A mass weighing 14 pounds stretches a spring 2 feet. The mass is attached to a dashpot device that offers a damping force numerically equal to β (β > 0) times the instantaneous velocity. Determine the values of the damping constant β so that the subsequent motion is overdamped, critically damped, and underdamped.
(If an answer is an interval, use interval notation. Use g = 32 ft/s2 for the acceleration due to gravity.)
(b) critically damped
Using the newton second lawk is the spring constanteb positive damping constantm mass attachedm(d^(2)x)/(dt^(2))=-kx-b(dx)/(dt)x(t) is the displacement from the equilibrium position(d^(2)x)/(dt^(2))+(b)/(m)(dx)/(dt)+(k)/(m)x=0Converting units of weights in units of mass (equation of motion)m=(W)/(g)=(14)/(32)=0.43" slug "From hook's law we can calculate the spring constant kk=(W)/(s)=(14)/(2)=7lb//ftIf we put m and k into the DE, we get (d^(2)x)/(dt^(2))+(b)/( 0.43 ... See the full answer