3 Coursework Measurement

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3 Coursework Measurement

• Breithaupt pages 219 to 239

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AQA AS Specification

• Candidates will be able to
• choose measuring instruments according to their
  sensitivity and precision
• identify the dependent and independent variables
  in an investigation and the control variables
• use appropriate apparatus and methods to make
  accurate and reliable measurements
• tabulate and process measurement data
• use equations and carry out appropriate
  calculations
• plot and use appropriate graphs to establish or
  verify relationships between variables
• relate the gradient and the intercepts of
  straight line graphs to appropriate linear
  equations.
• distinguish between systematic and random errors
• make reasonable estimates of the errors in all
  measurements
• use data, graphs and other evidence from
  experiments to draw conclusions
• use the most significant error estimates to
  assess the reliability of conclusions drawn

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SI Base Units
Physical Quantity Physical Quantity Unit Unit
Name Symbol Name Symbol
mass m kilogram kg
length x metre m
time t second s
electric current I ampere A
temperature interval ?T kelvin K
amount of substance n mole mol
luminous intensity I candela cd
SI comes from the French Le Système
International d'Unités Symbol cases are
significant (e.g. t time T temperature)
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Derived units (examples)Consist of one or more
base units multiplied or divided together
quantity symbol unit
area A m2
volume V m3
density D or ? kg m-3
velocity u or v m s-1
momentum p kg m s-1
acceleration a m s-2
force F kg m s-2
work W kg m2 s-2
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Special derived units (examples)All named after
scientists and/or philosophers to simplify
notation
physical quantity physical quantity unit unit unit
name symbol (s) name symbol base SI form
force F newton N kg m s-2
work energy W E joule J kg m2 s-2
power P watt W kg m2 s-3
pressure p pascal Pa kg m-1 s-2
electric charge q or Q coulomb C A s
p.d. (voltage) V volt V kg m2 A-1 s-3
resistance R ohm O kg m2 A-2 s-3
frequency f hertz Hz s-1
Note Special derived unit symbols all begin
with an upper case letter
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Some Greek characters used in physics
character name use character name use
a alpha radioactivity µ mu micro muons
ß beta radioactivity ? nu neutrinos
? gamma radioactivity p pi 3.142 pi mesons
d ? delta very small finite changes ? rho density resistivity
e epsilon emf of cells s S sigma summation
? kappa K mesons t tau tau lepton
? theta angles f phi work function
? ? lambda wavelength lambda particle ? O omega angular speed resistance
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Larger multiples
multiple prefix symbol example
x 1000 kilo k km
x 1000 000 mega M MO
x 109 giga G GW
x 1012 tera T THz
x 1015 peta P Ps
x 1018 exa E Em
also, but rarely used deca x 10, hecto x 100
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Smaller multiples
multiple prefix symbol example
10 deci d dB
100 centi c cm
1000 milli m mA
1000 000 micro µ µV
x 10-9 nano n nC
x 10-12 pico p pF
x 10-15 femto f fm
x 10-18 atto a as
Powers of 10 presentation
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Answers

1. There are 5000 mA in 5A
2. There are 8000 pV in 8 nanovolts
3. There are 500 µm in 0.05 cm
4. There are 6 000 000 g in 6 000 kg
5. There are 4 fm in 4 000 am
6. There are 5.0 x 107 kHz in 50 GHz
7. There are 3.6 x 106 ms in 1 hour
8. There are 0.030 MO in 30 k O
9. There are 4.0 x 1028 pC in 40 PC
10. There are 60 pA in 0.060 nA

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Mathematical signs complete
sign meaning sign meaning
gt v
less than mean value
much greater than lt x2 gt
vltx2gt root mean square value
proportional to
less than or equal to finite change
approximately equal to ? extremely small change
? ? sum of
equivalent to 8
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Mathematical signs answers
sign meaning sign meaning
gt greater than v square root
lt less than lt x gt mean value
much greater than lt x2 gt mean square value
much less than vltx2gt root mean square value
greater than or equal to a proportional to
less than or equal to ? finite change
approximately equal to ? extremely small change
? not equal to ? sum of
equivalent to 8 infinity
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Significant figures

