After a mass m is attached to a spring, it stretches it s units and then hangs at rest in the equilibrium position as shown in the following figure (b). After the spring/mass system has been set in motion, let x(t) denote the directed distance of the mass beyond the equilibrium position. As indicated in following figure (c), assume that the downward direction is positive, that the motion takes place in a vertical straight line through the center of gravity of the mass, and that the only forces acting on the system are the weight of the mass and the restoring force of the stretched spring. Use Hooke€™s law: The restoring force of a spring is proportional to its total elongation. Determine a differential equation for the displacement x(t) at time t > 0.

 

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x()<0 unstretched spring x= 0 x() >0 equilibrium position (b) (a) (c) wwwwwwa