A spring is stretched 2m when a force of 16nt is applied to it. A mass of 0.5kg

Question

A spring is stretched 2m when a force of 16nt is applied to it. A mass of 0.5kg is attached to the spring, and the damping force equals 4 times the velocity. Find the mass's equation of motion. Is the system overdamped, underdamped, or ciritically damped?

6. A spring is stretched 2 m when a force of 16 ntis applied to it. spring, and the damping force equals 4 A mass of 0.5 kg is attached to the overdamped, times the velocity. Find the mass's equation of Is the system underdamped, or critically damped? motion. (6 pts)

Explanation / Answer

let spring constant be k N/m.

then from hooke's law:

force=k*stretch

==>16=k*2

==>k=8 N/m

mass=m=0.5 kg

damping force=4*v

so equation of motion:

force=-k*x-4*v

==>m*(d^2x/dt^2)+4*(dx/dt)+k*x=0

==>0.5*(d^2x/dt^2)+4*(dx/dt)+8*x=0

==>(d^2x/dt^2)+8*(dx/dt)+16*x=0

comparing with standard equation of motion:

if natural frequency is w0 and damping factor is e,

then w0^2=16

==>w0=4

2*w0*e=8

==>e=8/(2*4)=1

so the system is critically damped.

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