Physics 103: Fall 2005
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?