Formulae With Prescribed Units
A physical quantity can be expressed as the product of a number and a physical unit. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is that all terms have the same dimension, meaning every term in the formula could be potentially converted to contain the identical unit (or product of identical units).
In the example above, for the volume of a sphere, we may wish to compute with r = 2.0 cm, which yields
There is vast educational training about retaining units in computations, and converting units to a desirable form, such as in units conversion by factor-label.
However, the vast majority of computations with measurements are done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning the numerical quantity is implicitly assumed to be multiplying a particular unit. The requirements about the prescribed units must be given to users of the input and the output of the formula.
For example suppose the formula is to require that, where tbsp is the U.S. tablespoon (as seen in conversion of units) and VOL is the name for the number used by the computer. Similarly, the formula is to require . The derivation of the formula proceeds as:
Given that, the formula with prescribed units is
The formula is not complete without words such as: "VOL is volume in tbsp and RAD is radius in cm". Other possible words are "VOL is the ratio of to tbsp and RAD is the ratio of to cm."
The formula with prescribed units could also appear with simple symbols, perhaps even the identical symbols as in the original dimensional formula:
and the accompanying words could be: "where V is volume (tbsp) and r is radius (cm)".
If the physical formula is not dimensionally homogeneous, and therefore erroneous, the falsehood becomes apparent in the impossibility to derive a formula with prescribed units. It would not be possible to derive a formula consisting only of numbers and dimensionless ratios.
Read more about this topic: Formula
Famous quotes containing the words formulae, prescribed and/or units:
“I don’t believe in providence and fate, as a technologist I am used to reckoning with the formulae of probability.”
—Max Frisch (1911–1991)
“... women can never do efficient and general service in hospitals until their dress is prescribed by laws inexorable as those of the Medes and Persians. Then, that dress should be entirely destitute of steel, starch, whale-bone, flounces, and ornaments of all descriptions; should rest on the shoulders, have a skirt from the waist to the ankle, and a waist which leaves room for breathing.”
—Jane Grey Swisshelm (1815–1884)
“Even in harmonious families there is this double life: the group life, which is the one we can observe in our neighbour’s household, and, underneath, another—secret and passionate and intense—which is the real life that stamps the faces and gives character to the voices of our friends. Always in his mind each member of these social units is escaping, running away, trying to break the net which circumstances and his own affections have woven about him.”
—Willa Cather (1873–1947)