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 7^{th} 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 |
10^{12} |
Giga |
G |
1 000 000 000 |
10^{9} |
Mega |
M |
1 000 000 |
10^{6} |
kilo |
k |
1 000 |
10^{3} |
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
The difference between a measured value and the true value.
This is a freak result – judged to not fit the main trend. It should be rejected
and the reading repeated.
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.
These cause readings to differ from the true value by a consistent amount each
time a measurement is made.
Sources 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.
The closer you are to the true value the more accurate the measurement:
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:
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. X^{3} = 8000,
so % error in X^{3}
= 3 x % error in X
= 15%
So, X^{3}
= 8000 ± 1200
OR,
X^{3}^{/2}
= 89, so % error in
X^{3}^{/2}
= 3/2 X 5% = 7.5%
so, X^{3}^{/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.
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/m^{3}
·
Density of water is about 1 000 kg/m^{3}
·
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