A mass-spring-damper system 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 mass-spring-damper system obeys

m\ddot x+c\dot x+kx=0

with

m=1\,\mathrm{kg}

c=2\,\mathrm{kg\,s^{-1}}

k=101\,\mathrm{N\,m^{-1}}

x(0)=0.05\,\mathrm{m}

\dot x(0)=0\,\mathrm{m\,s^{-1}}

. A symbolic solver returns

x(t)=e^{-t}\left(A\cos(10t)+B\sin(10t)\right)

with

t

in seconds and

x

in metres, but does not report

A

and

B

. ((a)) Find

A

and

B

(in metres) from the initial conditions. ((b)) Verify by direct substitution into

m\ddot x+c\dot x+kx=0

that the resulting

x(t)

is an exact solution, and state the value of the residual.