School Science Lessons
International system of units
2025-09-30

Measurement

Contents
6.1.0 Area
6.2.0 Concentration
6.3.0 Experiments
6.4.0 Length
6.5.0 Mass and Weight
6.6.0 Mathematics
6.7.0 Temperature
6.8.0 Units
6.9.0 Velocity
6.10.0 Volume
6.1.0 Area
6.1.1 Area, Imperial units used in land surveying
6.1.2 Area, (shape)
6.1.3 Fifth field, (map colouring topology)
6.2.0 Concentration
6.2.1 Concentration
6.2.2 Aspect ratio
6.2.3 Brix, sucrose concentration
6.2.4 Degrees proof, proof spirit
6.2.5 Ratio and proportion, concentration
6.3.0 Experiments
6.3.1 Experiments
6.3.2 Rate of reaction, k
6.3.3 Use measuring instruments
6.4.0 Length
6.4.1 Length
6.4.2 Ångström unit, A
6.4.3 Astronomical unit, au
6.4.4 Draft, (draught), of a ship
6.4.5 Micron, µ (mu)
6.4.6 Möbius strip
6.4.7 Spherometer
6.5.0 Mass, Weight
6.5.1 Measure your weight
6.5.2 Apothecaries' weight
6.5.3 Assay value of precious metals
6.5.4 Avoirdupois weight, English and US weights and measures
6.5.5 Carat
6.5.6 Mass, kilogram
6.5.7 Quark
6.5.8 Troy weight
6.5.9 United States lineal weights and measures
6.5.10 United States surface (land) weights and measures
6.5.11 Weight of one matchbox full of fertilizer
6.6.0 Mathematics
6.6.1 Angle, angular
6.6.2 Construct a rectangle from parts of a square
6.6.3 Ellipse
6.6.4 Energy conversion
6.6.5 Errors
6.6.6 Estimating
6.6.7 Fractions
6.6.8 Googol
6.6.9 Integers
6.6.10 Measurements
6.6.11 Table of numerals adding to 33
6.6.12 Order of accuracy
6.6.13 Order of magnitude
6.6.14 Perfect numbers
6.6.15 Pi, π
6.6.16 Prime numbers
6.6.17 Radians
6.6.17 Roman numerals
6.6.18 Significant figures and standard form
6.6.19 Standard deviation
6.6.20 Standard form, scientific notation
6.6.21 Standard deviation
6.6.22 Tests for divisibility
6.7.0 Temperature
6.7.1 Temperature
6.7.2 Celsius scale
6.7.3 Fahrenheit scale
6.7.4 Kelvin scale, absolute zero
6.7.5 Oven temperatures
6.7.6 Temperatures in different scales (F-32) / 9 = C / 5 | K = C + 273.15
6.7.7 Temperature and specific heat capacity
6.7.8 Triple point and ice point temperatures of water
6.8.0 Units
6.8.1 CGS. units (centimetre, gram, second)
6.8.2 Common measures
6.8.3 m.k.s. units
6.8.4 Newton, the newton, symbol N
6.8.5 Non-SI units
6.8.6 Radioactivity, radiation units, curie
6.8.7 Time, the present, seconds, leap second
6.10.0 Volume
6.10.1 Volume
6.10.2 Measuring cups, jugs, spoons
6.10.3 Tonnage, displacement
6.10.4 British liquid measures
6.10.5 American liquid measures
6.6.5 Errors
6.6.5.1 Accuracy and precision
6.6.5.2 Order of accuracy
6.6.13 Order of magnitude
6.6.19 Significant figures
6.6.21 Standard form, scientific notation
6.1.1 Imperial units used in land surveying
(1 hectare = 10, 000 m2, 1 km = 1, 000 m)
Table 3.11.0 Imperial units used in land surveying
Imperial Metric Imperial Metric
1 square mile 259.0 hectare (ha) 1 link 0.201 168 m (exact)
1 square mile 2.589 988 km2 1 foot 0.3 048 m (exact)
1 acre 4 046.856 m2 1 mile 1.609 344 m (exact)
1 acre
 0.4047 hectare (ha)
 1 perch
 25.2 929 m2
1 rood 1 011.714 m2 0.03 954 perch 1 m2

1 square centimetre, cm2 = 0.1550 square inch (in2)
1 square inch = 645.2 square mm
1 square metre (m2) = 10.76 square feet
1 square metre (m2) = 1.196 square yard
1 square metre (m2) = 0.0002471 acre (ac)
1 square mile = 1 U.S. "section"
1 hectare (ha) = 2.471 acre = 107 639 ft2
1 hectare (ha) = 0.00386 square mile
1 yard (yd) = 0.9 144 metre (m)
1 square foot = 0.92 903 square metre.
6.1.2 Area (shape)
Area, square metre (m2) hectare
1 km2 = 1 square kilometre = 1000 m × 1000 m × 1000 m.
It does not mean 1000 square metres.
Land: 100 metres (m) × 100 metres (m) = 10 000 square metres (m2) = (104 m2) = 1 hectare (ha) = 2.471 acre = 107 639 ft2
Imperial units used in land surveying (1 hectare = 10 000 m2, 1 km = 1 000 m)
Area of cloth for a dress, area of a bolt of cloth, floor cover, area of a fitted carpet.
Irregular shape area, use of graph paper.
Regular shape area, square, rectangle, circle
Area of the top of a matchbox: 20 cm2
Area of a square = length l2
Area of a rectangle = length l × width w
Area of a parallelogram = length l × vertical height / 2
of a circle =π × r2
cu. = cubic
Surface area of a sphere = 4π × r2
Volume of a sphere = 4 / 3π × r3.
6.2.4 Area (shape)
6.2.2 Aspect ratio
6.6.2 Construct a rectangle from parts of a square
6.1.3 Fifth field (map colouring topology)
See diagram 3.4.01 : The four fields
See the diagram of the four fields labelled A, B, C and D.
Each of the fields has a common boundary with all the other fields.
Can you draw a fifth field, E, which has a common boundary with all the other fields, A, B, C, and D?
It cannot be done!
You cannot draw a fifth field of any side or shape with common sides with four other fields.
If you colour the different States in a map of the United States of America, you need no more than four different colours to ensure that no two States with a common border have the same colour.
6.2.1 Concentration
Concentration is the quantity of dissolved substance to quantity of solvent.
Dilution is the volume of solvent in which a measured amount of solute is dissolved.
Different ways of expressing concentration, e.g. ppm, % weight for weight (w/w), % weight for volume (w/v).
Parts per million by mass (ppm, milligrams per kilogram, 0.0 001%) = about a grain of sugar in a cup of tea.
Parts per million, (ppm), 1 ppm = 1 mg per litre.
Parts per million, (ppm), usually refers to ppm by weight 1g solute per 1, 000, 000 g solution = 0.001 g per 1, 000 g solution = 1 mg solute per 1 kg solution.
If aqueous solution, where concentration of solute is so low that assume solution density = 1.00 g / mL, then ppm = 1 mg of solute per litre of solution.
So using this assumption, convert ppm in mg / Litre to molarity in mol / Litre.
If x ppm of Ca2+ ions (atomic weight of calcium = 40.8), then x ppm = x mg Ca2+ / Litre of solution = 0.00x g / Litre, 0.00x / 40.08 = mol / Litre.
6.2.2 Aspect ratio
Aspect ratio is the proportion between width and height, W: H format, e.g. for HD television channels the aspect ratio =16: 9, 1.77: 1.
Check the aspect ratio on your television set.
Each paper size in the A series of International Standard (I.S.O.), paper sizes is half the area of the next biggest sheet, half as wide, but same length, so 2 A4 sheets side to side = 1 A3 sheet.
If length = L and width = W, L /W = sqrt 2 = 1.41, the aspect ratio for the A series of paper sizes.
So one sheet of A3 can be reduced to one sheet of A4, with the photocopier control panel set to 71%, approximately 1/sqrt 2.
Similarly, for enlargements, the control panel is set to 141%.
This aspect ratio allows exact reductions or enlargement, but in USA and Canada, the American National Standards Institute (ANSI), uses two different aspect ratios in photocopiers.
A series
A1 paper 841 mm X 1189 mm, A2 half that, A3 half that, A4 half that, A8
ANSI E paper 34 inches X 44 inches.
6.2.3 Brix, sucrose concentration
One degree Brix, 1Bx = 1 g sucrose in 100 g as % w/w, as used in fruit juice, honey and wine industries.
Brix meters calculate mass fractions and give the value as Bx.
A sucrose solution with specific gravity 1.040 at 20 oC is 9.99325 Bx or 9.99249%.
6.2.4 Degrees proof, proof spirit
Proof is a standard of strength of distilled alcoholic liquors.
Proof spirit contains, in Britain 49.28% alcohol (ethanol) by weight, 57.10% by volume, relative density 0.920 at 10.6 oC (formerly specific gravity of 12 / 13 at 51 oF) in USA 50% by volume at15.6 oC.
This standard is quoted as 100 degrees of proof, 100o.
If a spirituous liquor is p% overproof (above standard strength, contains more alcohol than proof spirit), it contains as much alcohol in 100 vol as in 100 + p vol of proof spirit.
20o proof = 0.2 × 57.1% alcohol = 11.42% ALC / VOL, e.g. white wine.
Concentration of alcohol can also be measured with a hydrometer.
Formerly, proof spirit was an alcoholic beverage that, if poured over gunpowder and ignited, would ignite the gunpowder.
If the gunpowder did not ignite, the spirit was "under proof".
6.2.5 Ratio and proportion, concentration
Relative size of electron, atom, molecule, cell, man and woman, earth.
Pi, π = 22/7
63.2 Rate of reaction, k
6.2.4 Degrees proof, proof spirit
2.0.2 Golden mean (gif file)
6.3.1 Experiments
1. Relative positions between measured object and equipment +
When you read on a scale with a measured object directly touching with the equipment, be careful as their relative position will probably affect precision of your readings.
For example, if you measure temperature of liquid by a thermometer, you must immerse completely the measuring bulb in the liquid as you take readings.
2. Reaction time of the equipment
Some equipment reacts to measured quantities very quickly, such as meters for measuring electricity.
However, some equipment needs a certain reacting time, such as a mercury thermometer.
So you must take readings after the equipment stabilizes.
Even with equipment that reacts quickly you need to pay attention to such problems, e.g. when measuring electric potential, be certain that the pointer no longer moves before you read from the scale.
3. Line of vision
The angle between your line of vision and the object referred to can cause errors.
Your eye should be at right angles to the scale and directly opposite the part of the scale you are reading.
Reading a scale from the left side or the right side or above or below are all wrong, because they result in parallax error.
4. Record measurements in tables
Set up a table vertically if there is a possibility of additional requiring some extra space.
Include a title and table number on the top of a table to state what data the table contains.
The first column should contain data for the independent variable rather than the dependent variable.
The weight is the independent variable, because you decide its values, usually before doing the experiment.
The increase in length of spring is the dependent variable, because it depends on the weight added.
Express all data in standard form.
Increase in length of spring.
(Original length = 28.0 cm)
Table 3.3.4.1 Record measurements in tables
Weight
(N)		 Length of spring
(cm)		 Increase in length
(cm)
0.49 (0.5 kg) 32.8 4.8
0.98 (1 kg) 36.3 8.3
1.47 (1.5 kg) 39.4 11.4
1.96 (2.0 kg) 41.9 13.9

