Question 3.6: The damped mass–spring system shown in Fig. 3.12 has mass m ...

The natural frequency ω of the system is
ω = \sqrt{k}{m} = \sqrt{\frac{1000}{10}} = 10 rad/s

The critical damping coefficient C_{c} is given by
C_{c} = 2mω = 2(10)(10) = 200N · s/m

The damping factor ξ is given by
ξ = \frac{c}{C_{c}} = \frac{200}{200} = 1

Since ξ = 1, the system is critically damped and the solution is given by
x(t) = (A_{1} + A_{2}t)e^{−10t}
where the constants A_{1} and A_{2} can be determined from the initial conditions.