Physics 103: Fall 2005

**Oscillators**

When a stable equilibrium is disturbed, one typically sees *simple harmonic oscillation* about the
equilibrium point. In other words, one
sees sinusoidal motion at a definite frequency *f *or, equivalently, a definite angular frequency _{}, with some amplitude A, centered on the equilibrium
position. It is a remarkable simplicity
in Nature that this kind of motion happens everywhere. Of course real oscillations aren’t precisely
sinusoidal, because they run down, and in extreme cases they may run down so
fast that they don’t perform even one oscillation – that case probably
shouldn’t be called an oscillator. But
all real oscillators are very much alike, and, as we were saying, they are all
around us. This universality is what
makes oscillators so important.

We will look at an oscillator that is stripped down to
essentials, a mass on a spring. This is
the physicist’s metaphor for all the other oscillators. The spring is characterized by its “springiness”
*k*, which tells how hard it pushes to
restore equilibrium. Larger *k* implies higher frequency. The mass *m*
represents the inertia of the system, its tendency not to respond to the
spring, but rather just to keep doing what it is doing. Larger *m*
implies lower frequency.

The lab has two parts:

(1)
Determine *k*
for a spring from Hooke’s Law.

(2)
Determine *k*
for the same spring by measuring simple harmonic oscillations.

1. The definition of
springiness *k *is *F=-kx*, where *F* is the force exerted by the spring, and x is its extension from
equilibrium. Thus if you hang a weight *Mg* on a spring and let everything come
to equilibrium, the spring will stretch by an amount *x*, where *Mg=kx* That is, the extension *x* is proportional to *Mg*, with constant *k*. Get data using several masses
and find the slope of a line to get *k*.

2. Theoretically, the angular frequency of an oscillator is

_{}

Measure the angular frequency _{} for several values of *m*, and use a graphical method to
determine *k.* In doing this you will also be testing the
relationship between _{} and *m*.

For each part, 1 and 2, give a data table and a graph, and
explain the significance of the slopes and intercepts. Then write a few sentences on what is most
surprising in all this: the assertion
that the same property *k* shows up in
these two quite different ways, the first one static, the second dynamic. Does this seem to be experimentally true?