• Consider the number 3250.040
• It is quoted to SEVEN significant figures
• 3250.04 is SIX s.f.
• 3250.0 is FIVE s.f.
• 3250 is FOUR s.f. (NOT THREE!)
• 325 x 101 is THREE s.f. (as also is 3.25 x 103)
• 33 x 102 is TWO s.f. (as also is 3.3 x 103)
• 3 x 103 is ONE s.f. (3000 is FOUR s.f.)
• 103 is ZERO s.f. (Only the order of magnitude)

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Complete the table below
raw number to 3 s.f. to 1 s.f. to 0 s.f.
5672 5.67 x 103 104
18649 2 x 104
0.045632 0.0456 or 4.56 x 10-2 0.05 or 5 x 10-2 10-2
900
0.00200308 0.00200 or 2.00 x 10-3 0.002 or 2 x 10-3 10-3
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Answers
raw number to 3 s.f. to 1 s.f. to 0 s.f.
5672 5.67 x 103 6 x 103 104
18649 1.86 x 104 2 x 104 104
0.045632 0.0456 or 4.56 x 10-2 0.05 or 5 x 10-2 10-2
900 900 9 x 102 103
0.00200308 0.00200 or 2.00 x 10-3 0.002 or 2 x 10-3 10-3
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Results tables
Headings should be clear Physical quantities
should have units All measurements should be
recorded (not just the average)
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Reliability and validity of measurements

• Reliable
• Measurements are reliable if consistent values
  are obtained each time the same measurement is
  repeated.
• Reliable 45g 44g 44g 47g 46g
• Unreliable 45g 44g 67g 47g 12g 45g

• Valid
• Measurements are valid if they are of the
  required data OR can be used to obtain a required
  result
• For an experiment to measure the resistance of a
  lamp
• Valid current through lamp 5A p.d. across
  lamp 10V
• Invalid temperature of lamp 40oC colour of
  lamp red

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Range and mean value of measurements

• Range
• This equal to the difference between the highest
  and lowest reading
• Readings 45g 44g 44g 47g 46g 45g
• Range 47g 44g
• 3g

• Mean value lt x gt
• This is calculated by adding the readings
  together and dividing by the number of readings
• Readings 45g 44g 44g 47g 46g 45g
• Mean value of mass ltmgt (454444474645) / 6
• ltmgt 45.2 g

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Systematic and random errors

• Suppose a measurement should be 567cm
• Example of measurements showing systematic error
  585cm 583cm 584cm 586cm
• Systematic errors are often caused by poor
  measurement technique or incorrectly calibrated
  instruments.
• Calculating a mean value will not eliminate
  systematic error.

• Zero error can occur when an instrument does not
  read zero when it should do so. If not corrected
  for, zero error will cause systematic error. The
  measurement examples opposite may have been
  caused by a zero error of about 18 cm.
• Example of measurements showing random error
  only 566cm 568cm 564cm 567cm
• Random error is unavoidable but can be
  minimalised by using a consistent measurement
  technique and the best possible measuring
  instruments.
• Calculating a mean value will reduce the effect
  of random error.

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Accuracy and precision of measurements

• Accurate
• Accurate measurements are obtained using a good
  technique with correctly calibrated instruments
  so that there is no systematic error.
• Precise
• Precise measurements are those that have the
  maximum possible significant figures. They are as
  exact as possible.

