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The vibration of a building can be simplified and modelled as a mass spring system. Suppose that the roof mass is m = 20, 000 kg , the spring constant is k = 60, 000 N/m, and the damping constant is c = 40, 000 N s/m. Suppose that the building is initial not moving and is at its equilibrium position.

a) (4 pts) At time t = 0, an earthquake strikes the building with an external force of f(t) = 105 cos(t). Write down the equation and the initial conditions that model this situation.

b) (6 pts) Solve the above equation in part a). Note that you answer will still have in it. Identify the transient solution and the steady state solution from your answer. Then, explain the behavior of the building in the long run.

c) (6 pts) Write down the function for the amplitude of the steady state solution. Then, find the value that causes the practical resonance and compute the corresponding maximum amplitude.

d) (4 pts) If the building moves more than 0.5 m away from its equilibrium solution, it will collapse. Will this building collapse in this case and why? If the building collapse, should we make the damping constant c smaller or bigger to avoid this disaster? Explain your reasoning.

e) (Extra Credit, +3 pts to this exam) Find the value of the damping coefficient c such that when c < c the building will not collapse but when c > c the building will

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collapse