Vibrations and Waves · Problem 4-14

Coupled oscillators and normal modes — A.P. French

Physics — Waves · power-resonance-curve

Problem

The figure shows the mean power input

\bar{P}

as a function of driving frequency for a mass on a spring with damping. (Driving force

= F_0 \sin \omega t

, where

F_0

is held constant and

\omega

is varied.) The

Q

is high enough so that the mean power input, which is maximum at

\omega_0

, falls to half-maximum at the frequencies

0.98\omega_0

and

1.02\omega_0

. (a) What is the numerical value of

Q

? (b) If the driving force is removed, the energy decreases according to the equation

E = E_0 e^{-\gamma t}

What is the value of

\gamma

? (c) If the driving force is removed, what fraction of the energy is lost per cycle? A new system is made in which the spring constant is doubled, but the mass and viscous medium are unchanged, and the same driving force

F_0 \sin \omega t

is applied. In terms of the corresponding quantities for the original system, find the values of the following: (d) The new resonant frequency

\omega_0'

. (e) The new quality factor

Q'

. (f) The maximum mean power input

\bar{P}_m'

. (g) The total energy of the system at resonance,

E_0'

Related theory

Power In Ac Circuits And Power Factor

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