k k F '2 Two masses, m¡ and m2 are attached in series by two springs. A force, F is applied to the second mass. Each spring has spring constant, k. The position of each mass is described by x1 and x2, where Fspring 1 = -kx, and Fspring 2 = -k(x2 – x1) 1. write out the kinetic and potential energy in terms of x1 and x2 2. Use the Lagrange equations to write out two coupled ordinary differential equations here, the force F does virtual work, F8x2

1. A weight of 8 lbf is attached to the end of a vertical spring which stretches 4 inches before reaching equilibrium. If the weight is pulled down 6 inches below equilibrium point and given an initial upward velocity of 8 ft/sec, find the equation of motion. Note: In solving for the spring constant k, make sure it ends up as a positive number.

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 4 inches and another spring 2 inches. The springs are attached to a common rigid support and then to a metal plate. As shown in the figure, the mass is attached to the center of the plate in the double-spring arrangement. k₂ II k = 20 lb Determine the effective spring constant of this system. lb/ft Find the equation of motion x(t) if the mass is initially released from the equilibrium position with a downward velocity of 6 ft/s. (Use g for the acceleration due to gravity.) 32 ft/s² x(t) = ft