Part D

Find the equation for the block's position xnew$($t$)$

in the new coordinate system.

Express your answer in terms of L

$,$ xinit

$,\backslash$omega

$($Greek letter omega$),$ and t

$.$Part C

What is the equation x$($t$)$

for the block?

Express your answer in terms of t

$,\backslash$omega

$,$ and xinit

$.$Learning Goal:

To understand the application of the general harmonic equation to the kinematics of a spring oscillator.

One end of a spring with spring constant k

is attached to the wall. The other end is attached to a block of mass m

$.$ The block rests on a frictionless horizontal surface. The equilibrium position of the left side of the block is defined to be x$=0$

$.$ The length of the relaxed spring is L

$.($Figure $1)$

The block is slowly pulled from its equilibrium position to some position xinit$>0$

along the x axis. At time t$=0$

$,$ the block is released with zero initial velocity.

The goal is to determine the position of the block x$($t$)$

as a function of time in terms of $\backslash$omega

and xinit

$.$

It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is

x$($t$)=$Ccos$(\backslash$omega t$)+$Ssin$(\backslash$omega t$)$

$,$

where C

$,$ S

$,$ and $\backslash$omega

are constants. $($Figure $2)$

Your task, therefore, is to determine the values of C

and S

in terms of $\backslash$omega

and xinit

$.$