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Unformatted text preview: Math 251H Second Midterm Exam 75 minutes March 25, 2009 SAMPLE EXAM. 1 1. A 3 kg mass is suspended from a spring, which stretches the spring 5 m from its natural length. The system is placed into the liquid with damping constant 14 Newton-seconds per meter. At t = 0, the system is at rest at its equilibrium position, then receives an external force of 6cos( t ) Netwons. Assume is positive and g = 10 m/s 2 . (a) Set up an initial value problem that describes the motion of the mass. Be sure to explain any variables that appear in your equation. (b) For which value of will the system have resonance? (c) Find the steady-state solution. Solution: m= 3. From the equilibrium condition mg = KL , we have, K = 6. In addition, = 14. Therefore, the equation of motion is given by, 3 u + 14 u + 6 u = 6cos t,u (0) = 0 ,u (0) = 0 . The natural frequency is radicalbig K/m = 2. Resonance occurs when the frequency is close to the natural frequency....
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