3f-fall2010-exam_2_solutions

3f-fall2010-exam_2_solutions - Math 3F — Exam#2 Solutions...

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Unformatted text preview: Math 3F — Exam #2 Solutions Laney College, Fall 2010 Fred Bourgoin 1. A 3-kg mass is attached to a spring with stiffness k = 75 N/m. The mass is displaced upward 0.25 m and given a downward velocity of 1 m/s. The damping force is negligible. (a) Find the equation of motion of the mass. Answer. We need to solve the initial-value problem 3 u ′′ + 75 u = 0 , u (0) = − . 25 , u ′ (0) = 1 . The characteristic equation is 3 r 2 + 75 = 0, which has roots r = ± 5 i , so u ( t ) = c 1 cos 5 t + c 2 sin 5 t . The initial conditions yield u (0) = c 1 = − . 25 and u ′ (0) = 5 c 2 = 1 , so the solution is u ( t ) = − . 25 cos 5 t + 0 . 2 sin 5 t . (b) Find the amplitude, the period, and the frequency of the motion. Answer. The amplitude is R = √ ( − . 25) 2 + (0 . 2) 2 ≈ . 32 m; the period is 2 π 5 rad; and the frequency is 5 or 5 2 π , depending how you like your units. (c) How long after release does the mass pass through the equilibrium position? Answer. We are looking for the smallest positive value of t for which u ( t ) = 0. − . 25 cos 5 t + 0 . 2 sin 5 t = 0 = ⇒ tan 5 t = . 25 . 2 = 1 . 5 = ⇒ t = 1 5 tan − 1 (1 . 5) ≈ . 18 sec. 2. Solve: y ′′ − 2 y ′ + y = e t t Answer. First solve the complementary equation y ′′ − 2 y ′ + y = 0. The characteristic equation is r 2 − 2 r + 1 = 0 and has double root...
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3f-fall2010-exam_2_solutions - Math 3F — Exam#2 Solutions...

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