Questions: This problem is an example of critically damped harmonic motion. A mass m=5 kg is attached to both a spring with spring constant k=125 N / m and a dash-pot with damping constant c=50 N · s / m. The ball is started in motion with initial position x0=4 m and initial velocity v0=-23 m / s. Determine the position function x(t) in meters. x(t)= (4-3 t) e^-5 t set in motion with the same initial position and velocity, but with the dashpot disconnected (so c=0). Solve the resulting differential equation to find the position function u(t). In this case the position function u(t) can be written as u(t)=C0 cos (omega0 t-alpha0). Determine C0, omega0 and alpha0. C0= 4 omega0= 10 alpha0= 5.2185 (assume 0 leq alpha0<2 pi )

Transcript text: This problem is an example of critically damped harmonic motion. A mass $m=5 \mathrm{~kg}$ is attached to both a spring with spring constant $k=125 \mathrm{~N} / \mathrm{m}$ and a dash-pot with damping constant $c=50 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}$. The ball is started in motion with initial position $x_{0}=4 \mathrm{~m}$ and initial velocity $v_{0}=-23 \mathrm{~m} / \mathrm{s}$. Determine the position function $x(t)$ in meters. \[ x(t)= (4-3 t) e^{\wedge}(-5 t) \] set in motion with the same initial position and velocity, but with the dashpot disconnected (so $c=0$). Solve the resulting differential equation to find the position function $u(t)$. In this case the position function $u(t)$ can be written as $u(t)=C_{0} \cos \left(\omega_{0} t-\alpha_{0}\right)$. Determine $C_{0}, \omega_{0}$ and $\alpha_{0}$. \[ C_{0}= 4 \] \[ \omega_{0}= 10 \] \[ \alpha_{0}= 5.2185 \] (assume $0 \leq \alpha_{0}<2 \pi$ )