Question

A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping constant of 2 lb · s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 6 in/s, find its position u at any time t. (Use g = 32 ft/s2 for the acceleration due to gravity. Let u(t), measured positive downward, denote the displacement in feet of the mass from its equilibrium position at time t seconds.)a) u(t) =b) Plot u versus t.c) Determine when the mass first returns to its equilibrium position. (Round your answer to four decimal places.)d) Find the time ? such that |u(t)| < 0.01 inches for all t > ?. (Round your answer to four decimal places.)

Answer 1

A) U(t) = 1/(4√31)·e⁻²⁺Sin(2√31t)

B) See plot in picture workings

C) t = 0.2821 sec

D) t > 1.5927 s , |U(t)| < 0.01/12 ft

Explanation:

Given weight mg = 16 lb

L = 3 inches = 1/4 ft

r = 2

m = mg/g = 16/32 = 1/2 lb/ft/sec⁻²

k = mg/L = 16/1/4 = 16 x 4 = 64 lb/ft

See the picture below, Using the above derived data to solve each question in details.