6.3,2 Rate of reaction, k
For many chemical reactions, but not all, increasing the concentrationof reactants increases the rate of reaction.
The rate constant, k, is the constant for a given reaction at a given temperature.
H2 (g) + I2 (g) --> 2HI (g)
rate = k [H2 (g)] [I2 (g)], where [H2(g)] = concentration of hydrogen gas
If x = any substance, [x] = concentration of x.
If in a chemical reaction, [x] is doubled and the rate of reaction remains constant, then the rate of reaction is independent of [x].
If in a chemical reaction, [x] is doubled and the rate of reaction doubles, then the rate of reaction = k[x].
6.3.3 Use measuring instruments
Use measuring instruments, micrometer screw gauge, vernier calipers
See diagram 6.4.4 : Micrometer screw gauge allows very accurate measurement.
A wood screw is like a wedge wrapped around a cylinder so that turning the screw forces two pieces of wood together.
(The order of accuracy of an area is one half of the order of accuracy of the diameter.)
Two cylinders diameter 1 mm and 3 cm, must be measured to an accuracy of one part in fifty for the area of cross section.
If area must be correct to 1 in 50, diameter must be correct to 1 in 100.
For the 3 cm diameter cylinder, the order of accuracy is 0.03 cm.
To achieve that accuracy use a vernier calipers reading to 0.01 cm with 10 vernier divisions corresponding to 9 millimetre divisions on the main scale.
For the 1 mm diameter cylinder, the order of accuracy is 0.01 mm or 0.001 cm.
To achieve that accuracy, use a micrometer screw gauge with pitch 0.5 mm and drum divided into 50 equal parts, so that each division corresponds to 0.001 cm.
Carbon composite digital vernier caliper
It is ideal for general use and where the cost of precision stainless steel tool is not justified.
The digital display is calibrated in imperial and metric units with a corresponding vernier scale etched onto the caliper slide.
It is suited to a person who finds traditional vernier calipers hard to read.
6.4.1 Length
Length (l), kilometre (km), metre (m)
A metre is the length of a path travelled by light in a vacuum during a time interval of 1 / 299 792 458 of a second.
1 metre, m (SI unit)
1 centimetre, cm, also centimeter
1 kilometre, km, also kilometer
1 millimetre, mm, also millimeter
Originally, 1 metre = 1 / 10 000 000 of the meridian through Paris, between the North Pole and the Equator.
Length (l), the kilometre (km), metre (metre, m)
[Metre, m, unit of length, SI base unit, (American spelling: meter), and "meter" is in the names of measuring instruments.]
A metre is the length of a path travelled by light in a vacuum during a time interval of 1 / 299 792 458 of a second.
calipers, Vernier calipers, Vernier scale (Pierre Vernier 1580-1637), calipers are for measuring internal and external diameters.
Gauge, feeler gauges, micrometer screw gauges, are to find the thickness of one sheet of paper in a pile.
Rule, measuring timber for carpentry, tape measure, dressmaking measurements: circumference of the chest / waist / hips
Trundle wheels are used to measure the length of a crooked path.
1 kilometre, 1 km = 1 000 metres (Originally, it was a line from the north pole to the equator through Paris, was thought to be 10, 000 km.)
1 decimetre, 1 dm = 0.1 metre
1 centimetre, 1 cm = 0.01 metre
1 millimetre, 1 mm = 0.001 metre
1 micrometre (or micrometer), 1 µ m = 1 × 10-6 metre, one millionth of a metre, micron
Nanometre
1 nanometre, nanometer, nm = 10-9 metre, one billionth of a metre, (0.000000001 m), (1 X 10-9 m), (10 angstroms), (formerly 1 millimicron)
1 picometre, pm = 10-12 metre.
6.4.2 Ångström unit, A
One ångström unit, symbol A = 10-10 metre (one hundred-millionth of a centimetre), previously used as unit of measurement of wavelength, but nowadays use nanometre.
(Note: 1 nm = nanometre = 10 Angstrom units = 10-9 m.).
The unit, named after A. J. Ångström, Sweden (1814-1874), is still used in crystallography and to measure wavelengths of the electromagnetic spectrum.
6.4.3 Astronomical unit, au
One Astronomical unit = the mean distance between the Earth and the Sun, 149 597 871 km, but taken as 1.496 × 108km, (93 million miles).
It is used as a convenient way to measure distance in the solar system.
6.4.4 Draft, (draught), of a ship
The draft is the vertical distance between the waterline and the bottom of the hull or keel.
The draft usually varies along the length of the ship.
6.4.5 Micron, µ (mu)
The term "micron" was discarded by international agreement, but it was still used in industry and some sciences.
Then it was accepted again, because "micrometre" was confused with the "micrometer screw gauge", a measuring device containing a fine-pitched screw.
micron, µ (non-SI unit) = 1 micrometre, µm (British English) (1 micrometer, USA), a millionth of a metre
1 micrometre (1.000 µm) = 1.000 × 10-6 metre (m), one millionth of a metre, micron
(Micrometre is used to measure wavelength of infrared radiation.)
1 nanometre (1 nm) = 10-9 metre, one billionth of a metre, (10 angstroms) (formerly 1 millimicron)
1 millimicron (mµ) = 1 / 1000 of a micron = 10-9 metre = 10-3 micrometre = 10 Angstrom = SI unit nanometre (nm) (USA nanometer) (one billionth of a metre)
1 angstrom = 1.0 × 10-10 metres
(millisecond and microsecond are non-SI units).
6.4.6 Möbius strip
1. A Möbius strip is a two-dimensional surface with only one side.
It can be constructed in three dimensions as follows.
Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180 degree twist.
It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A.
2. Cut a strip of paper 2 cm wide on writing paper with lines on only one side.
Hold the strip by each end and half twist it, i.e. twist it by 180o.
Note that you could twist it to the left or to the right.
Use adhesive tape to stick the two ends together to make a loop.
Hold the paper strip against the point of a pencil then draw a line along the middle of the strip without taking the pencil off the paper.
Keep drawing the line until you get back to where you started.
Examine both sides of the strip of paper and note that you have drawn on both sides of the paper.
Use sharp scissors to cut along the line.
Note that you now have a new loop twice as long as the original loop.
3. Using a longer strip of writing paper 2 cm wide, repeat the above experiment with a full twist, 360o.
Again cut along the a line in the middle of the strip to produce two separate loops, the same size as before, but linked together like a link is in a chain.
4. This loop was discovered by August Ferdinand Mobius and Johann Benedict Listing in 1858, but the ancient Greeks may have known it.
The mobius strip seems to be useless, but it has been used in car fan belts, conveyor belts and continuous loop recording tapes to double the playing time.
5. Möbius joke: Q. "Why did the chicken cross the Möbius strip?" A. "To get to the same side".
6.4.7 Spherometer
A spherometer is an instrument for measuring the sphericity of curvature of a body or a surface.
It can measure the precise radius of a spherical object or the radius of curvature of a lens.
It works on the principle of a micrometer screw gauge.
A central screw has an attached scale so the distance can be measured of the length of the screw supported by three feet set in an equilateral triangle.
6.5.1 Measure your weight
Experiment
Use a bathroom scale or sling scale to record the weight in kilograms of each student on the same day each month, e.g. 15th day of the month.
Record the weights on a wall chart.
At the end of the year each student draws a graph of the recorded weights.
Calculate the average weight of the students each month.
Discuss how to measure the weight of your head.
6.5.2 Apothecaries' weight
Apothecaries' measures were formerly used in pharmacy and were usually adopted in formulas.
1 fluid ounce = 8 drachms = 489 minums.
The pound, ounce and grain are the same as in Troy weight.
In UK, the fluid drachm (fluidrachm) = 3.55 mL.
Table 3.2.4 Apothecaries' weight
pound ounce drachm scruple grain g
lb.
 oz.
 dr.
gr.
1 12 96 288 5 760 373.24
. 1 8 24 480 31 103
. . 1 3 60 3 888
. . . 1 20 1.30
. . . . 1 0.06

