MEASUREMENT

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MEASUREMENT
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(No Transcript)
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SIMPLE MEASURING

• Use the most sensible instrument (a trundle wheel
  is not good for measuring the length of a book)
• Measure to the nearest scale division (if the
  instrument maker thought the instrument could do
  better they would have added more divisions)
• The exception to this is when timing with a
  stopwatch, round the display reading to the
  nearest 0.1s. Your reaction time does not justify
  times to 0.01s.

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ACCURACY

• This means getting close to the right measurement
  by reducing systematic errors and random errors
• Systematic errors arise from the instrument or
  the person measuring
• Random errors result from an observer being
  unable to repeat actions precisely

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Systematic errors

• Check for zero errors (Have you got a ruler with
  the first 5cm sawn off? Did you re-zero the
  scales? Did your ammeter start at zero?)
• Avoid parallax errors by having your eye level
  with the quantity to be measured and
  perpendicular to the scale
• Check the measuring instrument by using another
  for the same measurement, especially when using
  Newtonmeters
• Count properly for oscillations lots of
  students count from one rather than zero. Also
  lots of students time half oscillations rather
  than full ones
• Repeating readings does not reduce systematic
  error!

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Random errors

• Repetition and averaging improves precision by
  reducing random errors
• Timing multiple oscillations improves precision
  (any inaccuracy in measurement is divided by the
  number of oscillations timed)
• If using a digital multimeter to measure a
  quantity you select the scale which gives the
  greatest number of significant figures

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Rounding Significant figures, sf

• For a single measurement the number of sf is
  determined by the smallest scale division. Eg
  with a metre rule marked in mm, a small length
  could be to 1sf, eg 8mm but a longer length could
  be to 3sf eg 268mm
• Single timings should be to 1 decimal place eg
  2.38s should be rounded to 2.4s
• Sensible rounding may be needed eg a height of a
  ball bounce may need to be rounded to the nearest
  cm, even if using a mm scale

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Sf and repeated readings

• If you repeat a measurement several times and
  average (find the mean of) the readings then, in
  general, you should quote your answer to the same
  number of sf as your measurements
• eg the average of 67, 62, 66, 68, 64, 65 is
  65.33333 You should round to 65
• For timing multiple events eg 10 oscillations,
  timed to the nearest 0.1s, you answer can be to
  the nearest 0.01s
• Eg 10 oscillations take 14.7s therefore one
  oscillation takes 1.47s

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UNITS

• Record readings with units
• Metric units are standard
• Watch out for tricky things like weight (this
  involves multiplying mass in kg by the strength
  of gravity weight is in Newtons)
• For derived units look at the formula eg1 speed
  distance/time so units are m/s (ms-1) eg2
  densitymass/volume so units are kg/m3 (kgm-3)
• Avoid non-standard abbreviations like sec instead
  of s, or cms instead of cm, or Ns instead of N.
• You will not achieve in your test if you get
  units wrong!

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A, M or E?

• For A you need to be able to take measurements,
  get near enough to the right answers and use the
  right units
• For M you need to also use techniques to improve
  accuracy
• For E you need to justify these techniques
  (explain why the techniques improve accuracy)
  These justifications are NOT just general
  statements

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LINEAR GRAPHS
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DRAW the GRAPH

• Label the axes and use the units
• Sensible scale please
• Plot the points
• Draw a best fit line
• If it looks like it should be a straight line
  then use a ruler
• If it looks like a curve then draw a best fit
  freehand curve

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WHATS the RELATIONSHIP?

• If the line is straight (use a ruler) then the
  relationship is called a linear one
• If it also goes through the origin it can be
  called a direct or proportional relationship.

