School Science Lessons
Topic 6 Measurement, SI, International system of units, accuracy and error
2012-04-20 SPwp
Please send comments to: J.Elfick@uq.edu.au
Table of contents
6.0.0 Measurement
3.3.1.0 Accuracy and error
6.3.01 Angle
6.3.05 Area (shape)
35.3.01 Assay value of precious metals
12.1.04 Atmospheric pressure water spray
3.2.1 Avoirdupois weight, English and United States weights and measures
6.3.04 Balances, Force, (weight) (Primary)
36.42.2 Cable, Knots, Latitude, nautical mile, knots, log, logbook
3.5.4 Common and miscellaneous measures, cup, spoon, matchbox
38.7.00 Computers
6.3.3.1a Conductivity
6.2.0 Different measurements, billion, trillion
3.5.4.1 Draft (draught) of a ship
36.36.0 Earth and celestial bodies
6.3.1.4 Electric current, ampere
3.3.1.0 Errors, Accuracy and error
6.4.0 Errors, theory of errors, addition of uncertainties
3.6.0 Estimating
6.2 Estimating experiments (Primary)
36.42.2 Fathom, Knots, Latitude, nautical mile, knots, log, logbook
6.3.04 Force, (weight) (Primary)
2.0.0 Graphs, Mathematics
6.3.1.2 Kilogram, mass
36.42.2 Knots, Latitude, nautical mile, knots, log, logbook
36.42.2 Latitude, nautical mile, knots, log, logbook
6.3.1.1 Length
6.3.1.7 Luminous intensity, candela, cp
6.3.1.2 Mass, kilogram
2.0.0 Mathematics, graphs
18.0 Measurement experiments (Primary)
36.36.0 Measurement of the Earth and celestial bodies
3.5.4a Measuring cups, jugs spoons
5.1.0 Mole, amount of substance
5.1.1 Mole, Prepare molar solutions
36.42.2 Nautical mile, Knots, Latitude, nautical mile, knots, log, logbook
12.1.04 Pressure, Atmospheric pressure
6.3.3.01 Radioactivity, radiation units
3.7.0 Ratio and proportion, concentration, rate of reaction, degrees proof
36.42.2 Shackle, Knots, Latitude, nautical mile, knots, log, logbook
6.3.0 SI, International system of units
6.3.1.5.0 Temperature, Celsius scale, Kelvin scale, Fahrenheit scale
6.3.1.3 Time, second
6.3.07 Tonnage, displacement
6.3.03 Velocity, (speed)
6.3.06 Volume
3.2.0.0 Weight, standards of weight
2.0.0 Mathematics, graphs
See Part 11. Interesting websites, Mathematics and computers
2.0.5 Conic sections, parabola, ellipse, hyperbola, (GIF file)
2.22 Copy with a rubber band (Primary)
6.15.3 Fractions
2.0.2 Golden mean, (GIF file)
3.3.5.0 Graphs
4.1.1 Graphs
4.2.2 Graphs Variation of temperature in water with the time and drawing a graph of the data
2.0.3 Greek alphabet
6.15.2 Integers
2.0.8 Mathematics for science teachers, (GIF file)
3.3.7.1 Mobius strip
2.0.6 Parabola equation, (GIF file)
6.15.0 Perfect numbers
3.2.2 Rank scaling tables
6.13.0 Roman numerals
2.0.7 Scale of a map, (GIF file)
3.3.7.0 Table of numerals adding vertically or horizontally or diagonally to 33
6.15.1 Tests for divisibility
3.2.0.0 Weight, standards of weight
3.2.4 Apothecaries' weight
3.2.1 Avoirdupois weight
3.2.2 Carat
6.3.1.2 Mass, kilogram
3.2.3 Troy weight
6.12.0 Weights of one matchbox full of fertilizer
3.3.1.0 Accuracy and error
6.4.0 Errors, theory of errors, addition of uncertainties
6.4.1 Significant digits and standard form, scientific notation
6.4.2 Order of magnitude (nearest power of ten, a factor or factors of ten)
6.4.3 Order of accuracy
6.4.4 Use measuring instruments, micrometer screw gauge, vernier calipers
3.3.3.0 Factors that affect readings, obtain data from the equipment
3.3.3.1 Relative positions between measured object and equipment
3.3.3.2 Reaction time of the equipment
3.3.3.3 Line of vision
3.3.4.1 Record measurements in tables
3.3.5.0 Graphs
3.3.7.0 Table of numerals adding vertically or horizontally or diagonally to 33
3.2.2 Rank scaling tables
3.7.0 Ratio and proportion, concentration, rate of reaction, degrees proof
6.6.3 Surface / volume ratio of soil particles
Surface / volume ratios of small block and large block. Surface / volume ratio of small cell and large cell.