6.5.3 Assay value of precious metals
An assay is a chemical analysis of a substance to find the proportion of a valuable constituent.
Assay value is measured in milligrams, mg, of precious metal per assay ton = troy ounces of precious metal per avoirdupois ton of ore.
An assay ton is equivalent to 29.160 g of precious metal per short ton, (2000 pounds).
6.5.4 Avoirdupois weight, English and US weights and measures
The imperial unit ounce may be a measure of mass or volume
1 avoirdupois weight pound (lb) = 16 ounces (oz.).
All chemicals were sold by avoirdupois weight.
(Latin: pondus (weight), 12 ounces of pure silver, 240 pennies, so cash to the value of 20 shillings sterling, symbol lb
(Latin: libra pondo (libra, scale, pondo, by weight)
A fluid dram is 1 ⁄ 8 of a fluid ounce, i.e. 3.696 mL USA and 3.551 mL UK.
In Scotland, a dram is a small volume of Scotch whisky.
Table 3.2.1 1 Avoirdupois weight
pound ounce drachm, dram grain (Troy) g
1 16 256 7 000 453.60
. 1 16 437.5 28.35
. . 1 27.34 1.771 845

6.5.5 Carat
Pure gold is rated at 24 carats, so 18 carat gold contains six parts of an alloy.
Gold leaf, 23-24 carat, is gold beaten into very thin sheets for gilding decoration and electrical contacts, e.g. gold leaf electroscope.
The concentration of gold in sea water is about 10-30 g / km2, but no profitable method of extraction is known.
The red-purple alloy guanin, prized by pre-Columbian Indians in the Cuba region, was an alloy of gold, silver and copper.
Note the colour, and density of the specimen.
For precious stones, 1 carat is about 1 / 142 of an ounce, formerly 3.17 grains, now standardised at 0.20 grams.
For gold, a carat is a ratio of 1 / 24.
Purity of gold is measured in carats.
24 carat gold is pure gold.
22 carat gold is 22 parts pure gold and 2 parts copper or other metal alloy.
14 carat gold is 14 parts pure gold and 14 parts copper or other metal.
The official mark stamped on gold and silver objects after being assayed is the hall mark (from Goldsmith's Hall, London).
For gold, the standard mark is a crown in England for 22 and 18 carat gold followed by the number of carats in figures.
Lower standards of gold have the number of carats in figures without the crown.
The term "carat" is said to come from Arabic quirat and Greek keration, the seed of Carob tree (Ceratonia siliqua).
It was used to weigh gold, at 0.18 g per seed, regardless of the shape of the seed.
Although this assumption was found to be inaccurate, for a long time the carob seed remained the only known way to weigh gold.
6.5.6 Mass, kilogram
The mass of a body is the quantity of matter it contains, as measured by its acceleration produced by a force, or by the force exerted on it by a gravitational field.
Every body or particle within a body has mass and is attracted towards the centre of the Earth by a force = mg, where m = mass, and g = the attractive force by which objects are attracted towards the centre of the Earth.
The intensity of gravity is measured by the acceleration caused by the gravitational force.
Inertial mass
The standard of mass is the kilogram, based the existence of a particular cylinder of platinum-iridium alloy.
This standard can be referred to as the inertial mass, mi..
So mass is defined by its inertia.
Gravitational mass
However, mass is conveniently measured by using the weight, W, of the body, i.e. the force of gravity attracting it to the Earth.
W = mg, where g is the acceleration of fall that varies slightly in different places on the surface of the Earth.
To define mass in terms of the gravitational force it can produce, i.e. gravitational mass, mg,
Use the formula: mg = Fd2 / MG where M is a standard body distance d from another body of mass mg,
F is the gravitational force between the bodies and G is the universal gravitational constant.
However, mi = mg.
The kilogram is the base unit of mass.
Originally, the gram was intended to be the mass of a cubic centimetre of pure water at 4 oC.
However, the gram was later defined as one thousandth part of a kilogram.
So the standard of mass is now the kilogram.
A kilogram is the mass of the international prototype kilogram kept in Sevres, France.
It is a 90% platinum and 10% iridium cylinder at the International Bureau of Weights and Measures.
A proposed alternative definition, called the Planck value:
A kilogram is such that the Planck constant is exactly 6.6 260 693 × 10-34 joule seconds.
Weight: 1000 grams (g) = 1 kilogram (kg), 1000 kg = 1 metric tonne (t).
MT metric ton t, T  metric ton, kilogram, kg (SI unit of mass) also kilogramme, kilo, microgram, µg, milligram, mg.
6.5.7 Quark
The name of the fundamental building block of matter, the quark, comes from the novel "Finnergans Wake" by James Joyce was given this name by Murray Gell-Mann.
It is generally pronounced "qwork" to rhyme with "pork".
6.5.8 Troy weight
Gold is still sold in troy ounces, as were precious metals.
1 troy weight pound, lb = 12 troy ounces.
1 grain = 6.479 × 10-5 kg.
Table 3.2.3 Troy weight
Latin: denarius, penny
pound ounce pennyweight, Dwt grain g
1 12 240 5 760 373.24
. 1 20 480 31.10
. . 1 24 1.56

