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 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 C and 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 xinit in terms of C, S, and (Greek letter omega).

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, (Greek letter omega), and t.

Explanation / Answer

part A: for doing this part, we will subsitute t= 0 in the general solution for the displacement from equilibrium of a harmonic oscillator

x(initial at t=0) = C cos ( w0) + S sin ( w0)

X (initial) = C cos 0 + S sin 0

x (initial) = C + 0 = C

You have to prvide the complete information to attempt part D

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