Suppose we have a spring - mass - damper system with mass m = c and spring constant k = 6. 3, damping constant 2' (a) Find the damping constant c such that the system will be critically damped and write down the general solution. [4 marks] (b) Assume that y(0) = 4 and solve for one constant in the general solution. [1 mark] (c) Find the initial velocity y'(0) such that that y(2) = 0 (we have a crossing of the horizontal axis). [4 marks] (d) Suppose we fix the damping constant at c = = 4, and introduce a forced input into the system, given by r(t) = 6 cos(3t). Find the (steady state) amplitude of the forced oscillations. [5 marks] (e) What should we change the input frequency to be to give maximum amplitude in the output oscillations, and what will this amplitude be? [5 marks]