Stroud System of Unitsby J.M. Redwood.<Text-field layout="Heading 1" style="Heading 1">Synopsis</Text-field>This worksheet describes Professor Stroud's system for handling units and illustrates it with an example.<Text-field layout="Heading 1" style="Heading 1">Introduction</Text-field> Before the International System (SI) of units came into use, there was great scope for errors when solving problems in mechanics. Professor H. Stroud, a physicist and chemist, devised a system for eliminating such errors during the late 1880s while teaching dynamics at Imperial College, London. The fundamentals of his system are:
a. Formulae consist solely of symbols;
b. When solving a particular problem, symbols in the formula must be substituted by the numerical value appropriate to the problem, together with the units for that value. For example, when replacing the gravitational constant, g, in a formula, the value and units used in the British system would be NiMqKC0lJkZsb2F0RzYkIiZ1QCQhIiQiIiIlI2Z0R0YpKiQlInNHIiIjISIi ; and
c. Units must be converted using "unity brackets".<Text-field layout="Heading 1" style="Heading 1">Unity Brackets</Text-field> A unity bracket in Professor Stroud's system is simply a dimension measured in two different units, expressed as a fraction and equated to unity. For example, a length of 25.4 millimeters is equal to a length of 1 inch, or NiMvNyMqKC0lJkZsb2F0RzYkIiRhIyEiIiIiIiUjbW1HRisqJkYrRislJWluY2hHRitGKkYr . Since they are all unity, any number of unity brackets may be multiplied, or divided to translate a dimension from one system of units to another, but without affecting anything else. The proper choice of unity brackets and whether to multiply, or divide them, is obvious by mentally cancelling units in the string of brackets and checking that the desired result will be obtained. For example, if a length, x, was measured as 402.5 millimeters, but had to be expressed in yards it could be written as NiMqLC0lJkZsb2F0RzYkIiVEUyEiIiIiIiUjbW1HRik3IyooRilGKSUlaW5jaEdGKSomLUYlNiQiJGEjRihGKUYqRilGKEYpNyMqKEYpRiklI2Z0R0YpKiYiIzdGKUYtRilGKEYpNyMqKEYpRiklJXlhcmRHRikqJiIiJEYpRjRGKUYoRik=. restart;x := 402.5*mm * (1*inch/(25.4*mm)) * (1*ft/(12*inch)) * (1*yard/(3*ft));<Text-field layout="Heading 2" style="Heading 2">Force and Mass</Text-field> Forces are often measured in units of weight; a unit of weight being the force experienced by a unit mass at rest close to the earth's surface. The unity bracket relating a force unit to the fundamental units of mass, length and time is obtained from Newton's 2nd law, NiMvJSJmRyomJSJtRyIiIiUiYUdGJw==. Thus, a mass of 1 kg close to the earth's surface experiences a force of 1 kilogram-force (kgf) and measurements show that this force will accelerate it at 9.80665 metres per second squared. So the unity bracket is NiMvNyMqKCIiIkYmJSRrZ2ZHRiYqLEYmRiYlI2tnR0YmLSUmRmxvYXRHNiQiJ2wxKSohIiZGJiUibUdGJiokJSJzRyIiIyEiIkYzRiY=, or NiMvNyMqKiIiIkYmJSRrZ2ZHRiYlInNHIiIjKigtJSZGbG9hdEc2JCInbDEpKiEiJkYmJSNrZ0dGJiUibUdGJiEiIkYm, and similarly NiMvNyMqKiIiIkYmJSRsYmZHRiYlInNHIiIjKigtJSZGbG9hdEc2JCInVDxLISIlRiYlI2xiR0YmJSNmdEdGJiEiIkYm is the unity bracket relating the pound-force (lbf) to the fundamental units of mass, length and time in the British system.
A unit of mass, used in the USA and common in aero-engineering, is the slug, which is the mass that when acted upon by a force of 1 pound-force will accelerate at 1 ft per second squared. The unity bracket relating the pound-force to the fundamental units of mass, length, and time in the US system is NiMvNyMqKiIiIkYmJSRsYmZHRiYlInNHIiIjKihGJkYmJSVzbHVnR0YmJSNmdEdGJiEiIkYm .
As before, these unity brackets may be multiplied, or divided together as often as necessary to translate from one system to another. For example, to convert a mass, m, measuring 9.75 slugs to pounds, it would be written as NiMvJSJtRyoqLSUmRmxvYXRHNiQiJHYqISIjIiIiJSVzbHVnR0YrNyMqKkYrRislJGxiZkdGKyUic0ciIiMqJkYsRislI2Z0R0YrISIiRis3IyooRi9GKyokRjBGMUYrKigtRic2JCInVDxLISIlRislI2xiR0YrRjNGK0Y0RjQ=, m := 9.75*slug * (lbf*s^2/(slug*ft)) *
(1/(lbf*s^2/(32.1741*lb*ft)));<Text-field layout="Heading 2" style="Heading 2">Checking Dimensions</Text-field> Unity brackets containing only the dimensions are useful for checking that the dimensions of a formula are correct. Using the example above about the slug, the dimensions involved are force [F], mass [M], length [L], and time [T], and the unity bracket can be written as NiMvNyMqJiUiRkciIiIqKCUiTUdGJyUiTEdGJyokJSJURyIiIyEiIkYuNyMqKEYmRidGK0YnKiZGKUYnRipGJ0Yu. From Newton's 2nd law, Force = Mass x Acceleration or, NiMvJSJGRyooJSJNRyIiIiUiTEdGJyokJSJURyIiIyEiIg==. With this substitution, the unity bracket is identically unity, so the unity bracket used for the slug is dimensionally correct.