Relative size of electron, atom, molecule, cell, man and woman, earth.
3.7.1 Concentration, parts per million (PPM)
3.7.2 Rate of reaction
3.7.3 Degrees proof, proof spirit
2.0.2 Golden mean, (GIF file
3.9.0.0 Non-SI units
3.5.3 American liquid measures, US measures, United States weights and measures, volume to liquid
3.5.6 angstrom
3.3.2.0 Astronomical unit (non-SI unit)
3.5.2 British liquid measures, imperial measures (fl. oz. = imperial fluid ounce) (ounce Latin: uncia, 12th part of a pound) 3.9.0 c.g.s. units (centimetre, gram, second)
3.13.1 Einstein was right, e = mc2
3.13.0 Energy conversion KJ, MJ, kWh, therm, BTU, calorie, horsepower
3.11.0 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, arcsecond (non-SI units)
3.5.7 micron, µ, micrometre, µm, millimicron, nanometre (millisecond and microsecond are non-SI units)
3.5.4 Miscellaneous measures
3.10.0 m.k.s. units
3.13.1.1 Quark
3.12.0 SI, c.c.s. and f.a.s.. conversion, metric conversion
3.5.1 Spoon volume
3.3.3.0 United States lineal weights and measures
3.3.4.0 United States surface (land) weights and measures
6.3.3.14 Viscosity, poise
6.3.0 SI, International system of units, SI units, Système international d'unités (French)
6.3.1 SI, The 7 base units
6.3.3.0 SI derived units
6.3.5 SI prefixes
6.3.6 SI, CGS, (cgs) and FPS, (fps) conversion, metric conversion
3.9.0.0 Non-SI units
6.3.3.1 Other derived units based on SI
6.3.3.5 Units used with SI units (Area, Mass, Pressure, Volume)
6.3.01 Angle
6.3.3.2 Angle, radian, degree, arc minute, arc second
3.3.3.3 Line of vision
2.0.1 Right-angled triangle
6.3.03 Velocity (speed)
28.1.1 Light rays, speed of light
14.1.01 Scalars and vectors
3.13.1 Speed of light, C, Einstein was right, e = mc2
4.24 Speed of reaction (Human physiology)
26.5.0 Speed of sound, (Velocity of sound in gases, liquids, and solids)
14.1.0 Velocity
5.23 Wind speed and direction (Primary)
6.3.04 Force (weight) (Primary)
2.26 Balance bottle tops (Primary)
2.28 Beam balance (Primary)
6.11 Coins on a slope (Primary)
2.29 Drinking straw balance (Primary)
3.24 Measure your weight (Primary)
3.17 Plumb bob (vertical test) (Primary)
2.25 Ruler balance (Primary)
2.27 Nail balance (Primary)
6.10 Pull with pulleys (Primary)
2.23 See-saw balance (Primary)
3.19 Single pan balance (Primary)
2.24 Steelyard balance (Primary)
6.3.05 Area (shape)
3.4.0 Area, square metre (m2) hectare
1.44 Area game (Primary)
1.22 Compare different shapes (Primary)
1.23 Make new shapes (Primary)
1.24 Seeds and seed pods (Primary)
4.17 Shapes game (Primary)
6.6.3 Surface / volume ratio of soil particles
1.25 Water pouring game (Primary)
6.3.06 Volume
3.5.3 American liquid measures, US measures, United States weights and measures, volume (liquid)
3.5.2 British liquid measures, imperial measures (fl. oz. = imperial fluid ounce)
3.5.4 Common and miscellaneous measures, cup, spoon, matchbox
4.20 Measure chest expansion (Primary)
1.29 Measuring cylinders / graduated cylinder
3.5.1 Spoon volume
6.6.3 Surface / volume ratio of soil particles
3.5.0 Volume (vol.) cubic metre (m3)
6.16 Volume of air breathed out (Primary)
3.23 Volume of a liquid (Primary)
2.1.6 Volume of liquid (meniscus diagram)
3.18 Volume of your fist (Primary)
1.25 Water pouring game (Primary)
6.3.1.1 Length
3.3.1.0 Angstrom unit
3.3.2.0 Astronomical unit
4.18 Diameter of a thread (Primary)
36.42.2 Latitude, nautical mile, knots, log, logbook
3.3.0 Length (l), the kilometre (km), metre (metre)
1.19 Length game (Primary)
2.14 Measure your body in hand spans (Primary)
3.15 Measure your height (Primary)
2.15 Measure with your body (Primary)
6.3.1.1 Metre
1.20 Pace distances (Primary)
4.15 Pace distances (Primary)
5.22 Rain gauge (Primary)
3.21 Trundle wheel (Primary)
6.3.1.3 Time, second
6.22 Pendulum tells the time (Primary)
4.24 Speed of reaction (Primary)
3.22 Throw up and fall down (Primary)
6.3.1.5.0 Temperature, Celsius scale, Kelvin scale, Fahrenheit scale
4.11 Air temperature (Primary)
6.17 Relative humidity (Primary)
6.14.0 Oven temperatures
6.3.1.5.1 Triple point and ice point temperatures of water
3.2.1 Avoirdupois weight, English and United States 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)
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
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.
