2018-10-29

Please send comments to: J.Elfick@uq.edu.au

6a Measurement

Table of contents

See: Measurement, (Commercial)

6.3.3.2 Angle, radian, degree, arc minute, arc second

35.3.01 Assay value of precious metals

6.15.1 Divisibility, Tests for divisibility

6.15.4 Ellipse

6.4.0 Errors, theory of errors, addition of uncertainties

6.15.3 Fractions

6.15.2 Integers

6.3.1.7 Luminous intensity, candela, cp

6.12.0 Matchbox, Weights of one matchbox full of fertilizer

6.3.3.7 Measure temperature and specific heat capacity

6.4.4 Measuring instruments, micrometer screw gauge, vernier calipers

6.3.3.02 Newton, The newton, symbol N

6.4.3 Order of accuracy

6.4.2 Order of magnitude (nearest power of ten, a factor or factors of ten)

6.14.0 Oven temperatures

6.15.0 Perfect numbers

6.3.3.4 Pi, π

6.15.01 Prime numbers

6.3.3.3 Radians

6.3.3.01 Radioactivity, radiation units, curie

6.13.0 Roman numerals

6.3.1.0 SI, The 7 base units

6.3.3.0 SI derived units

6.3.3.1 SI, Other derived units based on SI

6.3.3.5 SI, Units used with SI units (Area, Mass, Pressure, Volume)

6.4.1 Significant figures and standard form, scientific notation

6.3.3.6 Spherometer

6.4.1a Standard form, scientific notation

6.3.1.5.1 Triple point and ice point temperatures of water

6.3.1.5.1 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. 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

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

The combination 611 Pa, and 0.01

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

6.3.3.01 Radioactivity, radiation units, curie

See: Nuclear physics, (Commercial)

31.2.0 Electric charge, the coulomb, C, Coulomb's law

SI unit of activity, becquerel (Bq) = one nuclear disintegration per second, 1 s

Former unit, curie, Ci = 3.7 × 10

Simulate radioactive decay with dice

Each group starts with 80 dice and removes any sixes each throw.

Five throws is usully 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.

SI unit of absorbed dose of ionizing, gray, Gy = 1 joule of energy per kilogram of irradiated material.

Former unit, rad, rd = 10

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

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,

dose, rad × Q

SI dose equivalent, sieverts =10 rem

However, nowadays ionizing radiation is expressed in coulombs per kilogram.

6.3.3.2 Angle, radian, degree, arc minute, arc second

See diagram 6.3.3.2: Plane angle: (Symbol: rad)

Angle is the measurement of the inclination of one line to another.

Degrees,

Degree is the unit of angle.

One revolution = 360 degrees, 360

One right angle = 90 degrees, 90

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.

Second: The "second" refers to the second division of time into sixtieths after dividing the hour into minutes.

6.3.3.3 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.

6.3.3.4 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!

6.4.0 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 × 10

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.) 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.

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 ftigures.

Different rules may be taught in maths and physics classe and there may be a difference for 'rounding' rules.

6.4.1a 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 × 10

or 1.80 × 10

Express decimal fractions in standard form: 0.1 = 1 × 10

Standard form (scientific notation), e.g. 8.04 × 10

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 × 10

Express decimal fractions in standard form, e.g.

0.1 = 1 × 10

0.2 = 1 × 10

0.019 = 1.9 × 10

0.00 087 = 8.7 × 10

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 × 10

the radius of the Earth: 6.38 × 10

the radius of the Moon = 3.8 × 10

(3.8 is closer to 10

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:

Table 6.4.3 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 |

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.

6.4.4 Measuring instruments, micrometer screw gauge, vernier calipers

See: Micrometers, (Commercial)

See: Vernier Calipers, (Commercial)

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.

6.12.0 Weights 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.

6.13.0 Roman numerals

I = 1, V = 5, × = 10, L = 50, C = 100, D = 500, M = 1000.

6.15.0 Perfect numbers

See: Mathematics (Commercial)

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 = 2

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.

6.15.01 Prime numbers

See: Mathematics (Commercial)

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

6.15.1 Tests for divisibility

See: Mathematics (Commercial)

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, . . .

See: Mathematics (Commercial)

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

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.

6.15.4 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.

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).

6.14.0 Oven temperatures

See: Ovens, (Commercial)

Table 6.14.0 Oven temperatures

^{o}C |
^{o}F |
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., gro. = gross

k., 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

42

6.3.1.0 SI, The 7 base units

1. Length, 2. Mass, 3. Time, 4. Electric current, 5. Temperature, 6. Amount of substance, 7. 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.

Table 6.3.1 The 7 base units

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. thermodynamic temperature |
. | kelvin | K |

6. amount of substance | . | mole | mol |

7. luminous intensity | J | candela | cd |

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:

Table 6.3.3.0 SI derived units

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^{-1}T^{-2} |
pascal | Pa | N m^{-2} |

6. Energy (work) | M L^{2}T^{-2} |
joule | J | N m |

7. Power | M L^{2} 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^{-1}T^{-1
} |
poise |
P |
1 P = 0.1 NM^{-2}s |

15. Density | M L^{-3} |
. | kg m^{-3} |
. |

16. Plane angle | . | radian | rad | = 180^{o} / pi |

6.3.3.02 The newton, symbol N

A teaching toy called "Newton's Apple", weighs 102 g, so approximately 1 newton force

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 / s

So if 10

Table 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 |
^{-1} (s) |

Table 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 | 10^{4} m^{2} |

Mass | tonne | t | 10^{3} kg = Mg |

Pressure | bar | bar | 10^{5} N m^{-2} |

Volume | litre | l | 10^{-3} m^{3} = dm^{3} |

6.3.3.6 Spherometer

See: Spherometers, (Commercial)

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.

6.3.3.7 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.