with small amplitudes. What changes compared with the pendulum in air, and how? Discuss amplitude, period, energy.

Solution

The oil introduces strong viscous damping.

Amplitude: decays exponentially and rapidly, with $\gamma$ large, so the pendulum completes few oscillations before stopping:

$A(t) = A_0\, e^{-(\gamma/2)t}$

Energy: mechanical energy is dissipated as heat in the fluid, so it falls off rapidly.

Period: the damped angular frequency is slightly less than the natural one, so the period lengthens a little:

$\omega_d = \omega_0\sqrt{1 - \zeta^2}$

If the damping is very strong ($\zeta \geq 1$) the pendulum no longer oscillates: an aperiodic return to equilibrium.

$\ev{\text{amplitude decays rapidly; period barely altered (as long as } \zeta < 1)}$

Links

Topics: Oscillations and simple harmonic motion Concepts: Damped oscillations · Simple pendulum Skills: Analysis of limiting cases and thought experiments Objects: Simple pendulum