Question

It is known that a general solution for the displacement from equilibrium of a harmonic oscillator is x(t)=Ccos(ωt)+Ssin(ωt), where C, S, and ω are constants.

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 ω (Greek letter omega).

b) Find the value of S using the given condition that the initial velocity of the block is zero: v(0)=0.

c)What is the equation x(t) for the block? Express your answer in terms of t, ω, and xinit.

d)Find the equation for the block's position xnew(t) in the new coordinate system.
Express your answer in terms of L, xinit, ω (Greek letter omega), and t.

Answer 1

The characteristics of the expression of the simple harmonic motion allows to find the results for the expression of the mass- block system are:

     A) The constant Ces: C = xinit

     B) The ocsntna S is: S = 0

     C) The equation of the system is: x = xinit cos wt

     D) If the reference system is at some extreme, the equation is:

              $L - x_{init} = x_{init} \ cos \ wt$  

The simple harmonic movement is an oscillatory movement where the restoring force is proportional to the displacement, the general equation that describes this movement is indicated.

          x = C cos wt + S sin wt

Where x is the displacement C and S are constants. W the angular velocity and t the time.

A) The initial position of the body occurs when the time is zero, t = 0

We substitute.

           x = C cos 0 + S sin 0

           $x_{init}$ = C

B) The velocity of the particle is defined.

          $v= \frac{dx}{dt} \\ v= C w \ sin \ wt - Sw \ cos \ wt$  

The initial velocity occurred for time zero t = 0

          v = - S w

It indicates that the initial velocity is zero, since the angular velocity must be different from zero, it implies that the constant is valid.

          S = 0

C) The equation for the block remains.

           x (t) = $x_{init} \ cos \ wt$  

D) In ​​some cases it is measured with respect to another reference system, the most common are:

• For maximum compression it is the zero of the system.

• The maximum extension is the zero of the system.

In these cases, the change that must be made is

          x =  $L - x_{min}$ t

         

we substitute

          $L - x_{init} = x_{init} \ cos \ wt$  

          L = $x_{init}$  (1 + cos wt)

In conclusion, using the characteristics of the expression of the simple harmonic motion we can find the results for the expression of the mass- block system are:

    A) The constant Ces: C = xinit

     B) The ocsntna S is: S = 0

     C) The equation of the system is: x = xinit cos wt

     D) If the reference system is at some extreme, the equation is:

              $L - x_{init} = x_{init} \ cos \ wt$  

Learn more about simple harmonic motion here:  brainly.com/question/17315536