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 timet=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 ? and xinit.

It is known that a general solution for the displacement from equilibrium 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 values of Cand S in terms of ? and xinit.

Part A

Using the general equation for x(t) given in the problem introduction, express the initial position of the block x_init in terms of C, S, and ? (Greek letter omega).

Explanation / Answer

The block is slowly pulled from its equilibrium position to some position x_init > 0 along the x axis. At time t=0 , the block is released with zero initial velocity.

It is known that a general solution for the position of a harmonic oscillator is

x(t) = C*cos(omega*t) S*sin(omega*t)
where C, S, and omega are constants

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