periodic motion « Physics 1

We recall periodic motion in one dimension. The function $f(\omega t)$ is periodic if $f(\omega t)=f(\omega(t+T))$ for all t. A simple non-trivial example of this is the function

$x(t) = A \cos (\omega t + \theta)$

where $A$ and $\theta$ are constants. We see, that the first derivatives with respect to time are

$\dot{x} = -A\omega \sin(\omega t + \theta), \, \ddot{x} = -A\omega^2 \cos (\omega t + \theta)$

so that

$\ddot{x}=-\omega^2 x$

This is the equation of motion for small oscillations, and it implies periodic motion with period $\omega T =2\pi$ , so $T=2\pi/\omega$ . The amplitude is A, this means that $x$ varies between -A and +A. An example is a mass m attached to a spring; for the spring we have Newton’s 2nd law in the form $m\ddot{x}=-kx$ , and so the motion is periodic with $\omega=\sqrt{k/m}$ . Another important example which you should remember is the simple pendulum which has period $\omega =\sqrt{g/L}$ , where L is its length. This result holds for small oscillations as well.

Here are some examples of periodic motion (from our friend, Prof. Walter Lewin of MIT (course 8.01, Physics I: Classical Mechanics, fall 1999)).

Topics covered in this lecture:

Physical pendulum — Simple harmonic oscillations

(Young and Freedman, chap. 13)

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