Dimensional analysis
Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years.
There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,[4] although this does not invalidate the usefulness of dimensional analysis.
Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number—a numerical quantity and a corresponding dimensional unit.
Other relations can involve multiplication (often shown with a centered dot or juxtaposition), powers (like m2 for square metres), or combinations thereof.
This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor.
In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows.
Common dimensionless groups in fluid mechanics include: The origins of dimensional analysis have been disputed by historians.
[9][10] The first written application of dimensional analysis has been credited to François Daviet, a student of Joseph-Louis Lagrange, in a 1799 article at the Turin Academy of Science.
[10] This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the Buckingham π theorem.
Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize.
[17] The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units.
It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k, m, and T, because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant.
Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters.
If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon.
Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension.
As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs.
[23] This corresponds to the fact that under the natural pairing between a vector space and its dual, the dimensions cancel, leaving a dimensionless scalar.
Every possible way of multiplying (and exponentiating) together the measured quantities to produce something with the same unit as some derived quantity X can be expressed in the general form Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form Knowing this restriction can be a powerful tool for obtaining new insight into the system.
In chemistry, the amount of substance (the number of molecules divided by the Avogadro constant, ≈ 6.02×1023 mol−1) is also defined as a base dimension, N. In the interaction of relativistic plasma with strong laser pulses, a dimensionless relativistic similarity parameter, connected with the symmetry properties of the collisionless Vlasov equation, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential.
However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless.
For example, a quantity equation for displacement d as speed s multiplied by time difference t would be: for s = 5 m/s, where t and d may be expressed in any units, converted if necessary.
This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.
The three independent dimensionful constants: c, ħ, and G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c → ∞, ħ → 0 and G → 0.
[48] Mathematica also has a function that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations.
Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank
Huntley's concept of directed length dimensions however has some serious limitations: It also is often quite difficult to assign the L, Lx, Ly, Lz, symbols to the physical variables involved in the problem of interest.
Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts.
As an example, consider again the projectile problem in which a point mass is launched from the origin (x, y) = (0, 0) at a speed v and angle θ above the x-axis, with the force of gravity directed along the negative y-axis.
