To study the amplitude behavior of an anharmonic oscillator, and to compare experimental determinations of the oscillation periods for large and small amplitudes with theoretical predictions based on the potential energy function of the system.
The simple pendulum is a system for which the potential energy of the system is symmetrical for displacements from equilibrium. In both cases, for small displacements from equilibrium, the potential energy can be approximated by a parabola, so that the restoring force (or torque) is directly proportional to the displacement. Such a parabolic potential produces simple harmonic oscillation. A very important aspect of such motion is that the period of the oscillation is independent of amplitude. [See Chapter 13 of Serway.] While a symmetrical parabolic potential energy "well" is a good approximation for many physical systems, it is also common to encounter situations where the potential energy function is not simple and symmetrical.
In such a case, although the motion is not just a simple sine or cosine
function of time (as in Experiments 1 or 4), some simple physics still
tells us much about the motion. In particular, if we can assume that
frictional dissipation is not important, then conservation of
energy is useful. To the extent that energy is not lost to friction,
we expect
where E is the total energy, K = mv2/2 is the kinetic energy of the moving mass and U(x) is its potential energy.
In our experimental setup, we use a combination of gravitational forces and magnets to devise a potential energy function U(x) that is distinctly asymmetric (see Fig. 1).
On the right side, the gently sloping track provides a linear potential energy function with a small, constant restoring force to the left. Since the potential energy rise associated with raising a mass m by height h is mgh, the potential energy function on this side should be simply mgx sin q. (Note that an arbitrary constant can always be added to or subtracted from potential energy.)
On the left side, two repelling magnets provide a short-range restoring force of a very different character. In addition to producing a much steeper potential energy function on this side, the magnetic potential energy is distinctly nonlinear (for small repelling magnets the theoretical expectation is that the potential energy should vary approximately as r-n, where the exponent n is about 3.
Added together, these two contributions produce a total potential energy U(x) = Ugrav + Umag that is asymmetric but has a smooth minimum. If we place our mass at rest at this minimum, it is in equilibrium and will remain there. In this state, the kinetic energy is zero and the total and potential energies are at a global minimum. If we disturb the mass from this equilibrium by imparting to it some displacement or velocity, it will oscillate about the minimum position. However, since the potential energy well is not parabolic, the motion will not in general be simple harmonic.
A couple of limiting cases of our experiment can be explored without difficult calculation. Suppose first that we are looking at very large amplitude oscillations, where "large" is measured in comparison with the distance over which the magnetic force is appreciable. In this limit, the "magnetic side" of the well looks almost infinitely steep (see Fig. 2).
In this limit, the mass "free falls" under constant acceleration until it abruptly bounces off what amounts to a rigid wall. The situation is basically analogous to a bouncing ball; the greater the height from which it is dropped, the longer the fall time. The distance an object falls in time t under constant acceleration a is 1/2 at2, so we should expect (in this limit) the period of big oscillations (bounces) to be proportional to the square root of the amplitude.
The opposite limit — very small amplitudes — is also fairly simple. As illustrated in Fig. 3, for very small excursions about the minimum position xmin, the potential well looks like a symmetric parabola.
More than simply looking like a parabola, we can see
mathematically that it is parabolic for points sufficiently close
to xmin.
We can represent any smooth function like U(x)
in a neighborhood of a point (here, xmin)
in terms of the Taylor series expansion about that
point:
(2)Now note that the second term in this equation is automatically zero, since dU/dx = 0 at a minimum. For small displacements (x - xmin), succeeding terms in the Taylor series are smaller and smaller. For sufficiently small (x - xmin) the only significant terms are the constant and (x - xmin)2 terms; U(x) is a parabola! So while for big oscillations the period varies with amplitude, for very small amplitudes we would expect the period to "level off" at a constant value characteristic of simple harmonic motion.
The flow of air out the small holes in the track underneath the rider
allows the rider to glide on a thin, nearly frictionless cushion of air. If
the rider is unbalanced, however, the air may be forced preferrentially to
one end or the other. In this case, the uneven flow of air provides a "jet"
force that pushes the rider even on a level track. To balance the rider,
the position of the counterweight on the end opposite the magnet can be
adjusted until the rider does not jet on a level track.
