1.1  Fundamental (base) quantities and their units

 

Fundamental Quantity

SI Unit

Mass

Kilogram, kg

Length

metre, m

Time

second, s

Quantity of matter

mole, mol

Temperature

Kelvin, K

Electric current

Ampere, A

 

NB there is a 7th one for light intensity (the Candela) but its

 not on the AQA spec.

 

 

1.2 Prefixes

 

PREFIX

SYMBOL

MEANING

ORDER OF MAGNITUDE

Tera

T

1 000 000 000 000

1012

Giga

G

1 000 000 000

109

Mega

M

1 000 000

106

kilo

k

1 000

103

centi

c

1 /100

10-2

milli

m

1 /1 000

10-3

micro

μ

1 /1 000 000

10-6

nano

n

1 /1 000 000 000

10-9

pico

p

1 /1 000 000 000 000

10-12

femto

f

1 /1 000 000 000 000 000

10-15

 

 

1.3 Errors

 

·         measurement error

The difference between a measured value and the true value.

·         anomaly

This is a freak result – judged to not fit the main trend. It should be rejected and the reading repeated.

·         random error

These cause readings to be either above or below the true value.

Random errors are present when any measurement is made, and cannot be corrected.

The effect of random errors can be reduced by making more measurements and calculating a new mean.

·         systematic error

These cause readings to differ from the true value by a consistent amount each time a measurement is made.

Macintosh HD:Users:tonymead:Desktop:41wTXNB11xL._SY300_.jpgSources of systematic error can include the environment,

methods of observation or instruments used (typically a zero error)

 

 

Systematic errors cannot be dealt with by simple repeats. If a systematic error is suspected, the data collection should be repeated

 using a different technique or a different set of equipment, and the results compared.

·         accuracy

The closer you are to the true value the more accurate the measurement:

Macintosh HD:Users:tonymead:Desktop:AAAUATB0.JPG

·         precision

Precise measurements are ones in which there is very little spread about the mean value ie. not much random error –

  it gives no indication of how close results are to the true value.

A precise scale will have a high resolution:

Macintosh HD:Users:tonymead:Desktop:intervals.gif

The resolution of the instrument is the smallest change in the quantity that can be measured. For example the resolution of a stop clock is 0.01s,

    and the resolution of a digital voltmeter is 0.01V.

 

A sensitive instrument is one that responds to a small change of input with a large change in output.

·         Repeatability a measurement is repeatable if the same person repeats using the same method and apparatus

·         Reproducibility  a measurement is reproducible if another person does the experiment and gets the same results

      (or the same person uses a different method/apparatus).

1.4  Uncertainty

If measurements are repeated, the uncertainty is calculated by finding half the range of the measured values.

 

For example these measurements taken from a vernier caliper:

 

 

Range = max.reading – min.reading  = 10.32 – 10.22 = 0.10 , so uncertainty = 0.10 / 2 = 0.05

 

So we would quote: Mean distance: (10.26 ± 0.05) mm

 

If instead the measurements had all been the same:

 

Repeat

1

2

3

4

Distance/mm

10.26

10.26

10.26

10.26

 

 

Then to get the uncertainty we examine the resolution of the instrument, for Skinners’ vernier calipers this would be 0.02mm.

       So we would quote: Mean distance: (10.26 ± 0.02) mm

 

      NB. Comparing the two methods it is quite typical for the resolution of the instrument to produce a small percentage error.

 

1.5 Combining Errors

 

You must know the rules:

 

For sum and difference simply add the absolute errors

 

Eg.  X = 20 ± 1,   Y = 10 ± 2

 

sum X + Y = 30 ± 3,   difference  X  -  Y = 10 ± 3

 

For product or quotient you must add the percentage errors NOT the absolute errors:

 

Eg.  % error in X = 1/20 = 5%,      % error in Y = 2/10 = 20%

 

So % error in Product X.Y = 25%, % error in X / Y = 25% also,

 

So,  X.Y = 200 ± 25%  =  200 ± 50,    X/Y  =  2 ± 25%  =  2 ± 0.5

 

For powers, multiply the percentage uncertainty by the power:

 

Eg.  X3 = 8000,  so % error in X3  =  3 x % error in X  =  15%

 

So,  X3  =  8000 ± 1200


OR, X3/2 = 89,  so % error in X3/2  =  3/2 X 5%  =  7.5%

 

so, X3/2 = 89 ± 7  (approx.)

 

 

1.6 Error bars on graphs

 

·         Having correctly drawn your graph including a best fit line, you

       will need to insert error bars:

 

·         Next you need to draw a second graph line – a best fit line of maximum gradient as shown above (in

    the example above a 3rd line of minimum gradient has been drawn too - strictly this one is not needed.

·         Next you calculate the gradient of the best fit line AND also the gradient of the steeper line.

·         Now the percentage uncertainties in both the gradient and the y-intercept can be found:

·         ×

 

·        

 

1.7 Estimating

You will be expected to make reasonable estimations / approximations of some quantities.

 

Egs.

·         Mass of an apple ~ 100g  (from a weight of 1N)

·         Mass of a car 1 000 kg (from 1 tonne)

·         Mass of a person say, 75 kg

·         Volume of air in a room. To do this one, estimate dimensions of room in m, then multiply them together.

·         The above case could be extended to mass of air in room if you estimate the density of air to be 1 kg/m3

·         Density of water is about 1 000 kg/m3

·         One pace is about a metre

·         Electricity: air breakdown is about 1 million volts/m. So about 1 million volts would make a spark jump across a distance of 1m

·         If an apple falls a distance of 1m, then 1 J of work is done

·         Speed of light is about a million X more than the speed of sound.

·         Speed of air molecules is about same as the speed of sound (330 m/s)

·         Atmospheric pressure is about 100,000 Pa