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A spring (which behaves linearly both in tension and compression) of unstretched length 40 cm, when tested vertically in isolation as shown in Figure 1, is found to stretch by 0 cm when it supports a mass of m₁ kg. Ixo m₁ Figure 1 The spring extends by cm when supporting a m₁ kg mass The spring, along with a damper, is then attached horizontally to a trolley of mass 40 kg. which is constrained to move only in the x-direction (see Figure 2). The spring is stretched some distance from its equilibrium position before being released. Espring wwwww mg m damper % wwwww Fdamper (b)
Figure 2 (a) A frictionless trolley is attached by a spring, with spring constant k, of unstretched length 40 cm and a damper with a damping coefficient x. The trolley is shown in its natural rest position with its left-hand face at x = 0. (b) Free body diagram for the trolley shortly after being pulled to the right then released, with the spring under tension so, at this instant, its velocity is in the negative x-direction. (For clarity the damping force and normal reaction forces have been shown as acting at the point of the arrows - the spring force and weight are shown as acting at the tail of their arrows.) a. Assuming a mass m₁ = 8.0 kg and a displacement x = 2.0 cm in Figure 1, calculate the spring constant k. Assume that the acceleration due to gravity is 10 m s-2. (4 marks) b. Based on the free body diagram in Figure 2(b), apply Newton's second law to write a homogeneous second-order linear differential equation for the forces acting on the trolley in the x-direction. (Assume that you can ignore the fact that the damping force and the spring force do not act at the same point.) (8 marks) c. Calculate the damping coefficient X required to critically damp this mass-spring system. (7 marks) d. If the damper were disconnected, what would be the natural frequency of oscillation of the trolley? (6 marks)