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08Ma2aPracSol8

# 08Ma2aPracSol8 - Math 2a Practical Fall 2008 Solutions to...

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You are bound by the honor code to not look at this ﬁle if you did not take Math 2a in the Fall of 2008 1. A weight of 5 is attached to a spring with spring constant 2 and hangs at equilibrium in a medium with damping constant 1. At time t = 1 an impulse of magnitude 3 is imparted to the system. Find a formula for the displacement of the system at time t . Find the earliest time t 0 > 1 at which the mass returns to its equilibrium position, and determine the impulse that must be imparted at time t 0 to bring the system to rest again. 10pts Solution. The total mass times acceleration of the weight should equal the total of all forces applied to it. Thus we have 5 y 00 = - y 0 - 2 y + g ( t ), where g ( t ) is the driving force. The driving force is zero everywhere except t = 1, and the integral of the force function (the total impulse) is 3. Thus g ( t ) = 3 δ 1 . We therefore have the equation 5 y 00 + y 0 + 2 y = 3 δ 1 . The initial conditions are y (0) = 0 and y 0 (0) = 0 as the weight begins at equilibrium. Taking Laplace transforms, we have 5 s 2 F ( s ) + sF ( s ) + 2 F ( s ) = 3 e - s F ( s ) = 3 5 1 s 2 + 1 5 s + 2 5 e - s F ( s ) = 3 5 1 ( s + 1 10 ) 2 + 39 100 e - s F ( s ) = 10 39 3 5 39 10 ( s + 1 10 ) 2 + 39 100 e - s . The center term gives e - s/ 10 sin ± 39 10 s ² . We transfer the factor from in front, and then the e - s term indicates a shift of - 1 and multiplication by the heaviside function u 1 ( t ). Thus we obtain

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08Ma2aPracSol8 - Math 2a Practical Fall 2008 Solutions to...

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