s251Hex2_mockup(sp09)

s251Hex2_mockup(sp09) - Math 251H Second Midterm Exam 75...

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Math 251H Second Midterm Exam 75 minutes March 25, 2009 SAMPLE EXAM. 1

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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 6 cos( ωt ) Netwons. Assume ω 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 ω 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, γ = 14. Therefore, the equation of motion is given by, 3 u ′′ + 14 u + 6 u = 6 cos ωt, 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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