6.5.9 United States lineal weights and measures
foot (singular) feet (plural), yard (Old English: gerd, stick, rod) (mile: Latin: mille, 1 000, 1 000 paces, about 1 680 yards), (inch from ounce Latin: uncia, 12th part of a foot)
Table 3.3.3.0 United States lineal weights and measures
mile furlong rod yard foot inch
mi.
 fur.
 rd.
 yd.
 ft.
 in.
or "
1 8 320 1 760 5 280 63 360
-
 1 40 220 660 7 920
- - 1 5.5 16.5 198
- - - 1 3 36
- - - - 1 12

6.5.10 United States surface (land) weights and measures
1 square foot = 144 square inches
1 square yard = 9 square feet
1 square rod = 30.25 square yards
1 square rood = 40 square rods
1 acre = 4 square rods
1 square mile = 640 acres = 2 560 roods = 102 400 rods = 3 097 600 square yards = 27 878.400 square feet
Table 3.3.4.0 United States surface (land) weights and measures
acre rood rod yard foot
ac.
rd.
 yd.
 ft.
1 4 160 4 840 43 560
. 1 40 1 210 10 890
. . 1 30.25 272.25
. . . 1 9

6.5.11 Weight of one matchbox full of fertilizer
See diagram 3.1.2.6a : Matchbox
Ammonium sulfate (sulfate of ammonia) 26 g
Potassium sulfate (sulfate of potash) 40 g
Potassium chloride (muriate of potash) 24 g
Single superphosphate, "super" 22 g
Triple superphosphate, "super" 20 g
Sulfur 20 gm.
6.6.1 Angle, angular
See diagram 6.3.3.2 : Plane angle: (Symbol: rad)
See diagram 6.3.3.3 Angle tube syringe
Collect gas with an angle tube syringe.
See diagram 2.0.1 Right-angled triangle
Angle is the measurement of the inclination of one line to another.
Degrees, o: An angle is usually measured in degrees, 360 degrees (360o) = 1 revolution.
Degree is the unit of angle.
One revolution = 360 degrees, 360o.
One right angle = 90 degrees, 90o.
Degree can be divided into arc minutes, arcmin, such that 1 arcmin, 1' = 1 / 60 of a degree.
Degree can also be divided into arc seconds, arcsec, such that 1 arcsec, 1" = 1 / 3 600 of a degree.
Arc minutes and arc seconds are used in astronomy to measure the diameter or separation of astronomical objects.
6.6.2 Construct a rectangle from parts of a square
See diagram 3.4.2 : Square and rectangle
Draw the square in the diagram and draw the same lines to divide the square into four pieces.
Note the angle a.
Use scissors to cut out the four pieces.
Rearrange the four pieces to form a rectangle.
Why the increase in area?
Tan angle a in the square = 3/8 = 0.375.
Tan angle a in the square = 5/13 = 0.3846
So the pieces really do not fit together to form a rectangle!
6.6.3 Ellipse
| See diagram 6.15.5 : Section of a cone and ellipse
| See diagram 2.0.5 : Conic sections, parabola, ellipse, hyperbola (gif file)
| See diagram 6.15.4 : Draw an ellipse.
An ellipse is one of the conic sections, the cross-section of a cone cut by a plane having a smaller angle, a, with the base of the cone than the side of the cone makes, b.
An ellipse is a symmetrical closed curve traced by a point moving on a plane such that the sum of its distances from two other points is constant.
An ellipse has two foci.
To draw an ellipse, stick two thumbtacks (drawing pins), f1 and f2, down into a thick piece of paper on the table, about 15 cm apart.
They represent the two foci.
Tie the ends together of a 25 cm piece of string to make a loop.
Place the loop around the thumbtacks.
Hold the point of a pencil vertically inside the loop, but pulled tightly outwards.
Mark the paper at that point, p1.
Keep the loop of string tight and start moving the pencil around the two foci.
After travelling a few centimetres, stop the pencil and mark point p2.
Continue moving the pencil around the two foci until you return to point p1.
You have drawn an ellipse.
Measure p1f1, p1f2, p2f1, p2f2.
Note p1f1+ p1f2 = p2f1 + p2f2.
6.6.4 Energy conversion KJ, MJ, kWh, therm, BTU, calorie, horsepower
1 kilowatt (kW) = 1.341 horsepower (hp)
1 kilojoule (kJ) = 0.948 British Thermal Unit (Btu)
1 megajoule (mJ) = 948 Btu = 0.28 kWh = 0.37 horsepower hours
1 joule (J) = 0.239 calories (cal)
1 therm = 100 000 British Thermal Unit (BTU) = 106 mJ
1 British Thermal Unit (BTU) = 1.055 kilojoule
1 kilowatt hour, kilowatt-hour (kWh) = 3 412 Btu = 3.6 mJ
1 calorie (cal) = 4.186 J (if International Table calorie), however, the 15 oC calorie = 4.1855 J
1 horsepower (hp) = 746 watts, 0.7457 kilowatt
1 horsepower hour (hph) = 2.69 mJ watt, W.
6.6.5.1 Accuracy and precision
Accuracy and precision, possible error, least count
Errors by 10 students, standard error
Measurement errors, parallax error, zero error \ index error and correction, systematic error
Random errors and system errors, scale error, probable error
Significant figures - all the figures that can be read with meaning from an instrument
Standard form (scientific notation), e.g. 8.04 × 102 shows significant figures expressed unambiguously
The reading below, as shown by the arrow, is 98.5. The 9 and the 8 are certain figures.
The 5 is uncertain.
The absolute error is half the smallest division of the scale being read, i.e. 0.5.
So the reading in absolute error form is: 98 + or - 0.5.
100

99
->

98

97
6.6.6 Estimating
Estimating of parameters, prediction, size perception, relative size
Estimating height of people, tree, a house, bridge, mountain
Estimating distance from the roadside, of the car ahead.
6.6.7 Fractions
3 / 5 is a fraction, because it is < 1.
It is a rational number in the form p / q where p and q are integers and q is not = 0.
The denominator, 5, is the number of divisions of the whole ('fifths") and the numerator, 3, is the number of equal parts.
The vinculum, /, separates the numerator from the denominator, 3 / 5
A proper fraction is < 1, e.g. 3 / 5.
An improper fraction is > 1, or = 1, e.g. 6 / 4.
A fraction can be expressed as a mixed number, e.g. 31 / 2.
6.6.8 Googol
Googol, ten raised to the hundredth power, 1 then 100 zeroes, 10100.
The term "googol" is not in formal mathematics use.
The name "googol" is said to be invented by Milton Sirotta, 9 years old nephew of US mathematician Edward Kasner.
"Google", proprietary name of internet search engine, also transitive verb "to googlr" meaning to use Google to find information.
The name of the search engine "Google" is an accidental misspelling of "googol".
6.6.9 Integers
. . . -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 . . .
6.6.10 Different measurements, million, billion, trillion
Traditional counting units, a score, a dozen (doz.), common units, market units
Units and scale divisions, analogue units, digital units
b bit
B byte 
KB kilobyte
1 kB = 1 000 bytes = 103B
1 kB = 1 024 bytes = 210B
MB megabyte
1 MB = 1 0002 B = 106 B = 1 000 000 bytes
1 MB = 1 0242 B = 220 B = 1 048 576 bytes
Cardinal numbers, cardinal numerals, positive whole numbers, 1, 2. 3. . .
Hundred, 102, one hundred, 100
Thousand, 103, one thousand, 1 000
Million, a thousand thousand, one million, 1 000 000
Billion, a million million, 1012 in UK, (but a thousand million, 109, in USA, and now the most popular use in the world).
Trillion, a million million million, 1018, in UK, but a million million, 1012, in USA, now most popular use in the world
6.6.11 Table of numerals adding to 33
The traditional age of Jesus when he died is 33 years.
Table 3.3.7.0 Numerals adding vertically or horizontally or diagonally to 33
1 14 14 4
11 7 6 9
8 10 10 5
13 2 3 15

