# An object with mass 4 kg is attached to both a spring with spring constant 1 N/m and a dash-pot with damping constant γ (measured in units Ns/m).

(24 points) An object with mass 4 kg is attached to both a spring with spring constant 1 N/m and a dash-pot with damping constant γ (measured in units Ns/m). (a) (2 points) Let y(t) be the displacement of the object from equilibrium at time t. Write a second order linear diﬀerential equation for y(t) that describes this system. (b) (3 points) Convert the diﬀerential equation from part (a) into an equivalent ﬁrst order system of diﬀerential equations. (c) (4 points) For what positive value of γ does the system in part (b) have repeated eigenvalues? Call this value γ0. When γ = γ0, is the mass-spring system overdamped, underdamped, or critically damped? (d) (12 points) When γ = γ0, ﬁnd the general solution of the ﬁrst order system from part (b). (e) (3 points) When γ = γ0, sketch a phase portrait for the ﬁrst order system from part (b).

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