Physics and Physical
Measurement
Topic 1.2 Measurement and
Uncertainties
The S.I. System
Standards of Measurement
 SI units are those of the Système
International d’Unités adopted in 1960
 Used for general measurement in most
countries worldwide
Fundamental Quantities
 Some quantities cannot be measured in a
simpler form and for convenience they
have been selected as the basic quanitities
 They are termed Fundamental Quantities,
Units and Symbols
The 7 Fundamentals
Length
 Mass
 Time
 Electric current
 Thermodynamic temp
 Luminous Intensity
 Amount of a substance

metre
kilogram
second
ampere
Kelvin
candela
mole
m
kg
s
A
K
cd
mol
Derived Quantities
 When a quantity involves the measurement
of 2 or more fundamental quantities it is
called a Derived Quantity
 The units of these are called Derived Units
Derived Units
Examples…
 Acceleration
 Momentum
ms-2
kgms-1 or Ns
Some derived units have been given their
own specific names and symbols…
 Force
N = kg ms-2
 Joule
J = kgm2s-2
Standards of Measurement
 Scientists and engineers need to make
accurate measurements so that they can
exchange information
 To be useful a standard of measurement
must be Invariant, Accessible and
Reproducible
3 Standards (FYI – not tested)
 The Meter :- the distance traveled by a beam of
light in a vacuum over a defined time interval (
1/299 792 458 seconds)
 The Kilogram :- a particular platinum-iridium
cylinder kept in Sevres, France
 The Second :- the time interval between the
vibrations in the caesium atom (1 sec = time for 9
192 631 770 vibrations)
Conversions
 You will need to be able to convert from one
unit to another for the same quanitity
•
•
•
•
J to kWh (energy)
J to eV (energy)
Years to seconds (time)
And between other systems and SI
****Note: you should be able to do basic conversions now and
others will be developed throughout the year
SI Format
 The accepted SI format is
• ms-1 not m/s
• ms-2 not m/s/s
The IB will recognize work reported with “/”, but
will only use the SI format when providing info.
Uncertainity and error in
measurement
Errors
 Errors can be divided into 2 main classes
 Random errors
 Systematic errors
Mistakes

Mistakes on the part of an individual such as
•
•
•
•

misreading scales
poor arithmetic and computational skills
wrongly transferring raw data to the final report
using the wrong theory and equations
These are a source of error but are not
considered as an experimental error
Systematic Errors
 Cause a random set of measurements to be
affected in the same way
 It is a system or instrument issue
Systematic Errors result from
 Badly made instruments
 Poorly calibrated instruments
 An instrument having a zero error, a form of
calibration
 Poorly timed actions
 Instrument parallax error
 Note that systematic errors are not
reduced by multiple readings
Random Errors
 Are due to unpredictable variations in
performance of the instrument and the
operator
Random Errors result from
 Vibrations and air convection
 Misreading
 Variation in thickness of surface being
measured
 Using less sensitive instrument when a
more sensitive instrument is available
 Human parallax error
Reducing Random Errors
 Random errors can be reduced by taking
multiple readings, and eliminating
obviously erroneous result or by averaging
the range of results.
Accuracy
 Accuracy is an indication of how close a
measurement is to the accepted value
indicated by the relative or percentage
error in the measurement
 An accurate experiment has a low
systematic error
Precision
 Precision is an indication of the agreement
among a number of measurements made in
the same way indicated by the absolute
error
 A precise experiment has a low random
error
Reducing the Effects of Random
Uncertainties
 Take multiple readings
 When a series of readings are taken for a
measurement, then the arithmetic mean of
the reading is taken as the most probable
answer
 The greatest deviation from the mean is
taken as the absolute error
Absolute/fractional errors and
percentage errors
 We use ± to show an error in a
measurement
 (208 ± 1) mm is a fairly accurate
measurement
 (2 ± 1) mm is highly inaccurate
Absolute, fractional, and relative
uncertainty
Assume we measure something to be 208 ± 1
mm in length...
 1 mm is the absolute uncertainty
 1/208 is the fractional uncertainty (0.0048)
 0.48 % is the relative (percent) uncertainty
Combining uncertainties
To determine the uncertainty of a calculated
value...
For addition and subtraction, add absolute
uncertainities
 For multiplication and division add percentage
uncertainities
 When using exponents, multiply the percentage
uncertainty by the exponent