3.2.2 Carat
For precious stones, 1 carat is about 1 / 142 of an ounce. 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.
3.2.3 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.
pound ounce pennyweight, dwt grain g
1 12 240 5 760 373.24
. 1 20 480 31.10
. . 1 24 1.56
3.2.4 Apothecaries' weight, English and United States weights and measures
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.
pound ounce drachm scruple grain g
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
3.3.0 Length (l), the kilometre (km), metre (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.
Callipers, Vernier callipers, Vernier scale (Pierre Vernier 1580-1637), callipers are for measuring internal and external diameters.
Gauge, feeler gauges, micrometer screw gauges
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 to measure the length of a crooked path
1 kilometre, 1 km = 1 000 metres (originally, 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, 1 µ m = 1 × 10-6 metre, one millionth of a metre, micron
1 nanometre, 1 nm = 10-9 metre, one billionth of a metre, (10 angstroms), (formerly 1 millimicron)
1 picometre, 1 pm = 10-12 metre
3.3.1.0 One Angstrom unit, A = 10-10 metre, previously used as unit of measurement of wavelength, but nowadays use nanometre. (Note: 1 nm = nanometre = 10 Angstrom units = 10-9 m.), (Named after A. J. Angstrom, Sweden, 1814-1874), (The symbol is A with a little circle above it.), (The unit is still used in crystallography.)
3.3.2.0 One Astronomical unit, AU = the mean distance between the Earth and the sun, about 149 598 000 km (92 956 000 miles). It is used as a convenient way to measure distance in the solar system.
3.3.3.0 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)
mile furlong rod yard foot inch
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
3.3.3.1 Relative positions between measured object and equipment +
When you read on a scale with a measured object directly touching with the equipment, you must 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.
3.3.3.2 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.
3.3.3.3 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.
3.3.4.0 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
acre rood rod yard foot
1 4 160 4 840 43 560
. 1 40 1 210 10 890
. . 1 30.25 272.25
. . . 1 9
3.3.4.1 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)
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
3.3.5.0 Graphs
1. Area "under" a curve
Interpreting graphs. Linear graphs. Gradient (slope of a graph). Intercepts on a graph.
The graph of Y varies directly with X, e.g. Weight on spring.
The graph of Y varies inversely with X, e.g. PV of gas.
The graph of Y varies directly with X2, e.g. acceleration
A graph is a drawing that shows the relationship between variables.
Terminology: Axes of a graph, co-ordinates of a position on a graph, independent variable, dependent variable, line of best fit, area "under" a curve, interpreting graphs, linear graphs, gradient (slope of a graph), intercepts on a graph.
The graph of Y varies directly with X, e.g. weight on spring.
The graph of Y varies inversely with X, e.g. PV of gas.
The graph of Y varies directly with X2, e.g. acceleration.
2. Select scales
Select the scale of the axes to make the shape of the graph display the relation between data. The starting a point of the co-ordinate axis does not have to begin with zero and the scales of the two axes need not be the same. The variable you set up is the independent variable and is placed on the horizontal axis, the x axis. The variable that results from the independent variable is the dependent variable and is placed on the vertical axis, the y axis. If you investigate the cooling of a bucket of water, time is the independent
variable and temperature of the water is the dependent variable. When you say "Graph speed against time" or "Draw a velocity time graph", then "time" is the independent variable because you have mentioned it after the dependent variable "speed". You show the position of any plotted point as (XY).
