Dimensional analysis

Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of derived equations. Only like dimensioned quantities may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the "+" or "−" or "=" sign, that physical equation is not plausible, which might prompt one to correct errors before proceeding to use it. When like dimensioned quantities or unlike dimensioned quantities are multiplied or divided, their dimensions are likewise multiplied or divided.

Related Topics:
Physics - Chemistry - Engineering

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When dimensioned quantities are raised to a power or a power root, the same is done to the dimensions attached to those quantities.

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The dimensions of a physical quantity is associated with symbols, such as M, L, T which represent mass, length and time, each raised to rational powers. For instance, the dimension of the physical variable, speed, is distance/time (L/T) and the dimension of a force is mass×distance/time² or ML/T². In mechanics, every dimension can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of dimensions. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.

Related Topics:
Speed - Force - Electric charge

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The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but a length always has a dimension of L whether it is measured in meters, feet, inches, miles or micrometres. Dimensional symbols, such as L, form a group: there is an identity, 1; there is an inverse to L, which is 1/L, and L raised to any rational power p is a member of the group, having an inverse of 1/L raised to the power p. There are conversion factors between units; for example one meter is equal to 39.37 inches, but a meter and an inch are both associated with the same symbol, L.

Related Topics:
Unit - Group - Conversion factors

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In the most primitive form, dimensional analysis may be used to check the correctness of physical equations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. Moreover, the two sides of any equation must have the same dimensions. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is nearly 0.477.

Related Topics:
Exponential - Trigonometric - Logarithmic - Dimensionless number

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It should be noted that sometimes different physical quantities may have the same dimensions: work (or energy) and torque, for example, both have the same dimensions, M L2T-2. An equation with torque on one side and energy on the other would be dimensionally correct, but cannot be physically correct! However, torque multiplied by an angular twist measured in (dimensionless) radians is work or energy. (The radian is the mathematically natural measure of an angle and is the ratio of arc of a circle swept by such an angle divided by the radius of the circle. That ratio of like dimensioned quantities, length over length, is dimensionless.)

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The value of a dimensionful physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. Strictly, when like dimensioned quantities are added or subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. But, conceptually, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 meter added to 1 foot is a length, but it would not be correct to add 1 to 1 to get the result. A conversion factor, which is a ratio of like dimensioned quantities and is equal to the dimensionless unity:

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: 1 operatorname{ft} = 0.3048 operatorname{m} is identical to saying 1 = rac{0.3048 operatorname{m}}{1 operatorname{ft}}

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The factor 0.3048 rac{operatorname{m}}{operatorname{ft}} is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to identical units so that their numerical values can be added or subtracted.

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Only in this manner, it is meaningful to speak of adding like dimensioned quantities of differing units, although to do so mathematically, all units must be the same. It is not meaningful, either physically or mathematically, to speak of adding unlike dimensioned physical quantities such as adding length (say, in meters) to mass (perhaps in kilograms).

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The Buckingham π-theorem forms the basis of the central tool of dimensional analysis. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n-m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown.

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