Physics and Physical Measurement

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Physics and Physical Measurement

• Topic 1.2 Measurement and Uncertainties

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The S.I. system of fundamental and derived units
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Standards of Measurement

• SI units are those of the Système International
  dUnités adopted in 1960
• Used for general measurement in most countries

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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

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The Fundamentals

• Length metre m
• Mass kilogram kg
• Time second s
• Electric current ampere A
• Thermodynamic temp Kelvin K
• Amount of a substance mole mol

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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

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The Derived Units

• Acceleration ms-2
• Angular acceleration rad s-2
• Momentum kgms-1 or Ns
• Others have specific names and symbols
• Force kg ms-2 or N

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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

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3 Standards (for information)

• The Metre - 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)

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Conversions

• You will need to be able to convert from one unit
  to another for the same quanitity
• J to kWh
• J to eV
• Years to seconds
• And between other systems and SI

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KWh to J

• 1 kWh 1kW x 1 h
• 1000W x 60 x 60 s
• 1000 Js-1 x 3600 s
• 3600000 J
• 3.6 x 106 J

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J to eV

• 1 eV 1.6 x 10-19 J

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SI Format

• The accepted SI format is
• ms-1 not m/s
• ms-2 not m/s/s
• i.e. we use the suffix not dashes

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Uncertainity and error in measurement
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Errors

• Errors can be divided into 2 main classes
• Random errors
• Systematic errors

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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

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Systematic Errors

• Cause a random set of measurements to be spread
  about a value rather than being spread about the
  accepted value
• It is a system or instrument value

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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

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Random Errors

• Are due to variations in performance of the
  instrument and the operator
• Even when systematic errors have been allowed
  for, there exists error.

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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

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Reducing Random Errors

• Random errors can be reduced by
• taking multiple readings, and eliminating
  obviously erroneous result
• or by averaging the range of results.

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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

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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

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Limit of Reading and Uncertainty

• The Limit of Reading of a measurement is equal to
  the smallest graduation of the scale of an
  instrument
• The Degree of Uncertainty of a measurement is
  equal to half the limit of reading
• e.g. If the limit of reading is 0.1cm then the
  uncertainty range is ?0.05cm
• This is the absolute uncertainty

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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 or residual from the mean
  is taken as the absolute error

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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

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• In order to compare uncertainties, use is made of
  absolute, fractional and percentage
  uncertainties.
• 1 mm is the absolute uncertainty
• 1/208 is the fractional uncertainty (0.0048)
• 0.48 is the percentage uncertainity

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Combining uncertainties

• For addition and subtraction, add absolute
  uncertainities
• y b-c then y dy (b-c) (db dc)

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Combining uncertainties

• For multiplication and division add percentage
  uncertainities
• x b x c then dx db dc
• x b c

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Combining uncertainties

• When using powers, multiply the percentage
  uncertainty by the power
• z bn then dz n db
• z b

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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

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Uncertainties in graphs
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Plotting Uncertainties on Graphs

• Points are plotted with a fine pencil cross
• Uncertainty or error bars are required
• These are short lines drawn from the plotted
  points parallel to the axes indicating the
  absolute error of measurement

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Uncertainties on a Graph
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Significant Figures

• The number of significant figures should reflect
  the precision of the value or of the input data
  to be calculated
• 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

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Estimation

• You need to be able to estimate values of
  everyday objects to one or two significant
  figures
• And/or to the nearest order of magnitude
• e.g.
• Dimensions of a brick
• Mass of an apple
• Duration of a heartbeat
• Room temperature
• Swimming Pool

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• You also need to estimate the result of
  calculations
• e.g.
• 6.3 x 7.6/4.9
• 6 x 8/5
• 48/5
• 50/5
• 10
• (Actual answer 9.77)

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Approaching and Solving Problems

• You need to be able to state and explain any
  simplifying assumptions that you make solving
  problems
• e.g. Reasonable assumptions as to why certain
  quantities may be neglected or ignored
• i.e. Heat loss, internal resistance
• Or that behaviour is approximately linear

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Graphical Techniques

• Graphs are very useful for analysing the data
  that is collected during investigations
• Graphing is one of the most valuable tools used
  because

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Why Graph

• 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

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• You need to be able to give a qualitative
  physical interpretation of a particular graph
• e.g. as the potential difference increases, the
  ionization current also increases until it
  reaches a maximum at..

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Plotting Graphs

• Independent variables are plotted on the x-axis
• Dependent variables are plotted on the y-axis
• Most graphs occur in the 1st quadrant however
  some may appear in all 4

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Plotting Graphs - Choice of Axis

• When you are asked to plot a graph of a against
  b, the first variable mentioned is plotted on the
  y axis
• Graphs should be plotted by hand

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Plotting Graphs - Scales

• Size of graph should be large, to fill as much
  space as possible
• choose a convenient scale that is easily
  subdivided

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Plotting Graphs - Labels

• Each axis is labeled with the name and symbol, as
  well as the relevant unit used
• The graph should also be given a descriptive title

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Plotting Graphs - Line of Best Fit

• When choosing the line or curve it is best 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 dot to dot!

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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

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• The parameters of a function can also be obtained
  from the slope (m) and the intercept (c) of a
  straight line graph

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Gradients

• Gradient vertical run / horizontal run
• or gradient ?y / ?x
• uphill slope is positive and downhill slope is
  negative
• Dont forget to give the units of the gradient

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Areas under Graphs

• The area under a graph is a useful tool
• e.g. on a force displacement graph the area is
  work (N x m J)
• e.g. on a speed time graph the area is distance
  (ms-1 x s m)
• Again, dont forget the units of the area

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Standard Graphs - linear graphs

• A straight line passing through the origin shows
  proportionality

y ? x
y k x
k rise/run
Where k is the constant of proportionality
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Standard Graphs - parabola

• A parabola shows that y is directly proportional
  to x2

i.e. y ? x2 or y kx2 where k is the constant
of proportionality
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Standard Graphs - hyperbola

• A hyperbola shows that y is inversely
  proportional to x

i.e. y ? 1/x or y k/x where k is the constant
of proportionality
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Standard Graphs - hyperbola again

• An inverse square law graph is also a hyperbola

i.e. y ? 1/x2 or y k/x2 where k is the
constant of proportionality
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Non-Standard Graphs

• You need to make a connection between graphs and
  equations

If this is a graph of r against t2 plotted from
data having an expected relationship r at2/2
r0 where a is a constant
Then the gradient is a/2 and the y-intercept is
r0 - it is not the case that r ? t2, it is a
linear relationship
The intercept is therefore important too