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Unformatted text preview: October 3, 2011 11-1 11. Some applications of second order differ- ential equations The first application we consider is the motion of a mass on a spring. Consider an object of mass m on a spring suspended vertically as in the next figure. Assume that the unstretched spring has length and that gravity pulls the mass down with a force equal to October 3, 2011 11-2 mg in magnitude with g = 32 ft/sec 2 9 . 8 m/sec 2 . We also assume Hookes law which says that the force on the object exerted by the spring has magnitude kx where k is a positive constant and x is the displacement of the spring from its unstretched state. If the spring is extended, then the force is exerted toward the spring, while it is exerted away from the spring if the spring is compressed. Let L be the amount the spring is stretched when the mass is in equilibrium. Let u denote the displacement of the mass from equilibrium. At equilibrium, the force upward due to the spring must equal the force downward due to gravity, so we get kL = mg. By Newtons law of motion, we have that mass times acceleration = total force at time t on object at position u If we take the downward direction as positive, and ig-...
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