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GRADIENT

• This gives the mathematical relationship
• Gradient rise/run
• Pick start and end points far apart
• Find the rise (vertical)
• Find the run (horizontal)
• Calculate the gradient
• Try to work out the unit (unit of rise.unit of
  run-1)

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The gradient of the graph has been worked out for
you. Write down the units of the gradient
L (m)
L (m)








t2 (s2)
ms-2
6
ms-1/2
or
m
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Calculate the gradient of this graph
d (m)










18 16 14 12 10
8 6 4 2 0
1 2 3 4 5 6 7
8 9 10 t (s)
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INTERCEPT

• If the line does not go through the origin then
  look for where it crosses the vertical axis (it
  is usually a positive number)
• Write down the number (with its unit the same
  as the vertical axis unit)

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WORK OUT the EQUATION

• If it goes through the origin then
• dependent variable gradient x independent
  variable
• Eg distance 6 x time (for a distance vs time
  graph whose gradient is 6ms-1)
• Or d 6t

• If theres an intercept then
• dependent variable (gradient x independent
  variable) intercept
• Eg distance 6 x time 15 (for a distance
  vs time graph where you are given a 15m start)
• Or d 6t 15

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y
The equation of this graph is




y mx C
x
Now write down the equations of these graphs.
d
F








t
x
d mt C
F mx C
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Using the relationship

• This is where you have to make sense of what the
  graph is about
• You may have to say what the physical
  significance of the gradient or intercept is
• Eg the gradient might be a speed (or 1/speed!),
  an electrical resistance etc
• Eg the intercept might be a spring length
• You might be asked to calculate a quantity by
  using the mathematical relationship

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Non-linear graphs
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Non-Linear graphs

• Plot as normal and draw a freehand best-fit curve
• Look at the shape of the graph and guess the form
  of the relationship between the two variables
• Add a third table column with the independent
  variable changed according to the guessed
  relationship
• Plot a new graph if your guess was right your
  new graph will be a straight line

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Graph shapes
y is proportional to x2 ie a squared relationship
y is inversely proportional to x ie an inverse
relationship
y is inversely proportional to x2 ie an inverse
square relationship
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These are the 3 curves you are likely to meet
This is a square relationship. To get a straight
line you must plot y versus x squared
This is an inverse relationship. To get a
straight line you must plot y versus (1/x)
This is an inverse squared relationship. To get a
straight line you must plot y versus (1/x2)
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• This is a square root relationship. To get a
  straight line you must plot y vs vx or y2 vs x.

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y is proportional to x2

• Add a third table column with x2
• Now plot y versus x2
• If your guess was right you should now have a
  straight line through the origin
• Find the gradient of this line (m)
• The relationship is
• y mx2
• Eg Ek 12v2

x y x2
1 3 1
2 12 4
3 27 9
4 48 16
5 75 25
6 108 36
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y is inversely proportional to x

• Add a third table column with 1/x
• Now plot y versus 1/x
• If your guess was right you should now have a
  straight line through the origin
• Find the gradient of this line (m)
• The relationship is
• y m(1/x) or y m/x
• Eg P 26/V

x y 1/x
1 3.0 1.00
2 1.5 0.50
3 1.0 0.33
4 0.75 0.25
5 0.6 0.20
6 0.5 0.17
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y is inversely proportional to x2
x y 1/x2
1 50.0 1.00
2 12.5 0.25
3 5.6 0.11
4 3.1 0.06
5 2.0 0.04
6 1.4 0.03

• Add a third table column with 1/x2
• Now plot y versus 1/x2
• If your guess was right you should now have a
  straight line through the origin
• Find the gradient of this line (m)
• The relationship is
• y m(1/x2) or y m/x2
• Eg F 34/d2

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Dont forget

• We use x and y as general examples
• In reality the variables will use different
  letters
• Example pressure is inversely proportional to
  volume
• P 48/V
• DO NOT talk about x and y in your test.

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Units

• If the units of x are kg then the units of x2 are
  kg2
• If the units of x are kg then the units of 1/x
  are kg-1
• If the units of x are kg then the units of 1/x2
  are kg-2
• Make sure you put the right units in the table
  and on the graph

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Gradient units

• The general rule is
• Gradient unit is y unit.x unit-1
• Examples
• m plotted against s
• J plotted against m2
• m plotted against kg-1

• ms-1

• J(m2)-1 or Jm-2

• m(kg-1)-1 or m kg

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A, M or E?

• For A you can plot a straightforward graph with
  correct labels, units, scale etc. You can
  identify a relationship from the shape.
• For M you can find a gradient and intercept and
  use them to find a mathematical relationship and
  a physical quantity
• For E you can do the same with the reprocessed
  data for a non-linear graph (including correct
  units)