Understand how to determine the constants in the general equation for simple har

Question

Understand how to determine the constants in the general equation for simple harmonic motion, in terms of given initial conditions.

A common problem in physics is to match the particular initial conditions - generally given as an initial position x0 and velocity v) at t=0 - once you have obtained the general solution. You have dealt with this problem in kinematics, where the formula

has two arbitrary constants (technically constants of integration that arise when finding the position given that the acceleration is a constant). The constants in this case are the initial position and velocity, so "fitting" the general solution to the initial conditions is very simple.

For simple harmonic motion, it is more difficult to fit the initial conditions, which we take to be

x0, the position of the oscillator at t=0, and
v0, the velocity of the oscillator at t=0.

There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

where A, ?, C, and S are constants, ? is the oscillation frequency, and t is time.

Although both expressions have two arbitrary constants--parameters that can be adjusted to fit the solution to the initial conditions--Equation 3 is much easier to use to accommodatex0 and v0. (Equation 2 would be appropriate if the initial conditions were specified as the total energy and the time of the first zero crossing, for example.)

Part A

Find C and S in terms of the initial position and velocity of the oscillator.

Give your answers in terms of x0, v0, and ?. Separate your answers with a comma.

Explanation / Answer

Considering the general equation:

x(t)=Ccos(?t)+Ssin(?t), at t=0 we get initial position x(0)

=> x(0)= Ccos(0)+Ssin(0)= C

Hence, C=x(0) (initial position of oscillator)

Now, differentiating equation 3 wrt time t will give us the equation for velocity of the oscillator.

=> v(t)= d/dt( x(t))=d/dt(Ccos(?t)+Ssin(?t))

=>v(t)= Cw(-sin(wt))+Swcos(wt)

=>v(t)=w(Scos(?t)-Csin(?t))

=>at t=0, we get initial velocity

=> v(0)= w(Scos(0)-Csin(0))

=>v(0)= Sw

=> S=v(0)/w

as cos(0)=1 and sin(0) =0 for all the above calculations

Hence, in terms of x(0), v(0) and ? , we have

C=x(0)

S=v(0)/ ?

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