5-1. MODELING WITH HIGHER - MODELING WITH HIGHER-ORDER...

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MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS. 1. SPRING/MASS SYSTEMS: FREE UNDAMPED MOTION HOOKE’S LAW Suppose that a flexible spring is suspended vertically from a rigid support and then a mass m is attached to its free end. The amount of stretch, or elongation, of the spring will of course depend on the mass; masses with different weights stretch the spring by differing amounts. By Hooke’s law the spring itself exerts a restoring force F opposite to the direction of elongation and proportional to the amount of elongation s. Simply stated, F = ks , where k is a constant of proportionality called the spring constant. The spring is essentially characterized by the number k. For example, if a mass weighing 10 pounds stretches a spring ½ foot, then 10=k(1/2) implies k = 20 lb/ft. Necessarily then, a mass weighing, say, 8 pounds stretches the same spring only 2/5 foot. NEWTON’S SECOND LAW. After a mass m is attached to a spring, it stretches the spring by an amount s and attains a position of equilibrium at which its weight W is balanced by the restoring force ks. Recall that weight is defined by W = mg , where mass is measured in slugs, kilograms, or grams and g =32 ft / s 2 , 9.8 m/ s 2 , or 980 cm/ s 2 , respectively. The condition of equilibrium is mg = ks or mg - ks = 0. If the mass is displaced by an amount x from its equilibrium position, the restoring force of the spring is then k ( x + s ). Assuming that there are no retarding forces acting on the system and assuming that the mass vibrates free of other external forces— free motion —we can equate Newton’s second law with the net, or resultant, force of the restoring force and the weight: m d 2 x dt 2 =− k ( s + x ) + mg =− kx + mg ks =− kx (1) The negative sign in (1) indicates that the restoring force of the spring acts opposite to the direction of motion. Furthermore, we adopt the convention that displacements measured below the equilibrium position are positive.
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DE OF FREE UNDAMPED MOTION By dividing (1) by the mass m , we obtain the second-order differential equation d 2 x
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