Dimensional analysis of models and data pieces

James Farreneheit. Pricea)

Hardwoods Hole Oceanographic Institution, Forest Hole, Massachusetts 02543

Received 22 Might 2002; approved 4 November 2002 Dimensional analysis is a widely applicable and sometimes extremely powerful technique that is demonstrated here in a study of the simple, viscous pendulum. The п¬Ѓrst and essential step of dimensional examination is to deп¬Ѓne a very well idealized representation of a trend by real estate the relevant parameters, called the physical unit. The second stage is to understand consequences from the physical version and the basic principle that complete equations are in addition to the choice of products. The calculation that follows yields a basis set of nondimensional variables. The п¬Ѓnal stage is to interpret the non-dimensional basis placed in the light of observations or existing theory, and if essential to modify the foundation set to improve its energy. One strategy is usually to nondimensionalize the dependent adjustable by a climbing estimate. The remaining nondimensional variables can then be formed in ways that deп¬Ѓne element ratios or perhaps that correspond to the ratio of conditions in a governing equation. В© 2003 American Association of Physics Professors. DOI: 12. 1119/1. 1533057

I. REGARDING DIMENSIONAL EXAMINATION Dimensional examination is a impressive tool insofar as it can be used on virtually all quantitative models and data models. Topics in the recent materials include doughnuts, dinosaurs, 1 and the many fundamental hypotheses of physics. 2 In most cases dimensional research is very powerful; results range from the log-layer proп¬Ѓle of a thrashing boundary coating and the unreal slope inside the inertial subrange of isotropic turbulence, equally landmarks in п¬‚uid mechanics. 3 Often the result of dimensional analysis is known as a hint with the form of a remedy or a more effective way to show or assimialte a large data set. This type of results, though seldom full if used alone, are an essential component of many brought on. This conventional paper is an introduction to dimensional analysis that aims to make the method plus the results since accessible as is feasible. 3, four The plan should be to investigate the motion of the simple pendulum while putting an emphasis on the use of dimensional analysis as an complement to experimental, statistical, and assumptive methods. a few If the basic pendulum seems too familiar, skip forward to Sec. III. If the use of non-dimensional variables is usually familiar, neglect ahead to Sec. 4, where a standard method of computer a basis set of non-dimensional variables is usually presented. The effects of drag are considered in Sec. V, and concluding feedback are in Sec. VI. II. MODELS OF A SIMPLE PENDULUM Consider a pendulum that can be manufactured and discovered with simple tools; a little lead п¬Ѓshing sinker using a mass of some tens of grms suspended over a thin monoп¬Ѓlament line a couple of meters in length. The action of such a pendulum will be simply lightly damped by move with the encircling air and is characterized by two distinct period scalesвЂ”a frequent, fast time-scale oscillation using a period, L, and a slow, more1. Our objective or-less rapid decay using a time-scale, will be to learn how these kinds of time weighing scales and some various other variables, for example , tension in the line, differ with series length plus the mass of the bob. A. A physical model To analyze the motion of the pendulum, we all begin by record the factors that are presumed to be relevant to the 437 Am. L. Phys. 71 5, May possibly 2003 http://ojps.aip.org/ajp/

aspect of the motion that is of interest. To start, consider the fast time-scale, oscillatory action. The line will probably be idealized because rigid, in order that the bob must swing along a constant radius. The movement of the frank is then deп¬Ѓned by the angle of the range to the straight, (t), and its particular time derivatives; the angle is the reliant variable of the physical version and the period, t, is a only 3rd party variable. A lot of properties with the pendulum would appear to be importantвЂ”the mass of the bob, Meters, the length of the supporting series, L, and...