Classical Mechanics (Kibble & Berkshire) · Problem 5.7

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

Physics — Mechanics · free-linear-oscillations-fastest-decay

Problem

Fastest decay The oscillations of a galvanometer satisfy the equation

\ddot{x} + 2K\dot{x} + \Omega^2 x = 0.

The galvanometer is released from rest with

x = a

and we wish to bring the reading permanently within the interval

-\epsilon a \leq x \leq \epsilon a

as quickly as possible, where

\epsilon

is a small positive constant. What value of

K

should be chosen? One possibility is to choose a sub-critical value of

K

such that the first minimum point of

x(t)

occurs when

x = -\epsilon a

. [Sketch the graph of

x(t)

in this case.] Show that this can be achieved by setting the value of

K

to be

K = \Omega\left[1 + \left(\frac{\pi}{\ln(1/\epsilon)}\right)^2\right]^{-1/2}.

K

has this value, show that the time taken for

x

to reach its first minimum is approximately

\Omega^{-1}\ln(1/\epsilon)

when

\epsilon

is small.