Learning Goal:

To understand the application of the general harmonic equationto finding the acceleration of a spring oscillator as a function oftime.

One end of a spring with spring constant k isattached to the wall. The other end is attached to a block ofmass m. The block rests on a frictionless horizontalsurface. The equilibrium position of the left side of the block isdefined to be x=0. The length of the relaxedspring is L.(Figure 1)

The block is slowly pulled from its equilibrium position to someposition xinit>0 alongthe x axis. At time t=0 , theblock is released with zero initial velocity.

The goal of this problem is to determine the acceleration of theblock a(t) as a function of time interms of k, m,and xinit.

It is known that a general solution for theposition of a harmonic oscillator is

x(t)=Ccos(?t)+Ssin(?t),

where C, S,and ? are constants. (Figure 2)

Your task, therefore, is to determine the valuesof C, S, and ? intermsof k, m,and xinit andthen use the connectionbetween x(t) and a(t) tofind the acceleration.

QUESTION: Using the fact that acceleration is the secondderivative of position, find the acceleration of theblock a(t) as a function of time.

Express your answer in termsof ?, t,and x(t).