6.6.12 Order of accuracy
Calculation of possible error when the differences from the mean are known and the differences are small.
In the measurement of the diameter of an iron cylinder with a micrometer screw gauge, the readings along the length of the cylinder were as follows:
Table 6.3.11 Order of accuracy
Readings (cm)
 Residuals (reading - mean) (cm)
2.466
 0.002
2.461
 0.003
2.467
 0.003
2.463
 0.001
2.462
 0.002
2.465
 0.001
2.467
 0.003
2.464
 0.000
Mean = 2.464 cm
 Sum or residuals (Σ r) = 0.015 cm
All readings are between 2.461 and 2.467, so the true value of the diameter of the cylinder is unlikely to be outside these limits.
The probable value is the mean, 2.464 cm.
So the greatest probable error is 0.003 cm.
The diameter of the iron cylinder is 2.464 ± 0.003 cm.
However, this result probably overestimates the error.
The error is more accurately calculated by using the formula 3Σ r / n √n.
 [(3 × Σ r) / (8 × sqrt 8)]
3Σ r / n √n = (3 × 0.015) / (8 × √8) = 0.002
The diameter of the iron cylinder is 2.464 ± 0.002 cm.
Order of accuracy is usually expressed in round numbers: 0.002 in 2.464, 2 in 2, 464, 1 in 1, 232, 1 in 1, 200.
When the iron cylinder was measured, the order of accuracy of the measurements was 1 in 1, 200.
6.6.13 Order of magnitude (nearest power of ten, a factor or factors of ten)
Order of magnitude is a value expressed to the nearest power of ten.
Sometimes you are interested in knowing the approximate rather than the precise values, so you just use the nearest power of ten, e.g. speed of light: 3.0 × 108 ms-1 = (approx.) 108 ms-1, the radius of the Earth: 6.38 × 106 m = (approx.) 10 × 106 m= 107 m, the radius of the Moon = 3.8 × 108 = (approx.) 109 m.
(3.8 is closer to 100 (1) than to 101 (10), 3.8 is greater than 100 = 3.14).
6.6.14 Perfect numbers
A perfect number is equal to the sum of its factors, excluding itself, e.g. the factors of 6 are 1, 2, 3, and 6.
Excluding the last factor 6, 1 + 2 + 3 = 6.
A perfect number is a positive integer that is the sum of its positive divisors, excluding that number.
The first four perfect numbers are: 6, 28, 496, 8128.
If p = a prime number
Perfect number = 2p-1(2p-1), so if p =2, then 21(22-1) = 2 × 3 = 6
All known perfect numbers are even.
Euclid of Alexandria (325- 265 BC approximately) include a study of perfect numbers in his work on geometry, "The Elements", but they were known at much earlier dates.
Prime numbers of the form 2p-1 are called Mersenne primes, (Marin Mersenne 1588-1648).
Nobody has found any use for perfect numbers, but some Greek philosophers thought they had some sort of mystical properties.
6.6.15 Pi, π
Pi, π, is the sixteenth letter of the Greek alphabet.
Π, π
Pi is the ratio of the circumference of a circle to its diameter.
It is an irrational number (22/7) (3.14159).
By using the mnemonic:
"May I have a large cup of coffee", the number of letters in each word stands for a digit of pi, e.g. first digit 3 = "May".
Use string and a ruler to measure the circumference and diameter of different circular objects, then calculate the ratios of circumference to diameter.
A mathematician, when offered a slice of cake replied: "I prefer pi."
This reply is the shortest sentence palindrome!
6.6.16 Prime numbers
Prime numbers have only themselves and one as factors, so a number that is not prime is called a composite number.
Prime numbers < 1000:
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349
353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601
607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733
739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997
6.6.17 Radians
Angle is also measured in radians, an angle at the centre of a circle subtended by an arc equal to the radius of that circle, such that 2 π (pi) radians = 1 revolution.
Draw a big circle on the chalkboard.
Cut a piece of string with length of the radius.
Place the string on the circumference of the big circle to show a radian.
6.6.18 Roman numerals
I = 1, V = 5, × = 10, L = 50, C = 100, D = 500, M = 1000.
Year 2025 = MMXXV.
6.6.19 Significant figures
(1.) Observations should be as accurate as possible, unaffected by preconceived ideas.
Measurements are more precise if several measurements agree closely.
The accuracy of measurement is limited by the smallest unit on the measuring instrument, e.g. using a ruler marked in millimetres (mm), if the average of several measurements is 174.5 mm.
The reading is between 174 mm and 175 mm and the absolute error is +-0.5 mm.
(2.) Significant figures are all the figures that can be read with meaning from an instrument.
Significant figures of a number are the digits that contribute to its value.
Significant figures - all the figures that can be read with meaning from an instrument
For measurement, the significant figures are those you know with certainty plus the digit that is uncertain.
A "2 tonne truck" could weigh between 1.5 and 2.5 tonnes.
A reading of 25 cm could have a value between 24.5 and 25.5 cm.
So you say that the last digit is uncertain.
You count zeros between integers and zeros to the right of the decimal point following non zero integers.
You do not count other zeros.
The following examples each have four significant figures:
0.01 234
0.1 023
0.1 230
In the last case you are saying that the reading is closer to 0.1 230 than 0.1 229 or 0.1 231.
So be careful about zeros, especially the last zero.
(3.) If rounding off to 3 significant figures:
4.657 becomes 4.66, because 7 > 5.
4.655 becomes 4.66, because last digit is 5 and digit behind it is odd.
4.645 becomes 4.64, because last digit is 5 and digit behind it is even.
4.654 becomes 4.64, because 4 < 5. 3.
When adding or subtracting, all numbers must have the same number of digits after the decimal point.
This is equal to the least number of digits after the decimal point of any number in the addition or subtraction.
19.43 + 6.456 + 101.9 becomes 19.4 + 6.5 + 101.9 =127.8.
(4.) When multiplying or dividing numbers, the answer can have only as many significant figures as the number with the least number of significant figures.
17.9 × 4.3 = 76.97 Answer = 77 (4.3 has only 2 significant figures).
(5.) Significant figures are one way of indicating the uncertainty of a measurement, but many scientists are not bothered with them and use absolute uncertainty to indicate uncertainty.
Other scientists use an extra 'guard digit' in deciding the number of significant figures.
Different rules may be taught in maths and physics class and there may be a difference for 'rounding' rules.
6.6.20 Standard deviation
To calculate the standard deviation of 9, 2, 5, 4, 12, 7, 8, 11
1. Calculate the mean of these eight numbers.
2. For each number, subtract the mean and square the result.
3. Calculate the mean of those squared differences.
4. Calculate the square root of that mean.
In mathematics, mean and standard deviation are not used for less than 25 samples.
In biology, mean and standard deviation may be used on less than 10 samples.
6.6.21 Standard form, scientific notation
Standard form or scientific notation expresses a number as a product of a number between 1 and 10 and a power of 10.
It is a convenient way to express large and small numbers for easy comparison and show the number of significant figures.
So you can write 18 000 as 1.8 × 104 (2 significant figures, i.e. the value is between 1.7 and 1.9 × 104),
or 1.80 × 104 (3 significant figures, i.e. the value is between 1.79 and 1.81 × 104).
Express decimal fractions in standard form: 0.1 = 1 × 10-1, 0.019 = 1.9 × 10-2 Standard form (scientific notation), e.g. 8.04 × 102, shows the significant figures unambiguously.
The coefficient, 8.04, must be greater than or equal to 1 and less than 10.
The base number 10 is written in exponent form, so in 8.04 × 102, the number 2 is the exponent or power of  ten.
Express decimal fractions in standard form, e.g.
0.1 = 1 × 10-1
0.2 = 1 × 10-2
0.019 = 1.9 × 10-2
0.00 087 = 8.7 × 10-4.
6.6.22 Tests for divisibility
Divisible by 2, the number is even, i.e. it ends in 0, 2, 4, 6, 8
Divisible by 3, the sum of the digits is divisible by 3
Divisible by 4, the number formed by the last two digits is divisible by 4
Divisible by 5, the last digit is 5 or 0
Divisible by 6, the number is even and the sum of its digits is divisible by 3
Divisible by 7, no easy divisibility test
Divisible by 8, the number formed by its last 3 digits is divisible by 8
Divisible by 9, the sum of its digits is divisible by 9
Divisible by 10, the last digit is 0.
6.7.1 Temperature
See diagram 23.7.00 : Celsius temperature scale
The temperature of a body is its hotness or coldness with reference to a standard of comparison.
Temperature varies with the amount of heat energy in the body.
To convert between Fahrenheit scale and Celsius scale:
C = (F-32) X 5/9.
6.7.2 Celsius scale
The Celsius temperature scale, proposed in 1724 (Anders Celsius 1701-1744), has graduations on the thermometer
based on a lower fixed point of 0 oC, the freezing point of water, and an upper fixed point of 100 oC the boiling point of water.
So the fundamental interval is 100 Celsius degrees.
The Celsius scale was formerly called the centigrade scale, "100 steps"scale.
Some people still incorrectly quote temperatures in "degrees centigrade".
C = "Celsius" NOT "Centigrade".
The unit "degree Celsius" is equal to the unit "kelvin", and a temperature interval may also be expressed in "degrees Celsius".
To convert the Fahrenheit scale to the Celsius scale (F-32) / 9 = C / 5.
So 68 oF = 20 oC.
The Celsius and Fahrenheit scales have the same value at -40 oC, or -40 oF.
Human body temperature = 37 oC (Celsius).
0 degrees Celsius is the temperature at which ice melts.
100 degrees Celsius is the boiling point of water.
0 degrees Celsius is 273.15K.
The boiling point of water = 373.15 Kelvin.
6.7.3 Fahrenheit scale
The Fahrenheit temperature scale (Gabriel Daniel Fahrenheit 1686 - 1736), has thermometer graduations based on
a lower fixed point of 32 oF, the freezing point of water, and an upper fixed point of 212 oF, the boiling point of water.
So the fundamental interval is 180 Fahrenheit degrees,180 oF.
The Fahrenheit scale is still used in USA.
To convert the Fahrenheit scale to the Celsius scale (F-32) / 9 = C / 5.
So 68 oF = 20 oC.
The Celsius and Fahrenheit scales have the same value at -40 oC, or -40 oF.
Water freezes at 32 °F, and boils at 212 °F, so boiling and freezing point are therefore 180 °.
Normal body temperature = 98.6 °F.
Absolute zero = -459.67 °F.
6.7.4 Kelvin scale, absolute zero
The Kelvin scale (Lord Kelvin 1824 - 1907) is based on the idea of absolute zero.
Molecular motion, heat, approaches zero, the null point, as the temperature approaches -273.15 oC.
One kelvin degree, 1 K = 1 Celsius degree, 1 oC.
Absolute zero = -273.15 oC = 0 K, not "degree Kelvin".
To convert the Celsius scale to the Kelvin scale, add 273.15.
For example, 0 oC = 273.15 K, 100 oC = 373.15 K, and 10 oC = 283.15 K.
So this scale begins at absolute zero and increases in kelvins.
The Kelvin scale is the preferred scale for scientific experiments.
The temperature, kelvin, is the fraction 1 / 273.16 of the thermodynamic temperature of the triple point of water.
Proposed alternative definition of temperature, kelvin:
The kelvin is such that the Boltzmann constant is exactly 1.3806505 ×10-23joules per kelvin.
An approximate value of absolute zero may be obtained by plotting pressure versus temperature and extrapolating the line produced to zero pressure.
Experiment
Submerge the bulb of a pressure gauge successively in boiling water (100°C), ice water (0°C), and dry ice in alcohol (-78.5°C) and record the corresponding pressure indicated by the gauge.
6.7.5 Oven temperatures
Table 6.3.13 Oven temperatures
oC oF Gas mark Description
110 225 1 / 4 very cool, very slow
120 250 1 / 2 .
140 275 1 cool
150 300 2 .
170 325 3 very moderate
180 350 4 moderate
190 375 5 .
200 400 6 moderately hot
220 425 7 hot
230 450 8 .
240 475 9 very hot