Combining uncertainties
 If one uncertainty is much larger than
others, the approximate uncertainty in the
calculated result may be taken as due to
that quantity alone
Significant Figures
 The number of significant figures should
reflect the precision of the values used as
input data in a calculation
Simple rule:
 For multiplication and division, the number
of significant figures in a result should not
exceed that of the least precise value upon
which it depends
Uncertainties in graphs
Graphical Techniques
 Graphing is one of the most valuable tools
in data analysis because
• it gives a visual display of the relationship between
two or more variables
• shows which data points do not obey the relationship
• gives an indication at which point a relationship
ceases to be true
• used to determine the constants in an equation
relating two variables
 You need to be able to give a qualitative
physical interpretation of a particular
graph
Plotting Graphs
 Independent variables are plotted on the x-
axis
 Dependent variables are plotted on the yaxis
 Most graphs occur in the 1st quadrant
however some may appear in all 4
Plotting Graphs - Choice of Axis
 Experimentally speaking, the dependent
variable is plotted on the y axis and the
independent variable is plotted on the x
axis.
 When you are asked to plot a graph of a
against b, the first variable mentioned is
plotted on the y axis.
Plotting Graphs - Scales
 Size of graph should be large, to fill as
much space as possible…3/4 rule
 choose a convenient scale that is easily
subdivided
Plotting Graphs - Labels
 Each axis is labeled with the name of the
quantity, as well as the relevant unit used…
Temperature/K
speed/ms-1
 The graph should also be given a
descriptive title
Plotting Uncertainties on Graphs
 Error bars showing uncertainty are
required - short lines drawn from the
plotted points parallel to the axes
indicating the absolute error of
measurement
Plotting Graphs - Line of Best Fit
When choosing the best fit line or curve it is
easiest to use a transparent ruler
 Position the ruler until it lies along an ideal line
 The line or curve does not have to pass through
every point
 Do not assume that all lines should pass through
the origin
 Do not do play connect the dots!

Uncertainties on a Graph
y
Notice that the best
fitting line or curve is
one that passes
through the error
bars of the plotted
points. A straight line
could not accomplish
that with this data set
x
Analysing the Graph
 Often a relationship between variables will
first produce a parabola, hyperbole or an
exponential growth or decay. These can be
transformed to a straight line relationship
 General equation for a straight line is
y = mx + c
– y is the dependent variable, x is the independent
variable, m is the gradient and c is the y-intercept
Gradients
 Gradient = vertical run / horizontal run
gradient = y / x
 Don´t forget to give the units of the gradient
 In lab work, always report the maximum
and minimum gradient
Areas under Graphs
 The area under a graph is a useful tool.
For example…
• on a force vs. displacement graph the area is
work (N x m = J)
• on a speed time graph the area is distance
(ms-1 x s = m)
 Again, don´t forget the units of the area
Standard Graphs - linear graphs
 A straight line passing through the origin
shows proportionality
yx
y
k = rise/run
y=kx
Where k is the constant
of proportionality
x
Standard Graphs - parabola
 A parabola shows that y is directly
proportional to x2
y
y
x
x2
i.e. y  x2
or y = kx2
where k is the constant of proportionality
Standard Graphs - hyperbola
 A hyperbola shows that y is inversely
proportional to x
y
y
x
1/x
i.e. y  1/x
or y = k/x
where k is the constant of proportionality
Standard Graphs - hyperbola
again
 An inverse square law graph is also a
hyperbola
y
y
x
1/x2
i.e. y  1/x2
or y = k/x2
where k is the constant of proportionality