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3 Coursework Measurement

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3 CourseworkMeasurement

Breithaupt pages 219 to 239

- 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’ comes from the French ‘Le Système International d'Unités’

Symbol cases are significant (e.g. t = time; T = temperature)

Note – Special derived unit symbols all begin with an upper case letter

also, but rarely used: deca = x 10, hecto = x 100

Powers of 10 presentation

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

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)

Headings should be clear

Physical quantities should have units

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

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

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

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.

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

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

- 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

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 %

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

For any straight line:

y = mx + c

where:

m = gradient

= (yP – yR) / (xR – xQ)

and

c = y-intercept

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

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

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)

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

P

16

10

6

12

Q

8

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

- 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 = + 0.33; c = + 6

m = - 0.25; x-intercept = + 3; (c = + 0.75)

The micrometer is reading 4.06 ± 0.01 mm

The callipers reading is 3.95 ± 0.01 cm

NTNU Vernier Applet

Breithaupt chapter 14.3; pages 221 & 222

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.

- 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

- 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 1230.0456 to (a) 6 sf, (b) 3 sf and (c) 0 sf.

- 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