To understand the application of the general harmonic equation to the kinematics

Question

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 w and xinit . It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is , x(t)=Ccos(wt)+Ssin(wt) where C,S,andw are constants (Figure 2) Your task, therefore, is to determine the values of C and S in terms of w and xinit .

Explanation / Answer

= C*cos(0)+S*sin(0)

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