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Question: A mass weighing 4 N is attached to a spring whose constant is 2 N/m. The mass is initially released from a point 1 m above the equilibrium position and surrounding medium offers a damping force that is numerically equal to the instantaneous velocity. (a) Derive the system of differential equation describing the motion of the mass. (b) Find the equation of motion if the mass has a downward velocity of 8 m/s by using: (i) Laplace transform (ii) Systematic elimination

Answer 1

Given that: A mass weighing 4N is attached to a spring. Let us consider gravity g = 32 ft/ s^2 We divide the weight by gravity so, mass = 4/32 Mass = 1/8 s gs the spring that the mass is attached to has a spring constant of k = 2 lb/ft finally the median in parts a damping force that is proportional to the instantaneous velocity i.e., B = 1 We want to set up the differential eqn describe spring mass of the system the general eqn for a Mass of a spring Mass system damping is m x Prime + beta x prime + k x = 0 \r\n We have n = 1/8, k = 2, & B = 1 1/8 x Prime + x prime + 2x = 0 the following auxiliary eqn n^2 + 8n + 16 = 0 (n + 4)^2 = 0 n = -4 The roots as the following form x_c = c_1 e^n t + c_2 t e^n t the solution of the differential eq^n