<Text-field layout="Heading 1" style="Heading 1">Empirical Formulae</Text-field> Many formulae found in physics and engineering were obtained by measurement during experiments and depend upon the units in which the measurements were made. For example, experiments with water flowing over a particular weir might yield the result NiMvJiUiUUc2IyUibkcqJi0lJkZsb2F0RzYkIiQwJCEiJCIiIikmJSNIZEdGJiomIiImRi4iIiMhIiJGLg==, where NiMmJSJRRzYjJSJuRw== is the volumetric flow over the weir measured in cubic feet per minute and NiMmJSNIZEc2IyUibkc= is the head of water flowing over the weir measured in inches of water. That is, NiMmJSJRRzYjJSJuRw== is the number of cubic feet of water flowing per minute and NiMmJSNIZEc2IyUibkc= is the number of inches head of water over the weir. (The usual symbol for liquid head is H, but Maple's Unit modules, which will be used later, reserve this for magnetic inductance.)
Before Professor Stroud's system can be applied, empirical formulae must be normalized. In this example NiMmJSJRRzYjJSJuRw== and NiMmJSNIZEc2IyUibkc= must be replaced by Q and Hd, where NiMvJSJRRyomJkYkNiMlIm5HIiIiNyMqJiUjZnRHIiIkJSRtaW5HISIiRik= and NiMvJSNIZEcqJiZGJDYjJSJuRyIiIjcjJStpbmNoX3dhdGVyR0Yp, and the normalized formula isNiMvKiYlIlFHIiIiNyMqJiUjZnRHIiIkJSRtaW5HISIiRiwqJi0lJkZsb2F0RzYkIiQwJCEiJEYmKSomJSNIZEdGJjcjJStpbmNoX3dhdGVyR0YsKiYiIiZGJiIiI0YsRiY=. This can be re-arranged to show that the units in the square brackets form part of the numerical constant (0.305), and thus demonstrate that Q and Hd are symbols as required by Professor Stroud.
Professor Stroud's system can be applied to this normalized formula to obtain answers in any desired system of units.<Text-field layout="Heading 1" style="Heading 1">Example</Text-field> Estimate the volumetric flow in litres per second when there is a 15 mm head of water over the weir mentioned above.
The normalized formula iseq1 := Q/(ft^3/min) = 0.305*(Hd/inch_water)^(5/2); Inserting the head as 15 mm of watereq1 := eval(eq1,Hd=15*mm_water);Multiplying the left and right sides of this equation by the appropriate unity brackets,eq2 := lhs(eq1) * (1*ft^3/(28.3168*litre)) * (60*s/(1*min)) =
rhs(eq1) * (1*inch_water/(25.4*mm_water))^(5/2);eq2 := simplify(eq2) assuming mm_water>=0, inch_water >=0;solve(eq2,{Q}); Maple's Units Converter can be used on the normalized formula, but first it is necessary to define 1 mm head of water in terms of units known to Maple.Units:-AddUnit(mm_water,context=pressure,conversion=metre_water/1000);eq3 := 'Q/(convert(1,'units',ft^3/min,litre/s)) =
0.305*(Hd/(convert(1,'units',inch_water,mm_water)))^(5/2)';
eq3; This formula is not normalized. It is an empirical formula and yields the flow of water over the weir in litres per second for a given head measured in mm of water.Hd := 15;fsolve(eq3,{Q}); Maple's Units modules can also be used with the normalized formula, but again it is necessary to define a mm head of water in terms of the units that Maple recognises.restart;Units:-AddUnit(mm_water,context=pressure,conversion=metre_water/1000);with(Units[Natural]): On entering the normalized formula, Maple converts it thus,eq3 := Q/(ft^3/min) = 0.305 * (Hd/inch_water)^(5/2); This is still a normalized formula, though Maple will express the result in SI units.Hd := 15*mm_water;simplify(solve(eq3,{Q}));<Text-field layout="Heading 1" style="Heading 1">Observations</Text-field>1. Professor Stroud's system is easy to use, not only with pen and paper as originally intended, but also in a Maple worksheet. It shows very clearly that all the unit conversions needed in a particular problem have been applied correctly. Naturally, the numerical value of the answer obtained depends upon the correct numerical values being inserted in the unity brackets.
2. Maple users may find Professor Stroud's system of greatest use when making reality checks on worksheets involving units.Legal Notice: The copyright for this application is owned by the author. Neither Maplesoft nor the author are responsible for any errors contained within it and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. If you wish to use this application in for-profit activities, the author requests that you make a donation to Cancer Research UK (www.cancer.org.uk).