math4b-13 - Math 4B Lecture 13 Doug Moore One of the main...

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Math 4B Lecture 13 May 13, 2013 Doug Moore
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One of the main applications of linear second order differential equations with constant coefficients is to oscillations of mechan- ical and electrical systems. Suppose we imagine attaching a weight to the ceiling by means of the spring. We can let u ( t ) = height of the weight above equilibrium at time t, the equilibrium being measured when no weight is attached to the spring. Suppose that there are only two forces acting on the weight, a spring force F spring = - ku, in accordance with Hooke’s law, where k is the spring constant, and a gravitational force F gravitational = - mg, where m is the mass of the weight attached to the spring, and g is the acceleration of gravity, g = 32 ft/sec 2 or 9 . 8 m/sec 2 .
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These forces are in equilibrium when their sum is zero, - ku - mg = 0 , u = - mg k . This equation enables us to determine the relationship between the spring constant, the mass, and the corresponding displace- ment u from equilibrium. When the weight is not in equilibrium, we expect u ( t ) to vary with t , the variation being given by Newton’s second law of motion F total = m d 2 u dt 2 . Thus F total = F spring + F gravitational m d 2 u dt 2 = - ku - mg.
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The equation m d 2 u dt 2 + ku = - mg is a special case of a d 2 u dt 2 + b du dt + cu = f ( t ) , A particular solution can be found by the method of judicious guessing or undetermined coefficients. 1. If we seek a particular solution of the form u = A , where A is constant, we find that u = a. - g b. - g/k c. - mg d. - mg/k e. None of these Correct answer: d
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If we set x = u - A = u + mg k , then d 2 x dt 2 = d 2 u dt 2 , so the differential equation m d 2 u dt 2 + ku = - mg becomes m d 2 x dt 2 + kx = 0 . This is the same as the equation governing the motion of a cart containing mass m attached to a wall by means of a spring with spring constant k . We recognize this as the equation d 2 x dt 2 + k m x = 0 or d 2 x dt 2 + ω 2 x = 0 of simple harmonic motion, which has the general solution x = A cos( ωt ) + B sin( ωt ) , ω = k m , where A and B are constants.
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For the problem of a cart moving along a track attached to a wall by means of a spring, with x ( t ) = position of the cart to the right of equilibrium at time t, we consider not just the spring force F spring = - kx, where k
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