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# The Imperial System

A brief description of the Imperial system of measurements as used in some Codecogs worked examples.

## Overview

These notes are designed to help those of you who are not used to the Imperial System of measurements. Also shown is an easy method of conversion from one system of measurements to another. It is hoped that it will prove useful.

## The Imperial System.

This system can be traced back to the Roman Empire. It has of course been modified over time but is essentially based on unit sizes which have proved to be useful or convenient. May be the most interesting are the measurements of capacity. In general these ate based on a square based box. Doubling the size of the base and the volume increases by 4. Double the height and the increase is only 2. A similar system applies to measurements below 1 Inch, Smaller sizes are usually measured in fractions e.g. $\inline&space;\tfrac{1}{2}\;;\;\tfrac{1}{4}\;;\;\tfrac{1}{8}\;;\;\tfrac{1}{32}\;and\;\tfrac{1}{64}$ For more accuracy thousanths of an inch are used.

## Engineering In The Imperial System - The Slug

It is common to assume that engineers use the f.p.s. system (Foot Pound Second). The is quite wrong The Americans weigh themselves in Pounds and the British buy potatoes in pounds in both cases the pound is treated as a weight and is therefore a unit of Force. This gives problems with Newton's Second Law since a pound can not be simultaneously be both a mass and a weight. The result was the introduction of a unit of mass called the slug. All the worked examples, on Codecogs,calculated in Imperial units, are based on The Foot; Slug; Second system This can still cause confusion since in common usage, what was a pound of sugar ( a weight), is now sold in grams (a mass) and shops use 2.2 to convert from one to the other. It must be stressed that the Pound (lb.) can only, in engineering terms, be converted into Newtons.

By definition A force of 1 lb. acting on a mass of 1 slug will produce an acceleration of 1 ft./sec. squared

## Working With Imperial Equations.

When examining Imperial Calculations there are a few numbers which appear and which to those of us who use to the system, have a particular meaning:
• 32.2 is the acceleration due to gravity in feet/sec
• 550 is the number of foot pounds per second in 1 horse- power = 33,000 ft.lb.per min
• 14.7 is atmospheric pressure in pounds per square inch
• 62.4 is the weight in pounds of one cubic foot of water
• 1 gallon of water weighs 10 pounds

It is also useful to know that 60 miles per hour (mph) is equivalent to 88 feet per second.

As there are 12 inches in a foot, then when you find 12, 144, or 1728 in an equation it usually means that there is a conversion from feet to inches or via versa. On drawings it is normal to mark measurements made in inches by " . feet by ' and yards by ^x i.e. 2 foot 6 inches is written as 2'6"

For completeness:
• Water boils at $\inline&space;212^{\circ}$F or $\inline&space;672^{\circ}$R (at 14.7 psi)
• Water freezes at $\inline&space;-32^{\circ}$F, i.e $\inline&space;460^{\circ}$R ( at 14.7 p.s.i)

## The Conversion Of Units.

The following system has been found useful for converting quantities from one measurement system to another. The only proviso is that there must be the same zero. e.g. The system can not convert ^{0}F to $\inline&space;^{0}C$ but can convert $\inline&space;^{0}\,Rankin\;&space;to\;^{0}K$ since they both measure form absolute zero.

The system makes use of the fact that multiplying a quantity by unity does not alter the value. It is therefore assumed that as there are 25.4 mm in 1 inch, it is possible to write :-

$\left&space;[&space;\frac{25.4&space;mm}{1\;&space;inch}&space;\right&space;]$

and that the effective value of the above bracket is unity. In very simple terms. If it is necessary to convert 7 feet to mm. the bracket would be arranged as follows:-

$7&space;ft.&space;\left&space;[&space;\frac{12&space;inch}{1&space;ft.}&space;\right&space;]\times&space;\left&space;[&space;\frac{25.4&space;mm}{1\;&space;inch}&space;\right&space;]$

Now all that is necessary is to cancel out the words and all that is left is:-

$7\times&space;12\times&space;25.4&space;mm.$
i.e.7 feet = 2133.6 mm.

Note that the "Unity Bracket" can be put either way up and if necessary squared or cubed to fit the occasion.

## Imperial Units

The following are the units commonly used by engineers. A more complete list follows later:
• 12 inches (in.) = 1 foot (ft.)
• 3 feet = 1 yard (yd.)
• 1760 yards = 1 mile
• 2240 pounds = 1 ton

## English Weights And Measures

4 inches = 1 hand

1 cable's length = 100 fathoms = 600 feet
12 inches (in.) = 1 foot (ft.)

3 feet = yard (yd.)

1 ell = 3ft. 9ins

25 links = 16feet 6inches = 1 rod, pole or perch
5.5 yards = 1 rod, pole, or perch

40 poles (220 yards) = 1 furlong (furl)

10 chains = 22 yards, 1 furlong
8 furlongs (1760 yards) 1 mile (m.)

3 miles = 1 league

1 pole, Woodland measure = 6 yards
1 chain = 100 links (22 yards)

1 pole, Plantation measure = 7 yards
6 feet = 1 fathom

1 pole, Cheshire measure = 8 yards
6,080 feet or 1.1516 statute miles = 1 nautical mile

Scottish Standard Land Chain = 24.7065 yards
4,840 sq.yards = 1 English Acre

640 acres = 1 square mile