Extra
gr. and gro. = gross
k. and kt.= karat
knot
LT, L.T. = long ton
mph = miles per hour
n.m. = nautical miles
sq.= square
rpm = revolutions per minute
single hatch mark ' = foot or minute of longitude or latitude
single hatch mark " = second of longitude or latitude
5'6" = five feet, six inches
42o24'54"N = 42 degrees, 24 minutes, 54 seconds north.
6.7.6 Table 6.2.20 Equivalent temperatures in different scales (F-32) / 9 = C / 5 | K = C + 273.15
.
 Kelvin Celsius Fahrenheit
Absolute zero 0 oK -273 oC -459 oF
Freezing point of water 273 oK 0 oC 32 oF
Boiling point of water 373 oK 100 oC 212 oF

6.7.7 Temperature and specific heat capacity
1. Use digital and other measuring devices to collect data, ensuring measurements are recorded using the correct symbol, SI unit, number of significant figures and associated measurement uncertainty (absolute and percentage).
All experimental measurements should be recorded in this way.
2. Conduct an experiment that determines the specific heat capacity of a substance, ensuring that measurement uncertainties associated with mass and temperature are propagated.
Where the mean is calculated in this, and future experiments, determine the percentage and/or absolute uncertainty of the mean.
3. Conduct an experiment to investigate the initial and final temperature of two liquids before and after they are mixed.
Compare the final temperature data with a temperature value calculated theoretically by finding the percentage error.
4. Conduct an experiment that requires students to construct and interpret displacement / time and velocity / time graphs with resulting data.
Where appropriate, students should use vertical error bars when plotting data.
This ensures that they can determine the uncertainty of the gradient and intercepts using minimum and maximum lines of best fit.
6.7.8 Triple point and ice point temperatures of water
| See diagram 24.3.5.1 : Phase diagram, Most substances
| See diagram 24.3.5.2 : Phase diagram, Water, (Triple point of water)
1. Th19e triple point is the temperature at which the three phases of a substance can exist together.
The triple point temperature of water is the equilibrium point = 0.01 oC.
(273.16 K) and 611.2 Pa (N m-2) in a sealed vacuum flask.
It is an important fixed point for kelvin and thermodynamic scales of temperature.
The ice point temperature, 273.15 K, is the temperature when equilibrium exists between ice and water at standard pressure.
It is the lower fixed point of the Celsius scale.
2. For all substances, as pressure is lowered, the boiling temperature falls much more rapidly than does the freezing temperature.
For water, the freezing temperature rises slightly at low pressure.
The boiling temperature and freezing temperature are equal at the low pressure of 611 Pa (0.006 × times atmospheric pressure), pure water boils and freezes 0.01 oC.
The combination 611 Pa, and 0.01 oC is called the triple point of water, because at this pressure and temperature ice, liquid water, and steam can coexist in equilibrium.
This point is used to define the scale of temperature, i.e. the triple point of water occurs at 273.16 K, where K is the kelvin, 273.16 K = 0.01 oC.
6.8.1 CGS. units (centimetre, gram, second)
Table 3.9.0 The CGS. units (centimetre, gram, second)
Quantity CGS Unit
 Size
Length centimetre 1 cm = 10-2 m
Mass gram 1 g = 10-3kg
Area cm2 1 cm2= 10-4 m2
Volume cm3 1 cm3= 10-6 m3
Density g cm-3 1 g cm-3 = 10-3 kg m-3