Begin by familiarizing yourself with the apparatus, which consists of an air track with rider and a photogate with a digital timer. A few pointers regarding proper use of the air track are in order at this point. In order for energy conservation to be valid, the rider must not receive any unbalanced forcing from the air track flow. This requires the rider to be balanced carefully. The riders have been balanced prior to your arrival in lab; do not remove or move the weights on the rider or you will introduce a "rocket" effect that may adversely affect your results.
Handle the rider carefully, as any dents or warping of the rider caused by dropping will spoil its performance in the experiment.
Put the rider on the air track and remove any aluminum blocks (under the foot of the air track that is opposite the fixed magnet) to check that the track is level. Measure the thickness of the small aluminum blocks used to elevate the track's high end, using the provided vernier Calipers. Also measure the distance between the legs of the air track so that the angle of inclination q can be determined. Then place a single block under the foot of the air track opposite the fixed magnet. Allow the rider to settle down to its equilibrium position and place the photogate at this equilibrium position. Displace the rider and verify that the digital timer records the period of the oscillation.
Measure the period of oscillation and the amplitude. The amplitude is best determined by watching the edge of the rider and noting on the track scale the lower and upper extreme positions, x1 and x2. A convenient definition of the amplitude is then simply (x1 - x2)/2.
Since you must estimate x1 and
x2 while the rider is moving, the values may be somewhat
uncertain. However, there is another possible source of uncertainty.
You'll notice that because of some slight residual friction, the rider
doesn't return fully to its original x2 position on
successive oscillations. You should use the average of the "before"
and "after" values of x2 to determine your amplitude, and
use their difference to estimate the uncertainty in that value.
You will need of the order of ten well-chosen amplitudes to explore the full range of possible amplitudes. In choosing amplitudes, you should keep in mind that you will be plotting the logarithm of the period vs. the logarithm of the amplitude (see below). You would like to have the data spaced approximately evenly along the x-axis. Note that equal ratios are spaced evenly on a logarithmic scale. Use Kaleidagraph to plot your data as you go on a log-log plot. See the discussion of Power Laws.
Prepare a careful plot (with error bars!) of the logarithm of period on the y-axis versus the logarithm of the amplitude on the x-axis. If the period varies as the square root of amplitude, the logarithmic plot should yield a telltale linear trend. (of what slope?) Use this plot to discuss whether the results agree with your expectations at both the large and small amplitude limits.
Detail: Only a subset of your data are expected to fit a log-linear trend. There are a couple of ways for you to determine the slope and uncertainty of this subset. One way is to use a fitting program to obtain a linear fit to only the appropriate points. Or you can perform a "by eye" linear fit to obtain the slope; if you do this, be sure to use error bars also to judge the slope's uncertainty.
When the rider shown in Figure 1
is at its equilibrium position, the magnetic repulsion Fmag
between the magnets is just balanced
by the component of the gravitational force acting along the track. The
magnetic force can be given in terms of the magnetic potential energy
Umag associated with the repulsion between the magnets,
(3)
If we assume that the magnetic potential energy can be represented by a
simple power law of the form
(4)where x is the center-to-center magnet separation, then it is possible to solve for C and n by carefully measuring the equilibrium position for several values of the tilt angle q. Use different numbers of aluminum blocks under the of the air track to obtain several different tilt angles. Note that one of the blocks has half the thickness of the others, so you can explore "tilts" between 1/2 block and 3 1/2 blocks.
All you need to know to predict the period of small oscillations of a mass
m oscillating in a parabolic potential energy well is the effective
spring constant k of the potential well. Once you have m and
k, the expression
(5)gives the period.
If you know the potential energy function U(x), you can compute the effective spring constant k, which expresses how the spring force varies with position, by taking two derivatives of U with respect to x. The first derivative gives -F and the second gives the rate of change of F with respect to x, which is just k. Note that it is important to evaluate the derivatives at the equilibrium position (where F(x) = 0).
The Taylor expansion of Eq. (2) can be continued to the third-order term. When the magnitude of this term can no longer be neglected compared to the second-order term, we expect the motion to depart from simple-harmonic. Using your expression for Umag to evaluate the third-order term, estimate the range of motion for which this term is small compared to the second-order term. Compare this range to your experimental observations.
Updated 10/25/99 by Peter N. Saeta .