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Next: The Spring Pendulum Up: Simple Harmonic Motion Previous: Projecting the Wheel

Simple Harmonic Motion

The above mechanical example serves to introduce the idea of   $\cos(\theta)$  and   $\sin(\theta)$  as functions in the sense to which we have (I hope) now become accustomed. In particular, if we realize that (by definition)   $v_x \equiv \dot{x}$  and   $a_x \equiv \ddot{x}$,  the formulae for  vx(t)  and  ax(t) represent the derivatives of  x(t):
x = $\displaystyle r \, \cos(\omega t)$ (13.4)
$\displaystyle \dot{x}$ = $\displaystyle - \, r \, \omega \, \sin(\omega t)$ (13.5)
$\displaystyle \ddot{x}$ = $\displaystyle - \, r \, \omega^2 \, \cos(\omega t)$ (13.6)

-- which in turn tell us the derivatives of the sine and cosine functions:
$\displaystyle {d \over dt} \, \cos(\omega t)$ = $\displaystyle - \omega \, \sin(\omega t)$ (13.7)
$\displaystyle {d \over dt} \, \sin(\omega t)$ = $\displaystyle \omega \, \cos(\omega t)$ (13.8)

So if we want we can calculate the   $n^{\rm th}$  derivative of a sine or cosine function almost as easily as we did for our ``old'' friend the exponential function. I will not go through the details this time, but this feature again allows us to express these functions as series expansions:

\begin{displaymath}\begin{array}[c]{rcccccc}
\exp(z) =& 1 &+ z &+ \onehalf z^2  . . . 
 . . . \sin(z) =& & z & &- {1 \over 3!} z^3
& &+ \cdots
\end{array}\end{displaymath} (13.9)

where I have shown the exponential function along with the sine and cosine for reasons that will soon be apparent.



 
next up previous
Next: The Spring Pendulum Up: Simple Harmonic Motion Previous: Projecting the Wheel
Jess H. Brewer
1998-10-09