A block on a spring obeys with

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Oscillations, Waves & Acoustics · Scientific programming for physicists

Scientific Computing for Physicists, Ch Letting the computer do advanced algebra: operators, series and differentiated programs

Problem

A block on a spring obeys

\ddot{x}+\gamma\dot{x}+\omega_0^2x=(F_0/m)\cos(\omega t)

with

\omega_0=3\,\mathrm{rad\,s^{-1}}

\gamma=0.4\,\mathrm{s^{-1}}

and

F_0/m=2\,\mathrm{m\,s^{-2}}

; ignore the transient and keep only the steady state

x_p(t)=A\cos(\omega t-\varphi)

with

A=(F_0/m)/\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}

and phase

\varphi

(0,\pi)

fixed by

\cos\varphi=(\omega_0^2-\omega^2)/D

and

\sin\varphi=\gamma\omega/D

, where

D=\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2\omega^2}

. ((a)) Verify by direct substitution into the ODE that

x_p(t)

solves it for any driving frequency

\omega

, using the stated formulas for

A

and

\varphi

. ((b)) Evaluate

A

\omega=1.0

2.9

and

5.0\,\mathrm{rad\,s^{-1}}

. ((c)) Evaluating

\omega_0^2-\omega^2

suffers cancellation of relative size

\kappa=(\omega_0^2+\omega^2)/|\omega_0^2-\omega^2|

. Using the chapter's rule for testing generated code,

100\,\kappa\,\epsilon_{\rm mach}

with

\epsilon_{\rm mach}=2^{-52}

, give the relative tolerance to which a generated function for

A(\omega)

should agree with an independently coded evaluation of the same formula at

\omega=2.9\,\mathrm{rad\,s^{-1}}