(10 pt) A mass of 500 g stretches a spring 61.25 cm (beyond its natural length). Assume that the gravitational acceleration constant equals 9.8 m/s2. (a) Determine the spring constant. (b) Find the natural frequency of the mass-spring system. (c) Assume that the mass is set in motion by being released from an initial position of 50 cm (initial velocity is zero). Find the position of the mass at any time t 0. (Assume that there is no damping). (d) When will the mass first return to its equilibrium position? 1 (e) Assume that the mass is now attached to a damper with a damping coefficient of 5 N s/m. Find the position of the mass at any time t 0 using the same initial data, describe the behavior of the solution, and decide whether you get critical damping, overdamping, or underdamping. (f) Find the value of the damping coefficient, , that gives critical damping. Then, find the position of the mass at any time t 0 using the critical value for and describe the behavior of the solution. (Use the same initial data.) 2 In the last two parts, assume that = 0 (no damping) and that we have the same initial data as before. (g) Suppose that there is an external forcing term F (t) = 8 cos 4t. Find the position of the mass at any time t 0 and describe the behavior of the solution. Is this behavior expected? Explain. (h) Now suppose that the external forcing term is F (t) = cos 6t. Find the position of the mass at any time t 0 and describe the behavior of the solution. Is this behavior expected? Explain