A horizontal spring–mass system of mass and spring constant is driven by a periodic…

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

A horizontal spring–mass system of mass

m=0.50\,\text{kg}

and spring constant

k=200\,\text{N\,m}^{-1}

is driven by a periodic force while a light damping force

b\dot x

acts on the mass. The exact expression for the resonance angular frequency is

\omega_{\text{res}}=\sqrt{\omega_0^{2}-\dfrac{b^{2}}{2m^{2}}}

, where

\omega_0=\sqrt{\dfrac{k}{m}}

. (i) Show that

\displaystyle \lim_{\,b\to 0}\,\omega_{\text{res}}=\omega_0

. (ii) For

b=1.0\,\text{N\,s\,m}^{-1}

, calculate

\omega_0

and the corresponding

\omega_{\text{res}}

(correct to two significant figures). (iii) Determine the percentage decrease in the resonance frequency caused by the given damping.