Vibrations and Waves · Problem 4-8

Coupled oscillators and normal modes — A.P. French

Physics — Waves · damped-oscillator-no-spring

Problem

A mass

m

is subject to a resistive force

-bv

but no springlike restoring force. (a) Show that its displacement as a function of time is of the form

x = C - \frac{v_0}{\gamma}\,e^{-\gamma t}

where

\gamma = b/m

. (b) At

t = 0

the mass is at rest at

x = 0

. At this instant a driving force

F = F_0 \cos \omega t

is switched on. Find the values of

A

and

\delta

in the steady-state solution

x = A\cos(\omega t - \delta)

. (c) Write down the general solution [the sum of parts (a) and (b)] and find the values of

C

and

v_0

from the conditions that

x = 0

and

dx/dt = 0

t = 0

. Sketch

x

as a function of

t