3. Plot points and draw by hand
See diagram 2.0.4.1: Drawing a graph
When plotting points in the co-ordinate system by hand the symbols may be small dots surrounded by a circle or a thin cross shape. In the diagram, the computer using "brush" from the Windows XP Paint program has generated the symbols. When you have two graphs in one co-ordinate system, different symbols should express the points in different graphs. Do not graph if less than 6 points. Draw the graph by using the inner drawing method so that your wrist that is a centre to turn around in forming a smooth graph. Check the points that are far from the graph because measuring them again may be necessary. A dotted line should express a graph that you have deduced to distinguish from the graph obtained from experiment.
4. Graph the speed of two cars
See diagram 2.0.4.2: Speed of two cars
Suppose you mark a straight road every 10 metres and can use a stopwatch to record when a car reaches each mark. The following table shows your data for 2 cars, car A and car B. In the graph the points for Car A are almost in a straight line. You can say that the line of best fit is a straight line. However, the graph line does not go exactly through each point because some experimental error can occur when reading the stopwatch, recording the data and plotting the graph. However, if you assume that the graph line is properly straight then you can say that each quantity is proportional to the other, distance = speed (velocity) × time, d = vt. Car A was moving with constant speed 4.2 m / s.
Estimate how far Car A had moved after 8 seconds, by interpolation = 33 m. See the P on the graph.
Estimate how far Car A had moved after 8 seconds, by calculation, d = vt, d =4.2v × 8 t = 33.6 m.
Car A
Distance
(m)		 Elapsed time
(seconds)		 Speed
(m / s)
0 0 0
10 2.3 4.3
20 4.9 4.1
30 7.1 4.2
40 9.7 4.1
50 12.0 4.2
. . Average speed = 4.18 = 4.2
In the graph the points for Car B are not in a straight line. The line of best fit is a curve so the speed is constantly changing. The graph shows the method of calculating the instantaneous speed at two distances 15 m and 35 metres. Draw a tangent to the point on the graph corresponding to the distance. Construct a right angle triangle with the tangent as hypotenuse then read the corresponding values for distance and time from the two sides of the triangle then calculate the speed, v = d / t. At 15 m, the instantaneous speed
was s / t, 10 d / 3.2 t = 3.125 m / sec. = 3.2 m / sec. At 35 m, the instantaneous speed was s / t, 10 d / 7.8 t = 6.25 = 6.2 m / sec.
Car B
Distance
(m)		 Elapsed time
(seconds)		 Instantaneous speed
(m / sec.)
0 0 .
10 4.0 (15 m, 3.2 m / sec
20 7.0 .
30 8.9 (35 m. 6.2 m / sec)
40 10.5 .
50 12.0 .
3.3.7.0 Table of numerals adding vertically or horizontally or diagonally to 33, the traditional age of Jesus when he died.
1 14 14 4
11 7 6 9
8 10 10 5
13 2 3 15
3.3.7.1 Mobius strip
1. 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. 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.
2. 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 links in a chain.
3.4.0 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
Area of a circle = π × r2
Surface area of a sphere = 4π × r2
Volume of a sphere = 4 / 3 π × r3
3.5.0 Volume (vol.) cubic metre (m3)
See diagram 2.1.6: Liquid volume
Volume in a measuring cylinder, meniscus
Volume of a bucket, fish tin, coconut, cups, a tablespoon, a teaspoon, of cooking oil to be used 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
Volume, solid: 1 centimetre (cm) × 1 centimetre (cm) × 1 centimetre (cm) = 1 cubic centimetre (1 cc, 1 cm3) = 1 millilitre, 1 mL
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 mol. dm-1 = 1 mole per cubic decimetre = 1 mole per litre = 1 mol. L-1
3.5.1 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)
1 teaspoon, tsp, US = 5 mL (for US food labels))
1 measuring spoon for medicines and some fertilizers = 5 mL
(1 salt spoon, saltspooon = ¼ teaspoon).
3.5.2 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
3.5.3 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)
3.5.4 Common and miscellaneous measures, cup, spoon, matchbox
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 = 18 inches (English) 17.5 inches (Roman) 21 inches (Egyptian) (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 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 37oC (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 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 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
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 quart (liquid) = 0.9463 litre
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
3.5.4a Measuring cups, jugs spoons
Commercial
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
3.5.4.1 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.
