Vibrations and Waves · Problem 3-15

Forced vibrations and resonance — A.P. French

Physics — Waves · quality-factor-oscillators

Problem

Many oscillatory systems, although the loss or dissipation mechanism is not analogous to viscous damping, show an exponential decrease in their stored average energy with time,

\bar{E} = \bar{E}_0 e^{-\gamma t}

. A

Q

for such oscillators may be defined using the definition

Q = \omega_0/\gamma

, where

\omega_0

is the natural angular frequency. (a) When the note ``middle C'' on the piano is struck, its energy of oscillation decreases to one half its initial value in about 1 sec. The frequency of middle C is 256 Hz. What is the

Q

of the system? (b) If the note an octave higher (512 Hz) takes about the same time for its energy to decay, what is its

Q

? (c) A free, damped harmonic oscillator, consisting of a mass

m = 0.1

kg moving in a viscous liquid of damping coefficient

b

(

F_{\text{viscous}} = -bv

), and attached to a spring of spring constant

k = 0.9

N/m, is observed as it performs damped oscillatory motion. Its average energy decays to

1/e

of its initial value in 4 sec. What is the

Q

of the oscillator? What is the value of

b

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