Transcript Slide 1

MEASUREMENT
SIMPLE MEASURING
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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.
ACCURACY
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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
Systematic errors
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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!
Random errors
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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
Rounding & Significant figures, sf
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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
Sf and repeated readings
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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
UNITS
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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
density=mass/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!
A, M or E?
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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
LINEAR GRAPHS
DRAW the GRAPH
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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
WHAT’S the RELATIONSHIP?
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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.
GRADIENT
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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)
The gradient of the graph has been worked out for you.
Write down the units of the gradient
L (m)
L (m)
L
2
t
t2 (s2)
= 5 ms-2
t
L
= 6 ms-1/2
t
or
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m/ s
s
Calculate the gradient of this graph
d (m)
18
-1
16
Gradient = 13m = 2.2 ms
6s
14
Gradient = 2.2ms-1 (2 sig.fig.)
12
13 m
10
8
6
6s
4
2
0
1
2
3
4
5
6
7
8
9
10 t (s)
INTERCEPT
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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)
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
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If there’s 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
d = mt + C
x
F = mx + C
Using the relationship
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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
Non-linear graphs
Non-Linear graphs
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Plot as normal and draw a freehand bestfit 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
Graph shapes
ie a squared relationship
y
y is proportional to x2
x
y
y is inversely proportional to x
ie an inverse relationship
y
x
y is inversely proportional to x2
ie an inverse square relationship
x
x
This is an inverse squared
relationship. To get a
straight line you must plot y
versus (1/x2)
x
This is an inverse
relationship. To get a straight
line you must plot y versus
(1/x)
y
y
This is a square relationship.
To get a straight line you
must plot y versus x
squared
y
These are the 3 curves you are likely to meet
x
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This is a square root relationship. To
get a straight line you must plot y vs
√x or y2 vs x.
y is proportional to x2
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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
y is inversely proportional to x
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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
y is inversely proportional to x2
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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
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
Don’t forget
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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.
Units
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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
Gradient units
The general rule is:
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Gradient unit is ‘y unit’.x unit-1’
Examples:
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ms-1
m plotted against s:
J plotted against
m2:
m plotted against
kg-1:
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J(m2)-1 or Jm-2
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m(kg-1)-1 or m kg
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A, M or E?
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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)