3.5.6 angstrom = 0.1 nanometres, (nm), 10 Angstrom units = 10-9 m.
3.5.7 micron, µ, micrometre, µm, millimicron, nanometre
micron, µ, (non-SI unit) = 1 micrometre, µm (British English), (1 micrometer, USA), a millionth of a metre
(However, a "micrometer" is also a measuring device containing a fine pitched screw.)
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)
3.6.0 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
3.7.1 Concentration, parts per million (ppm) 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%) is about equivalent to 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 assumptions can 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
3.7.2 Rate of reaction
For many chemical reactions, but not all, increasing the concentration of 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].
3.7.3 Degrees proof, proof spirit
Proof spirit contains, in Britain 49.28% alcohol (ethanol) by weight, 57.10% by volume, relative density 0.920 at 10.6oC (formerly specific gravity of 12 / 13 at 51oF) in USA 50% by volume at 15.6oC. This standard is quoted as 100 degrees of proof, 100o. If a spirituous liquor is p% overproof (above standard strength) 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 that which if poured over gunpowder and ignited would ignite the gunpowder. If the gunpowder did not ignite, the spirit was under proof.
3.9.0 c.g.s. units (centimetre, gram, second)
Quantity c.g.s. Unit Size
Length centimetre 1 cm = 10-2 m
Mass gram 1 g = 10-3 kg
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
3.10.0 The 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).
3.11.0 Imperial units used in land surveying (1 hectare = 10,000 m2, 1 km = 1,000 m)
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
3.12.0 SI, c.g.s. and f.p.s. conversion, metric conversion
c.g.s. = centimetre, gram, second
f.p.s.. = foot, pound, second
MK or MESA = metre, kilogram, second (ampere)
Physical quantity c.g.s. unit FPS unit
length, metre (m) 1 centimetre (cm) = 0.3937 inch (in)
 1 inch (in) = 25.4 millimetre (mm)
" 1 metre = 3.2808 feet (ft)
 1 foot (ft) = 0.3048 metre (m)
"
 1 metre (m) = 1.094 yard (yd)
 1 yard (yd) = 0.9144 metre (m)
" 1 kilometre (km) = 0.6213 mile
 1 mile = 1.6093 kilometre (km)
mass, kilogram (kg) 1 gram (g) = 10-3 kilogram (kg)
1 gram (g) = 0.0353 ounce (oz)
1 kilogram (kg) = 2.205 pounds (lb)
 1 ounce (oz) = 28.35 gram (g)
1 pound (lb) = 0.4536 kilogram (kg)
" 1 tonne (t) = 1.102 US short ton, 2000 pounds (lb)
1 tonne (t) = 0.9843 US long ton, 2,205 pounds (lb)
 1 US hundredweight (cwt) = 100 pound (lb)
1 US cwt = 45.36 kilogram (kg)
1 US short ton = 0.9072 tonne (t)
1 US long ton = 1.016 tonne (t)
volume, litre (L)
1 cm3 = 10-6 m3
1 litre (L) =10-3 m3
1 millilitre (mL) = 1 cm3
 1 cubic centimetre (cc) = 0.0610 cubic inch
1 millilitre (mL) = 0.3382 fluid ounce
1 litre (L) = 0.2642 US liquid gallon
1 litre (L) = 0.02838 US bushel
 1 cubic inch = 16.38 cubic centimetre (cc)
1 fluid ounce = 29.47 millilitre (mL)
1 US liquid gallon = 3.785 litre (L)
1 US bushel = 35.24 litre (L)
1 quart (liquid) = 0.9463 litre (L)
density
 1 g cm-3 = 10-3 kg m-3
1 kilogramme / hectolitre = 0.7770 pound / US bushel
1 pound / US bushel = 1.287 kg / hL
velocity or speed 1 cm s-1 = 10-2 m s-1 .
" 100 km / hour 62.5 miles / hour
force
 dyne, 1 dyne = 10-5 newton (N) .
pressure, stress 1 dyne cm2 = 10-1 pascal (Pa) .
