7.3 Applications  (Page 2/13)

It is convenient to rearrange this equation and introduce a new variable, called the angular frequency , $\omega \text{.}$ Letting $\omega =\sqrt{k\text{/}m},$ we can write the equation as

This differential equation has the general solution

which gives the position of the mass at any point in time. The motion of the mass is called simple harmonic motion    . The period of this motion (the time it takes to complete one oscillation) is $T=\frac{2\pi }{\omega }$ and the frequency is $f=\frac{1}{T}=\frac{\omega }{2\pi }$ ( [link] ).

Writing the general solution in the form $x(t)=c_1\text{cos}\left(\omega t\right)+c_2\text{sin}\left(\omega t\right)$ has some advantages. It is easy to see the link between the differential equation and the solution, and the period and frequency of motion are evident. This form of the function tells us very little about the amplitude of the motion, however. In some situations, we may prefer to write the solution in the form

Although the link to the differential equation is not as explicit in this case, the period and frequency of motion are still evident. Furthermore, the amplitude of the motion, A , is obvious in this form of the function. The constant $\varphi$ is called a phase shift and has the effect of shifting the graph of the function to the left or right.

To convert the solution to this form, we want to find the values of A and $\varphi$ such that

We first apply the trigonometric identity

to get

Thus,

If we square both of these equations and add them together, we get

Thus,

Now, to find $\varphi ,$ go back to the equations for $c_1$ and $c_2,$ but this time, divide the first equation by the second equation to get

Then,

We summarize this finding in the following theorem.