• The precision of a measuring instrument is equal
  to the smallest possible non-zero reading it can
  yield.
• The precision of a measurement obtained from a
  range of readings is equal to half the range.
• Example If a measurement should be 3452g
• Then 3400g is accurate but not precise
• whereas 4563g is precise but inaccurate

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Uncertainty or probable error

• The uncertainty (or probable error) in the mean
  value of a measurement is half the range
  expressed as a value
• Example If mean mass is 45.2g and the range is
  3g then
• The probable error (uncertainty) is 1.5g
• Uncertainty is normally quoted to ONE significant
  figure (rounding up) and so the uncertainty is
  now 2g
• The mass might now be quoted as 45.2 2g
• As the mass can vary between potentially 43g and
  47g it would be better to quote the mass to only
  two significant figures
• So mass 45 2g is the best final statement
• NOTE The uncertainty will determine the number
  of significant figures to quote for a measurement

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Uncertainty in a single readingOR when
measurements do not vary

• The probable error is equal to the precision in
  reading the instrument
• For the scale opposite this would be
• 0.1 without the magnifying glass
• 0.02 perhaps with the magnifying glass

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Percentage uncertainty

• It is often useful to express the probable error
  as a percentage
• percentage uncertainty probable error x 100
  
  measurement
• Example Calculate the uncertainty the mass
  measurement 45 2g
• percentage uncertainty 2g x 100
  
  45g
• 4.44

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Combining uncertainties

• Addition or subtraction
• Add probable errors together, examples
• (56 4m) (22 2m) 78 6m
• (76 3kg) - (32 2kg) 44 5kg
• Multiplication or division
• Add percentage uncertainties together, examples
• (50 5m) x (20 1m) (50 10) x (20 5)
  1000 15 1000 150 m2
• (40 2m) (2.0 0.2s) (40 5) (2.0
  10) 20 15 20 1.5 ms-1
• Powers
• Multiply the percentage uncertainty by the power,
  examples
• (20 1m)2 (20 5)2 (202 (2 x 5))
  (400 10) 400 40 m2
• v(25 5 m2) v(25 20) v(25 (0.5 x
  20)) (5 10) 5 0.5 m

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The equation of a straight line graph

• For any straight line
• y mx c
• where
• m gradient
• (yP yR) / (xR xQ)
• and
• c y-intercept

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Direct proportion

• Physical quantities are directly proportional to
  each other if when one of them is multiplied by a
  certain factor the other changes by the same
  amount.
• For example if the extension, ?L in a wire is
  doubled so is the tension, T
• A graph of two quantities that are proportional
  to each will be
• a straight line
• AND passes through the origin
• The general equation of the straight line in this
  case is y mx, with, c 0

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Linear relationships - 1

• Physical quantities are linearly related to each
  other if when one of them is plotted on a graph
  against the other, the graph is a straight line.
• In the case opposite, the velocity, v of the body
  is linearly related to time, t. The velocity is
  NOT proportional to the time as the graph line
  does not pass through the origin.
• The quantities are related by the equation v
  u at. When rearranged this becomes v at u.
• This has form y mx c
• In this case m gradient a
• c y-intercept u

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Linear relationships - 2

• The potential difference, V of a power supply is
  linearly related to the current, I drawn from the
  supply.
• The equation relating these quantities is V e
  r I
• This has the form y mx c
• In this case
• m gradient - r (cell resistance)
• c y-intercept e (emf)

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Linear relationships - 3

• The equation relating these quantities is EKmax
  hf f
• This has the form y mx c
• In this case
• m gradient h (Planck constant)
• c y-intercept f (work function)
• The x-intercept occurs when y 0
• At this point, y mx c becomes
• 0 mx c
• x x-intercept - c / m
• In the above case, the x-intercept, when EKmax
  0
• is f / h

The maximum kinetic energy, EKmax, of electrons
emitted from a metal by photoelectric emission is
linearly related to the frequency, f of incoming
electromagnetic radiation.
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Calculating the y-intercept

• The graph opposite shows two quantities that are
  linearly related but it does not show the
  y-intercept.
• To calculate this intercept
• 1. Measure the gradient, m
• In this case, m 1.5
• 2. Choose an x-y co-ordinate from any point on
  the straight line. e.g. (12, 16)
• 3. Substitute these into y mx c, with (P y
  and Q x)
• In this case 16 (1.5 x 12) c
• 16 18 c
• c 16 - 18
• c y-intercept - 2