" 1 bar = 105 pascal (Pa) 1 bar = 750.07 mm Hg
" millibar = 100 pascal (Pa) .
energy, work (J = joule) 1 erg = 10-7 joule (J)

 .
power (W = watt) 1 erg S-1 = 10-7 watt (W) 1 horsepower (hp) = 745.7 watt (W)
viscosity
 1 poise (P) = 10-1 NM-2s .
temperature
 0oC (Celsius) = 32oF (Fahrenheit)
100oC = 212oF
 32oF (Fahrenheit) = 0oC (Celsius)
212oF = 100oC
thermal energy

 1 calorie (cal) = 4.186 joule (J) British thermal unit, 1 BTU = 1.055 × 103 J
3.13.0 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 (kWh) = 3 412 Btu = 3.6 mJ
1 calorie (cal) = 4.186 J (if International Table calorie), however, the 15oC calorie = 4.1855 J
1 horsepower (hp) = 746 watts, 0.7457 kilowatt
1 horsepower hour (hph) = 2.69 mJ
3.13.1 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 to in equations called quantum chromodynamics to has been fiendishly 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
3.13.1.1 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".
6.2.0 Different measurements, billion, trillion
Traditional counting units, a score, a dozen, common units, market units
Units and scale divisions, analogue units, digital units
Cardinal numbers, cardinal numerals, positive whole numbers, 1, 2, 3, ...
Hundred, 102, one hundred, 100
Thousand, 103, one thousand, 1000
Million, a thousand thousand, one million, 1,000,000
Billion, a million million, 1012 in UK, but a thousand million, 109, in USA, now 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
Googol, 1 googol = 1.0 × 101000 (The name of the search engine "Google" is a mispelling of "googol".)
6.3.07 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!
6.3.1 SI, The 7 base units
Length, Mass, Time, Electric current, Temperature, Amount of substance, Luminous intensity
The of measurement that form the basis of any system of measurement are the defined mechanical units of mass, length and time. Some fundamental systems also include a unit of electricity. Coordinate systems are used to define the position of a point on a plane using two co-ordinates or in space using three co-ordinates,
e.g. Cartesian coordinate system.
Quantity Dimension Name of SI unit Symbol
1. Length
 L metre m
2. Mass M kilogram kg
3. Time T second s
4. Electric current I ampere A
5. Temperature
 . kelvin K
6. Amount of substance . mole mol
7. Luminous intensity J candela cd
6.3.1.1 Length, metre
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. Length (l) the kilometre (km), metre (metre)
6.3.1.2 Mass, kilogram
See 36.110: Mass, inertial mass and gravitational mass
The gram was intended to be the mass of a cubic centimetre of pure water at 4oC. Later, it was defined as one thousandth part of a kilogram. The standard of mass is now the kilogram. A kilogram is the mass of the international prototype kilogram kept in Sevres, France, as a 90% platinum and 10% iridium cylinder at the International Bureau of Weights and Measures. A proposed alternative definition, called the Planck value is that 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)
6.3.1.3 Time, second
A second is the time equal to the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
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
6.3.1.4 Electric current, ampere
An ampere is the current which, if maintained in two parallel straight conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in a vacuum, would produce between these conductors a force equal to 2 × 10-7 newton per metre of length. Proposed alternative definition: The ampere is such that the elementary charge is exactly 1.60 217 653 × 10-19 coulombs.
(1 coulomb = 1 ampere second)
6.3.1.5.0 Temperature, Celsius scale, Kelvin scale, Fahrenheit scale
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.
1. The Fahrenheit temperature scale (Gabriel Daniel Fahrenheit 1686 - 1736) has graduations on the thermometer based on a lower fixed point of 32oF, the freezing point of water, and an upper fixed point of 212oF, the boiling point of water. So the fundamental interval is 180 Fahrenheit degrees, 180oF. The Fahrenheit scale is still used in USA.
2. The Celsius temperature scale, proposed in 1724, (Anders Celsius 1701 -1744), has graduations on the thermometer based on a lower fixed point of 0oC, the freezing point of water, and an upper fixed point of 100oC, 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".
To convert the Fahrenheit scale to the Celsius scale (F-32) / 9 = C / 5. The Celsius and Fahrenheit scales have the same value at -40oC or -40oF. Human body temperature = 37oC (Celsius) or 98.6oF (Fahrenheit)
3. 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 -273.15oC. One kelvin degree,
1 K = 1 Celsius degree, 1oC. Absolute zero = -273.15oC = 0K, not "degree Kelvin". To convert the Celsius scale to the Kelvin scale, add 273.15. For example, 0oC = 273.15 K, 100oC = 373.15 K, and 10oC = 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 × 10-23 joules per kelvin.
