Ex1. A 32 lb weight stretches a spring 2 feet. Determine the amplitude and period of motion if the weight is released 1 foot above equilibrium position with an initial upward velocity of 2ft/sec. How many complete vibrations will the weight have completed at the end of seconds?

Solution:

Note from w=mg, mass=weight/gravity, so m=32/32=1 slug. The spring constant k is found by f(s)=ks, so this says 32=k(2) and k=16. Now our k/m=16/1 and the DE with IC is

The general solution is y=Ccos(4t)+Dsin(4t). y(0)=-1 forces C=-1

y’=-4Csin(4t)+4Dcos(4t) and y’(0)=-2 forces D=-1/2 and we have the solution

y=-cos(4t)-(1/2)sin(4t)

We can then write in the form form, and we get

The amplitude is and the period is . seconds would be 8 periods, so there will be 8 complete vibrations in seconds.