A force of 8 pounds stretches a spring 1 foot. A mass weighing 3.2 pounds is attached to the spring, and the system is then immersed in a medium that offers a damping force numerically equal to 0.8 times the instantaneous velocity. (a) Find the equation of motion if the mass is initially released from rest from a point 1 foot above the equilibrium position. X(t) = ft (b) Express the equation of motion in the form x(t) = Ae"M sin( \\/ (1)2 Azt + (,0), which is given in (23) of Section 3.8. (Round {p to two decimal places.) x(t) = ft ' (c) Find the first time at which the mass passes through the equilibrium position heading upward. (Round your answer to three decimal places.) 5 A lkilogram mass is attached to a spring whose constant is 21 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneom velocity. Determine the equations of motion if the following is true. (a) the mass is initially released from rest from a point 1 meter below the equilibrium position x(t) = 71e_3t %e_7t m u! (b) the mass is initially released from a point 1 meter below the equilibrium position with an upward velocity of 11 m/s x(t) = Sie_3t 16," m 2 2 X A mass of 1 slug is suspended from a spring whose spring constant is 9 Ib/ft. The mass is in ially released from a point 1 foot above the equilibrium position with an upward velocity of J3 ft/s. Find the times for which the mass is heading downward at a velocity of 3 ft/s. (Enter your answers as a comma-separated list. Let n represent an arbitrary integer.) 2m: 1: t: 3 + 18 s A mass weighing 4 pounds is attached to a spring whose constant is 2 lb/ft. The medium offers a damping force that is numerically equal to the instantaneous velocity. The mass is initially released from a point 1 foot above the equilibrium position with a downward velocity of 14 ft/s. Determine the time (in s) at which the mass passes through the equilibrium position. (Use g = 32 ft/s2 for the acceleration due to gravity.) S X Find the time (in s) after the mass passes through the equilibrium position at which the mass attains its extreme displacement from the equilibrium position. 0.355 s X What is the position (in R) of the mass at this instant? 0.6165 Ft x