Equivalent temperatures in different scales
.
 Kelvin Celsius Fahrenheit
Absolute zero 0oK -273oC -459oF
Freezing point of water 273oK 0oC 32oF
Boiling point of water 373oK 100oC 212oF
6.3.1.5.1 Triple point and ice point temperatures of water
See diagram 24.3.5: Triple point of water
The 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°C (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.
6.3.1.7 Luminous intensity, candela, cp
A candela is the intensity in a given direction, of a light source that emits monochromatic radiation of frequency 540 × 1012 hertz with a radiant intensity in that direction of 1 / 683 watts per steradian. It is the unit of luminous intensity equal to 1 / 60 of the luminous intensity per square centimetre of the surface of a black body at the temperature of solidification of platinum. The previous unit was the candlepower, about 0.98 of a candela, that was defined in various ways, including the light from a standard whale oil candle. However, people liked to continue to use the term candlepower, so nowadays 1 candlepower = 1 candela. The zirconium wire in a camera flash cube ignites to release a 2 000 modern candlepower burst of light for about 30 millionths of a second.
6.3.3.0 SI derived units
Acceleration, Angle (plane angle), Density, Electric capacitance, Electric charge, Electric potential difference, Electric resistance, Energy (work), Force, Frequency, Momentum, Power, Pressure (stress), Radioactivity, Velocity (speed), Viscosity
These units are physical quantities formed from the base units. Some of these units are as follows:
Quantity Dimension Unit name Symbol Equivalent
1. Velocity (speed) L T-1 . v
 m s-1
2. Acceleration L T-2 . a
 m s-2
3. Momentum M L T-1 . Mv
 kg m s-1
4. Force
 M L T-2.
 newton.
 N kg m s-2 = J m-1
5. Pressure (stress) M L-1T-2 pascal Pa N m-2
6. Energy (work) M L2T-2 joule J N m
7. Power M L2 T-3 watt W J s-1
8. Electric charge . coulomb C A s
9. Electric potential difference .
 volt
 V
 W A-1.
10. Electric capacitance . farad F C V-1
11. Electric resistance . ohm ω V A-1
12. Frequency T-1 hertz Hz s-1
13. Radioactivity T-1 becquerel Bq s-1
14. Viscosity
 M L-1T-1
 poise
 P
 1 P = 0.1 NM-2s
15. Density M L-3 . kg m-3 .
16. Plane angle . radian rad = 180o / pi
6.3.3.01 Radioactivity, radiation units
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
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
SI dose equivalent, sieverts =10 rem
However, nowadays ionizing radiation is expressed in coulombs per kilogram.
6.3.3.1 Other derived units based on SI
Physical quantity Name of unit Symbol
Surface tension newton per metre N m-1
Electric field strength volt per metre V m-1
Magnetic field strength ampere per metre A m-1
Specific heat capacity joule per kilogram kelvin J kg-1 K-1
Concentration mole per cubic metre mol m-3
Conductivity
 mho or siemans
 Ω,-1 (s)
6.3.3.1a Conductivity: mho (ohm backwards) = 1 siemens (s) = Ω,-1 (ohms) = amps / volts
[ Ω, Greek letter omega]
6.3.3.2 Angle, radian, degree, arc minute, arc second
See diagram 6.3.3.2: Plane angle: Radian (Symbol: rad)
Angle is the measurement of the inclination of one line to another. Measured in degrees, such that 360 degrees (360o) = 1 revolution.
Angle is also measured in radians, such that 2π 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.
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.
6.3.3.5 Units used with SI units (Area, Mass, Pressure, Volume)
Physical quantity Name of unit Symbol Definition of unit
Area hectare ha 104 m2
Mass tonne t 103 kg = Mg
Pressure bar bar 105 N m-2
Volume litre l 10-3 m3 = dm3
6.3.5 SI prefixes
Decimal fractions and multiples
Symbol
 Prefix
 Decimal
 Short scale
 10n
Y
 yotta
 1 000, 000, 000, 000, 000, 000, 000, 000
 septillion
 1024
Z
 zetta
 1 000, 000, 000, 000, 000, 000, 000
 sextillion
 1021
E
 exa
 1 000, 000, 000, 000, 000, 000
 quintillion
 1018
P
 peta
 1 000, 000, 000, 000, 000
 quadrillion
 1015
T tera
 1 000, 000, 000, 000 trillion
 1012
G giga
 1 000, 000, 000 billion
 109
M mega
 1 000, 000 million
 106
k kilo
 1 000 thousand
 103
h hecto
 100 hundred
 102
da deca
 10 ten
 10
.