6.8.2 Common measures
6.10.2 Measuring cups, jugs spoons
1 barleycorn, 1 / 3 inch, 0.84667 cm (old British unit)
1 barrel, bbl. of crude oil = 42 US gallons, = 34.97 Imperial gallons (about 159.1 litres)
1 barrel (petroleum) = 35 imperial gallons (about 159 L)
1 barrel (beer cask) = 32 imperial gallons
1 cubic inch = 16.38 cubic centimetres
1 cubit = (English cubit 46 cm) (Roman cubit 44 cm) (Egyptian cubit 53 cm) (Hebrew 56 cm), traditional from the tip of the elbow to the tip of the longest finger)
1 cup, cupful = 284 mL
1 cup, teacup (the cup you use with a saucer) = 200 - 250 mL
(1 / 4 cup of butter, half fill a cup with water, add butter until water rises to the 3 / 4 level)
1 dash = what you pick up between your thumb and first two fingers
1 drachma = 1 / 8 oz
1 ell = 45.5 cm (English), 37 cm (Scotch), 54 cm (French) (cloth measure from elbow to finger tips)
1 English wine bottle 750 mL
1 fluid ounce = 29.57 millilitres, mL
1 Foolscap printing paper = 13.5 × 17 inches
1 Foolscap writing paper = 13.25 × 16.5 inches
1 glass, wine glass = 1 / 4 cup
1 hair breadth = 1 inch / 48
1 human's body temperature 37 oC (Celsius
1 hundredweight, British hundredweight, 112 pounds, 1 Cwt. 1 / 20 ton
("long hundredweight"), 50.80 kg
1 hundredweight, US hundredweight, cwt, 100 pounds (lb) ("short hundredweight"), 45.36 kg
1 hundredweight, metric hundredweight 50 kg
1 Imperial fluid ounce = 28.42 cc
1 jeroboam = 4 English wine bottles = 4 × 262 / 3 fluid ounces
1 jerrican = 4½ gallons (used for military fuel)
1 jigger = 1.5 fl oz
1 journey-weight of gold = 15 pounds troy (701 sovereigns)
1 kati, caddy = 1 lb, 5 oz, 2 dr, weight still used in Malaysia
(1 kati said to be 12 / 16 British pound in Hong Kong)
1 league, about 2-4 miles
1 matchbox volume = 25 mL, Area of the top of a matchbox = 20 cm2
1 magnum = 2 English wine bottles (2 "reputed" quarts)
1 nail = formerly a weight of 8 pounds or a length of 2.25 inches
1 peck = 2 dry gallons, a quarter of a bushel
1 penny weight = 1 / 20 fl. oz
1 pinch = what you pick up between your thumb and first two fingers
(½ pinch = to what you can pick up between your thumb and one finger)
1 pipe = 105 gallons of wine
1 quart (liquid) = 0.9463 litre
1 quarto sheet of paper, folded twice to give 4 leaves, 8 pages, about as high as wide
1 quintal, q, 100 kg = 220.5 pounds
1 rehoboam = 6 English wine bottles
1 US bushel = 35.24 litres
1 US liquid gallon = 3.785 litres
1 US short ton = 0.9072 tonne
1 US long ton = 1.016 tonne.
6.8.3 m.k.s. units
The metric system of units based on metre, kilogram, second.
Also, the electrical unit was the ampere and magnetic constant was 4 pi × 10-7 Hm-1 (henry = H, now SI unit of inductance).
6.8.4 The newton, symbol N
A teaching toy called "Newton's Apple", weighs 102 g, so approximately 1 newton force
In "Memoirs of Sir Isac Newton's Life", by William Stukeley:
"Why should that apple always descend perpendiculary to the ground, thought he to himself ..."
The newton is the SI unit of force, equal to the force that would give a mass of one kilogram an acceleration of one metre per second per second.
1 N is the force of Earth's gravity on a mass of about 102 g.
A mass of 1 kg exerts a force about 9.81 N down on the Earth.
1.0 kilogram-force = 9.80665 N
F = ma, F = 0.102 kg x 9.8 m / s2 = 1 newton
So if 102 g is elevated through one metre = approximately one joule of energy.
6.8.5 Non-SI units
6.10.4 American liquid measures, US measures, United States weights and measures, volume to liquid
6.2.3 Ångström unit, A
6.4.3 Astronomical unit, au (non-SI unit)
6.2.28 British liquid measures, imperial measures
(fl. oz. = imperial fluid ounce) (ounce Latin: uncia, 12th part of a pound)
6.8.1 CGS. units (centimetre, gram, second)
6.6.4 Energy conversion KJ, MJ, kWh, therm, BTU, calorie, horsepower
href="#6.2.1 H">6.2.1 Imperial units used in land surveying (1 hectare = 10 000 m2, 1 km = 1 000 m)
36.42.2 Knots, Latitude, nautical mile, knots, log, logbook
36.14.2 Light year, parsec, minute of arc, arc second (non-SI units)
6.4.5 Micron, µ (mu)
6.8.3 m.k.s. units
6.5.7 Quark
6.5.9 United States lineal weights and measures
6.5.10 United States surface (land) weights and measures
6.8.6 Radioactivity, radiation units, curie
31.2.0 Electric charge, the coulomb, C, Coulomb's law
1. SI unit of activity, becquerel (Bq) = one nuclear disintegration per second, 1 s-1
Former unit, curie, Ci = 3.7 × 1010 nuclear disintegrations per second (was supposed to be the activity if 1 gram of radium 226).
SI unit of absorbed dose of ionizing, gray, Gy = 1 joule of energy per kilogram of irradiated material.
Former unit, rad, rd = 10-2 Gy.
2. SI unit of exposure to ionizing radiation
Former unit of exposure to X-ray or γ radiation, roentgen, R, = radiation producing ions with total charge of 2.58 × 10-4 coulombs per kilogram of air.
Also, 1 rem = the approximate effect of 1 roentgen of X-rays on human tissue.
Use of a quality factor, Q, where X-ray, γ-ray, or β-radiation, Q = 1, α particles, Q = 20, neutrons Q = 10, dose equivalent, rem, = dose, rad × Q.
3. SI dose equivalent, sieverts =10 rem.
However, nowadays ionizing radiation is expressed in coulombs per kilogram.
Experiment
Simulate radioactive decay with dice
Each group starts with 80 dice and removes any sixes each throw.
Five throws are usually enough for calculation.
Calculate triplicates and uncertainty to give a lnN vs t graph to confirming exponential decay.
The calculate theoretical half life and % error.
Repeat the activity by removing 5 and 6 for a shorter half life.
6.8.7 Time, the present, seconds, leap second
* The "present"
Archaeologists define "the present" at 1 January 1950, a date before nuclear explosions contamination of the Earth.
So an event said to occur "10, 000 years ago" occurred 10, 000 years before 1 January 1950.
* Seconds
The "second" refers to the second division of time into sixtieths after dividing the hour into minutes.
A second is the time equal to the duration of 9 192 631 770 cycles of oscillation (periods of the radiation) corresponding to the transition between the two hyperfine levels of the ground state of the Caesium 133 atom.
Before 1956, the mean solar day was 24 hours, at 60 minutes per hour, and 60 seconds per minute = 84, 400 seconds.
60 seconds = 1 minute
60 minutes = 1 hour
24 hours = 1 day
7 days = 1 week
365 days = 1 year, 366 days = 1 leap year
10 years = 1 decade
100 years = 1 century
1000 years = 1 millennium
a.m. (ante meridiem) = morning
p.m. (post meridiem) = afternoon
1.00 p.m. = 1300 hours.
* Leap second
1. One extra second is added to Universal Coordinated Time (UTC), about every one and a half years to keep UTC close to mean solar time.
The leap second is added, because the rotation of the around its axis Earth's may become slower, because of climatic and geological changes.
However, atomic clocks used to define UTC keep almost constant speed.
Since 1972, 24 leap seconds have been added and the next leap second will be added in June 2012.
Only the days on which the leap seconds are added have 86, 401 seconds instead of 86, 400 seconds.
The International Earth Rotation and Reference System Service (IERS), compares the Earth's rotation to atomic time and add a leap second when the differences approach 0.9 seconds.
Experts at the International Telecommunication Union (ITU), have been unable to decide whether to abolish the leap second, so have deferred the decision until 2015.
While the US, Japan, Italy, Mexico and France have argued that leap seconds were causing problems for communication and navigation systems, the UK, Germany and Canada, want to keep adding the leap second every few years. 2. ABC / Reuters, Posted Sun Jul 1, 2012 1: 26pm AEST (Australia)
An extra second has been added to the world's atomic clocks in an adjustment to keep them in step with the slowing rotation of the Earth.
The so-called "leap second" was added to electronic clocks at midnight universal time on Saturday.
At that time, atomic clocks read 23 hours, 59 minutes and 60 seconds before they moved on to Greenwich Mean Time.
Super-accurate atomic clocks are the ultimate reference point by which the world sets its wrist watches.
But their precise regularity, which is much more constant than the shifting movement of the Earth around the sun that marks out our days and nights, brings problems of its own.
If no adjustments were made, the clocks would move further ahead and after many years the sun would set at midday.
Leap seconds have a similar function to the extra day in each leap year, which keeps the calendar in sync with the seasons.
The last so-called leap seconds happened in 2008, 2005 and 1998.
Adjustments to atomic clocks are more than a technical curiosity.
A collection of the highly accurate devices are used to set Coordinated Universal Time, which governs time standards on the world wide web, satellite navigation, banking computer networks and international air traffic systems.
There have been calls to abandon leap seconds, but a meeting of the International Telecommunications Union, the UN agency responsible for international communications standards, failed to reach a consensus in January.
Opponents of the leap second want a simpler system that avoids the costs and margin for error in making manual changes to thousands of computer networks.
Supporters argue it needs to stay to preserve the precision of systems in areas like navigation.
A decision is not urgent.
Some estimate that if the current arrangement stays, the world may eventually have to start adding two leap seconds a year.
But that is not expected to happen for another hundred years or so.
6.9.0 Velocity (speed)
6.9.1 Speed of light, C
14.8.0 Scalars and vectors
4.24 Speed of reaction, Catching the ruler, (Experiments)
26.5.0 Speed of sound, (Experiments) 14.1.0 Velocity and speed, (Experiments)
6.9.1 Speed of light, C
Einstein was right, e = mc2
Albert Einstein's celebrated formula e = mc2 has finally been corroborated, thanks to a mighty computational effort by French, German and Hungarian physicists.
A brain power consortium led by Laurent Lellouch, of France's Centre for Theoretical Physics, using some of the world's most powerful supercomputers, has set down the calculations for estimating the mass of protons and neutrons, the particles at the nucleus of atoms.
According to the conventional model of particle physics, protons and neutrons comprise smaller particles known as quarks, which are bound by gluons.
The odd thing is the mass of gluons is zero and the mass of quarks is 5 per cent.
Where is the missing 95 per cent?
The answer, according to the study published in US journal Science, comes from the energy from the movements and interactions of quarks and gluons.
In other words, energy and mass are equivalent, as Einstein proposed in his "Special Theory of Relativity" in 1905.
The e = mc2 formula shows that mass can be converted into energy, and energy can be converted into mass.
By showing how much energy would be released if a certain amount of mass were to be converted into energy, the equation has been used many times, most famously as the basis for atomic weapons.
Resolving e = mc2 at the scale of sub-atomic particles in equations called quantum chromodynamics to has been difficult.
"Until now, this has been a hypothesis," France's National Centre for Scientific Research said proudly in a statement.
"It has now been corroborated for the first time."
For those keen for more, the computations involve "envisioning space and time as part of a four-dimensional crystal lattice, with discrete points spaced along columns and rows".
AAP (Australian Associated Press)The Australian (newspaper), November 22-23, 2008.
Einstein wrote: "If a body emits the energy L in the form of radiation, its mass decreases by L/V2.
Here it is obviously inessential that the energy taken from the body turns into radiant energy, so we are lead to the more general conclusion.
The mass of a body is a measure of its energy content; if the energy changes by L, the mass changes in the same sense by L/9 x 1020 if the energy is measured in ergs and the mass in grams."
In these units, speed of light = 3 X 1010 cm per second, and speed of light squared = 9 X 1020.
Substitute E for L, and c for V, then mass, m, decreases by E/c2, E= mc2.
6.10.1 Volume
See diagram 2.1.6 : Liquid volume (gif)
Measuring cylinders / graduated cylinder: 1.29
6.6.3 Surface / volume ratio of soil particles
2.1.6 Volume of liquid (meniscus diagram)