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Answers

1. Quantity P is related to quantity Q by the
   equation P 5Q 7. If a graph of P
   against Q was plotted what would be the gradient
   and y-intercept?
2. Quantity J is related to quantity K by the
   equation J - 6 K/3. If a graph of J
   against K was plotted what would be the gradient
   and y-intercept?
3. Quantity W is related to quantity V by the
   equation V 4W 3. If a graph of W against
   V was plotted what would be the gradient and
   x-intercept?

m 5 c 7
m 0.33 c 6
m - 0.25 x-intercept 3 (c 0.75)
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Analogue Micrometer

• The micrometer is reading 4.06 0.01 mm

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Analogue Vernier Callipers

• The callipers reading is 3.95 0.01 cm
• NTNU Vernier Applet

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Further Reading

• Breithaupt chapter 14.3 pages 221 222

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Internet Links

• Unit Conversion - meant for KS3 - Fendt
• Hidden Pairs Game on Units - by KT - Microsoft
  WORD
• Fifty-Fifty Game on Converting Milli, Kilo Mega
  - by KT - Microsoft WORD
• Hidden Pairs Game on Milli, Kilo Mega - by KT -
  Microsoft WORD
• Hidden Pairs Game on Prefixes - by KT - Microsoft
  WORD
• Sequential Puzzle on Energy Size - by KT  -
  Microsoft WORD
• Sequential Puzzle on Milli, Kilo Mega order -
  by KT  - Microsoft WORD
• Powers of 10 - Goes from 10E-16 to 10E23 -
  Science Optics You
• A Sense of Scale - falstad
• Use of vernier callipers - NTNU
• Equation Grapher - PhET - Learn about graphing
  polynomials. The shape of the curve changes as
  the constants are adjusted. View the curves for
  the individual terms (e.g. ybx ) to see how they
  add to generate the polynomial curve.

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Core Notes from Breithaupt pages 219 to 239
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Notes from Breithaupt pages 232 236

1. Copy table 1 on page 232
2. What is the difference between a base unit and a
   derived unit? Give five examples of derived
   units.
3. Convert (a) 52 kg into g (b) 4 m2 into cm2 (c)
   6 m3 into mm3 (d) 3 kg m-3 into g cm-3
4. How many (a) mg in 1 Mg (b) Gm in 1 TM (c) µs
   in 1 ks (d) fV in 1 nV am in 1 pm?
5. Copy and learn table 2 on page 236
6. Try the summary questions on pages 233 237

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Notes from Breithaupt pages 219 to 220, 223 to
225 233

1. Define in the context of recording measurements,
   and give examples of, what is meant by (a)
   reliable (b) valid (c) range (d) mean
   value (e) systematic error (f) random error
   (g) zero error (h) uncertainty (i) accuracy
   (j) precision and (k) linearity
2. What determines the precision in (a) a single
   reading and (b) multiple readings?
3. Define percentage uncertainty.
4. Two measurements P 2.0 0.1 and Q 4.0 0.4
   are obtained. Determine the uncertainty (probable
   error) in (a) P Q (b) Q P (c) P x Q
   (d) Q / P (e) P3 (f) vQ.
5. Measure the area of a piece of A4 paper and state
   the probable error (or uncertainty) in your
   answer.
6. State the number 1230.0456 to (a) 6 sf, (b) 3 sf
   and (c) 0 sf.

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Notes from Breithaupt pages 238 239

1. Copy figure 2 on page 238 and define the terms of
   the equation of a straight line graph.
2. Copy figure 1 on page 238 and explain how it
   shows the direct proportionality relationship
   between the two quantities.
3. Draw figures 3, 4 5 and explain how these
   graphs relate to the equation y mx c.
4. How can straight line graphs be used to solve
   simultaneous equations?
5. Try the summary questions on page 239