1
 one
 100
d deci
 0.1 tenth
 10-1
c centi
 0.01 hundredth
 10-2
m milli
 0.001 thousandth
 10-3
mu, µ micro
 0.000 001 millionth
 10-6
n nano
 0.000 000 001 billionth
 10-9
p pico
 0.000 000 000 001 trillionth
 10-12
f
 femto
 0.000 000 000 000 001 quadrillionth
 10-15
a
 atto
 0.000 000 000 000 000 001 quintillionth
 10-18
z
 zepto
 0.000 000 000 000 000 000 001 sextillionth
 10-21
y
 yocto
 0.000 000 000 000 000 000 000 001 septillionth
 10-24
Some people also want "hella" for 1027.
Myria (10,000) is no longer used.
Water is measured in kilolitres, megalitres and gigalitres, e.g. gigalitres per year, GL / Y.
A Short Scale "trillion" = a Long Scale, (now discarded), "billion", and a Short Scale "trillionth" = a Long Scale, (now discarded), "billionth".
6.3.6 SI, CGS, (c.c.s) and FPS, (f.a.s.) conversion, metric conversion
CGS, (c.c.s) = centimetre-gram-second (system)
FPS, (f.a.s.) = foot-pound-second (system) (British Weights and Measures Act, 1824)
The f.a.s.. unit of mass is the slug, i.e. the mass that will accelerate at 1 ft sec-2 when a one pound force acts on it.
MK or MESA = metre, kilogram, second, (ampere)
Physical quantity c.c.s. unit f.a.s.. unit
Length (m = metre) centimetre, 1 cm = 10-2 m
 foot, 1 ft = 0.3048 m
" . inch, 1 in = 2.54 × 10-2 m
" . mile, 1 mile = 1.61 km
Mass (kg = kilogram) gram, 1 g = 10-3 kg
 pound, 1 lb = 0.4536 kg
" . ounce, 1 oz = 2.835 × 10-2 kg, 28.35 g
" . ton, 1 ton = 1.016 × 103 kg
Volume 1 cm3 = 10-6 m3 1 ft3 = 2.832 × 10-2 m2
".
 1 litre, L =10-3 m3
.
 1 in3 = 1.639 × 10-5 m2
1 cubic inch = 16.39 cubic centimetres
" 1 millilitre, 1 mL = 1 cm3
Density 1 g cm-3 = 10-3 kg m-3 .
Velocity or speed 1 cm s-1 = 10-2 m s-1 .
" 100 km / hour 62.5 miles / hour
Force dyne, 1 dyne = 10-5 N .
Pressure, stress 1 dyne cm2 = 10-1 Pa .
" bar, 1 bar = 105 Pa 1 bar = 750.07 mm Hg
" millibar = 100 Pa .
Energy, work (J = joule) erg, 1 erg = 10-7 J .
Power (W = watt) 1 erg S-1 = 10-7 W horsepower, 1 hp = 745.7 W
Viscosity poise, 1 P = 10-1 NM-2s .
Thermal energy International table calorie = 4.186 J British thermal unit, 1 BTU = 1.055 × 103 J
6.4.0 Errors, theory of errors, addition of uncertainties
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 the 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
6.4.1 Significant figures and standard form, scientific notation
1. 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. 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.
2. 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. 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 it can 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 expressed 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
6.4.2 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 10 ½ = 3.14).
6.4.3 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, the readings along the length of the cylinder were as follows:
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 3Σ r / n √n.
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.
6.4.4 Use measuring instruments, micrometer screw gauge, vernier calipers
(The order of accuracy of an area is generally taken to be 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.
6.12.0 Weights of one matchbox full of fertilizer
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
6.13.0 Roman numerals
I = 1, V = 5, × = 10, L = 50, C = 100, D = 500, M = 1000
6.14.0 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
6.15.0 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: 5, 14, 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.
6.15.1 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
6.15.2 Integers
... -5, -4, -3, -2, -1, 0, 1,2, 3, 4. 5, ...
6.15.3 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 = 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.
35.3.01 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).