3 Coursework Measurement

Breithaupt pages 219 to 239

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

SI Base UnitsPhysical Quantity Unit

Name Symbol Name Symbol

mass m kilogram kg

length x metre m

time t second s

electric current I ampere A

temperature interval ΔT 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)

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

Special derived units (examples)All named after scientists and/or philosophers to simplify notation

physical quantity 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 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

Some Greek characters used in physics

character name use character name use

α alpha radioactivity μ mu micro

& muons

β beta radioactivity ν nu neutrinos

γ gamma radioactivity 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

Larger multiplesmultiple prefix symbol example

x 1000 kilo k km

x 1000 000 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

Smaller multiplesmultiple 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

Answers :1. There are 5000 mA in 5A2. There are 8000 pV in 8 nanovolts3. There are 500 μm in 0.05 cm4. There are 6 000 000 g in 6 000 kg5. There are 4 fm in 4 000 am6. There are 5.0 x 107 kHz in 50 GHz7. There are 3.6 x 106 ms in 1 hour8. There are 0.030 MΩ in 30 k Ω9. There are 4.0 x 1028 pC in 40 PC10. There are 60 pA in 0.060 nA

Mathematical signs – complete:sign meaning sign 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 ∞

Mathematical signs – answers:sign meaning sign 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

Significant figuresConsider 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)

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

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

Results tables Headings should be clear

Physical quantities should have units

All measurements should be recorded (not just the ‘average’)

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; 46gUnreliable: 45g; 44g; 67g; 47g; 12g; 45g

ValidMeasurements 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 = 10VInvalid: temperature of lamp = 40oC; colour of lamp = red

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 < 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> = (45+44+44+47+46+45) / 6

<m> = 45.2 g

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.

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.

PrecisePrecise 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 3452gThen 3400g is accurate but not precisewhereas 4563g is precise but inaccurate

Uncertainty or probable errorThe uncertainty (or probable error) in the mean value of a measurement is half the range expressed as a ± valueExample: 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 ± 2gThe mass might now be quoted as 45.2 ± 2gAs the mass can vary between potentially 43g and 47g it would be better to quote the mass to only two significant figuresSo mass = 45 ± 2g is the best final statement

NOTE: The uncertainty will determine the number of significant figures to quote for a measurement

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

Percentage uncertainty

It is often useful to express the probable error as a percentagepercentage uncertainty = probable error x 100% measurement

Example: Calculate the % uncertainty the mass measurement 45 ± 2g

percentage uncertainty = 2g x 100% 45g = 4.44 %

Combining uncertaintiesAddition 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

The equation of a straight line graph

For any straight line:y = mx + c

where:m = gradient

= (yP – yR) / (xR – xQ)

andc = y-intercept

Direct proportionPhysical 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

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

Linear relationships - 1Physical 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 + cIn this case m = gradient = ac = y-intercept = u

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

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)

Linear relationships - 3The equation relating these quantities is: EKmax= hf – φThis has the form: y = mx + cIn this case:m = gradient = h (Planck constant)c = y-intercept = – φ (work function)

The x-intercept occurs when y = 0At this point, y = mx + c becomes:0 = mx + cx = x-intercept = - c / mIn the above case, the x-intercept, when EKmax = 0is = φ / 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.

Calculating the y-interceptThe 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, mIn this case, m = 1.52. 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) + c16 = 18 + cc = 16 - 18

c = y-intercept = - 2

6

8

16

P

Q12

10

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)

Analogue Micrometer

The micrometer is reading 4.06 ± 0.01 mm

Analogue Vernier Callipers

The callipers reading is 3.95 ± 0.01 cmNTNU Vernier Applet

Further Reading

Breithaupt chapter 14.3; pages 221 & 222

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.

Core Notes from Breithaupt pages 219 to 239

Notes from Breithaupt pages 232 & 2361. Copy table 1 on page 2322. 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 2366. Try the summary questions on pages 233 &

237

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) √Q.

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.

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