Volume (vol.) cubic metre (m3)
Volume in a measuring cylinder, meniscus
Volume of a bucket, fish tin, coconut, tablespoon, teaspoon, cooking oil for food, of agricultural chemical to be used on a farm
Volume of water used at home or school, reading a water meter
Volume of petrol (gasoline) used by a motor vehicle
Volume of irregular shapes, volume of small quantity of sand or glass beads.
Displaced volume, overflow vessels
Volume of regular shapes, a cube, a block, cylinder, sphere, cone
Volume of gas used at home or school, reading a gas meter
Millilitre
Millilitre, one millilitre (mL), the basic unit of capacity, is equivalent to one cubic centimetre (cc or cm3).
Volume, solid: 1 centimetre (cm) × 1 centimetre (cm) × 1 centimetre (cm) = 1 cubic centimetre (1 cc, 1 cm3) = 1 millilitre, 1 mL
[millilitre mL (SI unit), also milliliter, ml, mℓ] (a thousandth of a litre of capacity)
1 cubic decimetre, 1 dm3 = 1 litre, 1 L = 1000 mL = 1000 cm3 = 1000 cc
Volume, liquid: 1 000 millilitres = 1 litre (L)
Mole, 1 mole, 1 M = 1 mole per cubic decimetre = 1mole per litre = 1 mol. L-1.
6.10.2 Measuring cups, jugs, spoons
Jug plastic, translucent, graduated with multiple measuring units, 1000 mL
Measuring cylinders / graduated cylinder: 1.29
Plastic measuring spoon set, 1.25 mL, 2.5 mL, 5 mL, 20 mL, set / 4
Plastic measuring cups, ¼, 1 / 3, ½, 1 cup, set / 4
Polypropylene beakers, opaque, unsuitable for heating, graduated with multiple measuring units, 1000 mL.
Spoon volume
1 tablespoon (tbsp) (spoon to serve with, the biggest spoon):
15 mL (most countries) to 20 mL (Australia) (0.5 fl oz)
1 dessertspoon (the spoon you eat with) = 10 mL (2 teaspoons)
1 teaspoon, tsp. (the smallest spoon)
1 teaspoon, tsp. UK = 4.5 to 5 mL (0.2 fl oz) (UK 4 mL) (1 fluid dram)
1 teaspoon, tsp. US = 1 / 3 tablespoon, 1 / 6 U.S. fl. oz, 1 / 48 of a cup
1 / 768 of a U.S. liquid gallon (1 / 3 of a cubic inch, cu. in.)
1 teaspoon, tsp. US = 5 mL (for US food labels))
1 measuring spoon for medicines and some fertilizers = 5 mL
(1 salt spoon, saltspoonful = ¼ teaspoon).
1 tablespoon, tbsp. = 3 teaspoon, tsp.
6.10.3 Tonnage, displacement
Measurement of the volume of a boat for registration and fees, e.g. Panama canal fees
Gross tonnage, GT = KV where V= total volume in cubic metres and K = 0.2 + 0.02 log10V
Net tonnage, NT, is the total cargo space
The displacement, the volume of the hull below the waterline × specific gravity of water, is expressed in metric tons (not tonnage!).
6.10.4 British liquid measures
imperial measures (fl. oz. = imperial fluid ounce)
These measures were usually adopted in formulas.
1 fluid ounce = 28.42 mL (0.96 US oz)
1 imperial pint = 568.3 mL (20 fl oz)
1 quart = 1140 mL (40 fl oz) (38.5 US oz)
1 imperial gill = 0.132 L (5 fl oz)
1 imperial gallon = 4.54 609 litres, 4.55 L
1 fluid drachm = 60 minims
1 fluid ounce = 8 fluid drachms
1 pint = 20 fluid ounces
1 gallon = 8 pints.
1 hogshead (of beer) = 54 imperial gallons (245.48 litres).
6.10.5 American liquid measures
US measures, United States weights and measures, volume to liquid
1 liquid US pint = 473.1 mL (473.179 cc) (16 fl oz)
1 dry US pint = 550.6 mL (19 fl oz)
1 US fluid ounce = 29.56 mL (29.574 cc)
1 US gill = 0.118 L
1 US gallon = 3.79 L (3 785.435 cc)
1 pint = 4 gills
1 quart = 2 pints
1 gallon = 4 quarts (231 cubic inches)
pt. pint
qt. quart
gal. gallon.