A mass weighing 11 lb stretches a spring 4in. The mass is pulled down an additional 3 in and is then set in motion with an initial upward velocity of 5 ft/s. No damping is applied.
(a) Determine the position u of the mass at any time t. Use 32 ft/s² as the acceleration due to gravity. Pay close attention to the units.
(b) Determine the period, amplitude and phase of the motion.

a) , b) , ,

Explanation:

a) The system mass-spring is well described by the following equation of equilibrium:

After some handling in physical and mathematical definition, the following non-homogeneous second-order linear differential equation:

The solution of this equation is:

The velocity function is:

Initial conditions are:

Equations at are:

The spring constant is:

After some algebraic handling, amplitude and phase angle are found:

The position can be described by this function:

b) The period of the motion is:

The amplitude is:

The phase of the motion is: