1. 3 Coursework Measurement
Breithaupt pages 219 to 239

2. 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

3. SI Base Units Physical Quantity Unit Name Symbol mass m kilogram kg
length x
metre
time t
second s
electric current I
ampere A
temperature interval ΔT
kelvin K
amount of substance n
mole
mol
luminous intensity
candela cd
‘SI’ comes from the French ‘Le Système International d'Unités’
Symbol cases are significant (e.g. t = time; T = temperature)

4. 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 ρ
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

5. Special derived units (examples) All named after scientists and/or philosophers to simplify notation
physical quantity
unit
name
symbol (s)
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
kg m2 A-1 s-3
resistance R
ohm Ω
kg m2 A-2 s-3
frequency f
hertz Hz
s-1
Note – Special derived unit symbols all begin with an upper case letter

6. Some Greek characters used in physics
name
use α
alpha
radioactivity μ mu
micro
& muons β
beta ν nu
neutrinos γ
gamma π pi
3.142…
& pi mesons
δ Δ
delta
very small & finite changes ρ
rho
density & resistivity ε
epsilon
emf of cells
σ Σ
sigma
summation Κ
kappa
K mesons τ
tau
tau lepton θ
theta
angles φ
phi
work function
λ Λ
lambda
wavelength
& lambda particle
ω Ω
omega
angular speed
& resistance

7. Larger multiples multiple prefix symbol example x 1000 kilo k km
mega M MΩ
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

8. Smaller multiples multiple prefix symbol example ÷ 10 deci d dB ÷ 100
centi c cm
÷ 1000
milli m mA ÷
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

9. Answers : There are 5000 mA in 5A There are 8000 pV in 8 nanovolts
There are 500 μm in 0.05 cm
There are g in kg
There are 4 fm in am
There are 5.0 x 107 kHz in 50 GHz
There are 3.6 x 106 ms in 1 hour
There are MΩ in 30 k Ω
There are 4.0 x 1028 pC in 40 PC
There are 60 pA in nA

10. Mathematical signs – complete:
meaning
> √
less than
mean value
much greater than
< x2 >
√<x2>
root mean square value ≥
proportional to
less than or equal to
finite change
approximately equal to ∂
extremely small change ≠ ∑
sum of ≡
equivalent to ∞

11. Mathematical signs – answers:
meaning
>
greater than √
square root
<
less than
< x >
mean value
much greater than
< x2 >
mean square value
much less than
√<x2>
root mean square value ≥
greater than or equal to α
proportional to ≤
less than or equal to ∆
finite change ≈
approximately equal to ∂
extremely small change ≠
not equal to ∑
sum of ≡
equivalent to ∞
infinity

12. Significant figures Consider the number 3250.040
It is quoted to SEVEN significant figures
is SIX s.f.
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)

13. 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
0.0456
or 4.56 x 10-2
0.05
or 5 x 10-2
10-2
900
or 2.00 x 10-3
0.002
or 2 x 10-3
10-3

14. Answers: raw number to 3 s.f. to 1 s.f. to 0 s.f. 5672 5.67 x 103
104
18649
1.86 x 104
2 x 104
0.0456
or 4.56 x 10-2
0.05
or 5 x 10-2
10-2
900
9 x 102
103
or 2.00 x 10-3
0.002
or 2 x 10-3
10-3

15. Results tables Headings should be clear
Physical quantities should have units
All measurements should be recorded (not just the ‘average’)

16. 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

17. Range and mean value of measurements
This equal to the difference between the highest and lowest reading
Readings: 45g; 44g; 44g; 47g; 46g; 45g
Range: = 47g – 44g
= 3g
Mean value < x >
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 <m> = ( ) / 6
<m> = 45.2 g

18. 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.

19. 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

20. 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

21. Uncertainty in a single reading OR 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

22. 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% g
= 4.44 %

23. 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
√(25 ± 5 m2) = √(25 ± 20%) = √(25 ± (0.5 x 20%)) = (5 ± 10%) = 5 ± 0.5 m

24. 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

25. Direct proportion
The graph below shows how the extension of a wire, ∆L varies with the tension, T applied to the wire.
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

26. Linear relationships - 1
The graph below shows how the velocity of a body changes when it undergoes constant acceleration, a from an initial velocity u.
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

27. 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 = ε – r I
This has the form: y = mx + c
In this case:
m = gradient = - r (cell resistance)
c = y-intercept = ε (emf)

28. Linear relationships - 3
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.
The equation relating these quantities is: EKmax= hf – φ
This has the form: y = mx + c
In this case:
m = gradient = h (Planck constant)
c = y-intercept = – φ (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 = φ / h

29. 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 =
c = y-intercept = - 2 6 8 16 P Q 12 10

30. Answers
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?
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?
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 = ; c = + 6
m = ; x-intercept = + 3; (c = )

31. Analogue Micrometer
The micrometer is reading 4.06 ± 0.01 mm

32. Analogue Vernier Callipers
The callipers reading is 3.95 ± 0.01 cm
NTNU Vernier Applet

33. Further Reading
Breithaupt chapter 14.3; pages 221 & 222

34. 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 10E+23 - 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. y=bx ) to see how they add to generate the polynomial curve.

35. Core Notes from Breithaupt pages 219 to 239

36. Notes from Breithaupt pages 232 & 236
Copy table 1 on page 232
What is the difference between a base unit and a derived unit? Give five examples of derived units.
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
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?
Copy and learn table 2 on page 236
Try the summary questions on pages 233 & 237

37. Notes from Breithaupt pages 219 to 220, 223 to 225 & 233
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
What determines the precision in (a) a single reading and (b) multiple readings?
Define percentage uncertainty.
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) √Q.
Measure the area of a piece of A4 paper and state the probable error (or uncertainty) in your answer.
State the number to (a) 6 sf, (b) 3 sf and (c) 0 sf.

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