laplace transform

Question

A mass of 1 slug is suspended from a spring whose spring constant is 9 lb/ft. The mass is initially released from a point 1 foot above the equilibrium position with an upward velocity of sqrt3 ft/s. Using Laplace transform, find the equation of motion.

Answer 1

The sum of forces on the mass = ma=-ky-mg where m= mass =1, a=acceleration=y'', y=position with up +, g = 9.81 m/s². First use the transform of the force equation and simplify to find y(s)= (sy(s)+y'(0)-g/s)/(s²+k/m). Use partial fractions to get y(s)=(1-g/9)s/(s²+9) +y'(0)/(s²+9) -g/9/(s²+9). Then find the inverse transform