Under some circumstances when two parallel springs, with constants k₁ and k₂, support a single mass, the effective spring constant of the system is given by k = 4k₁k₂/(k₁ + k₂). A mass weighing 20 pounds stretches one spring 6 inches and another spring 2 inches. The springs are attached to a common rigid support and then to a metal plate. As shown in Figure 5.1.16, the mass is attached to the center of the plate in the double-spring arrangement. Determine the effective spring constant of this system. Find the equation of motion if the mass is initially released from the equilibrium position with a downward velocity of 2 ft /s.

Figure 5.1.16

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