We'll manipulate the following equations on the basis of initial conditions.

x(t)=Acos(ωt+ϕ)

x(t)=Ccos(ωt)+Ssin(ωt)

Derivation rule:

$\overline{)\frac{\mathbf{d}}{\mathbf{d}\mathbf{t}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{\left(}}{\mathbf{a}}{\mathbf{x}}{\mathbf{\right)}}{\mathbf{=}}{\mathbf{}}{\mathbf{-}}{\mathbf{a}}{\mathbf{\xb7}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{\left(}}{\mathbf{a}}{\mathbf{x}}{\mathbf{\right)}}}$

The following trigonometric identies are useful:

$\overline{){\mathbf{cos}}{\mathbf{}}{\mathbf{(}}{\mathbf{a}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{\mathbf{b}}{\mathbf{)}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{cos}}{\mathbf{}}{\mathbf{a}}{\mathbf{\xb7}}{\mathbf{cos}}{\mathbf{}}{\mathbf{b}}{\mathbf{}}{\mathbf{-}}{\mathbf{}}{\mathbf{sin}}{\mathbf{}}{\mathbf{a}}{\mathbf{\xb7}}{\mathbf{sin}}{\mathbf{}}{\mathbf{b}}}$

Learning Goal:

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

x(t)=x_{0}+v_{0}t+(1/2)at^{2}

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

x_{0}, the position of the oscillator at t=0, and

v_{0}, 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:

x(t)=Acos(ωt+ϕ) and

x(t)=Ccos(ωt)+Ssin(ωt),

where A, ϕ, C, and S are constants, ω 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 accommodate x0 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 x_{0}, v_{0}, and ω. Separate your answers with a comma.

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Intro to Simple Harmonic Motion (Horizontal Springs) concept. You can view video lessons to learn Intro to Simple Harmonic Motion (Horizontal Springs). Or if you need more Intro to Simple Harmonic Motion (Horizontal Springs) practice, you can also practice Intro to Simple Harmonic Motion (Horizontal Springs) practice problems.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Jerousek's class at UCF.