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CAMBRIDGE A LEVEL
PHYSICS
MEASUREMENTTECHNIQUES
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L E A R N I N G O U T C O M E SNUMBER LEARNING OUTCOME
i U n d e r s t a n d t h e c o n c e p t o f m e a s u r e m e n t
ii B e a b l e t o m e a s u r e p h y s i c a l q u a n t i t i e s
iii U n d e r s t a n d t h e u s e o f c a l i b r a t i o n c u r v e s
iv D i f f e r e n t i a t e b e t w e e n r a n d o m a n d
s y s t e m a t i c e r r o r s
v S h o w d i s t i n c t i o n b e t w e e n a c c u r a c y a n d
p r e c i s i o n
vi L e a r n t o h a n d l e u n c e r t a i n t i e s i n
m e a s u r e m e n t s
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C O N C E P T O F
M E A S U R E M E N T
W h a t i s a m e a s u r e m e n t ?
L o o s e l y d e f i n e d , a m e a s u r e m e n t
i s t h e b e s t e s t i m a t e o f a
p h y s i c a l d i m e n s i o n .
E x a m p l e s :
T h e h e i g h t o f a p e r s o n i s a
m e a s u r e m e n t .
T h e t i m e l a p s e b e t w e e n t h e
b e g i n n i n g a n d e n d o f a 1 0 0 m r a c e
i s a l s o a m e a s u r e m e n t .
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C O N C E P T O F
M E A S U R E M E N T
W h y b e s t e s t i m a t e ?
A m e a s u r e m e n t i s t h e b e s t e s t i m a t ed u e t o p r e s e n c e o f u n c e r t a i n t i e s i nt h e m e a s u r e m e n t .
W h a t c a u s e s u n c e r t a i n t i e s ? U n c e r t a i n t i e s i n m e a s u r e m e n t sa r e d u e t o l i m i t a t i o n s i n h u m a n
o b s e r v a t i o n a n d t h em e a s u r e m e n t t o o l .
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C O N C E P T O F
M E A S U R E M E N T
H o w t o p r e s e n t a m e a s u r e m e n t ?
measurement = best estimate absolute uncertainty
=
or
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C O N C E P T O F
M E A S U R E M E N TH o w t o o b t a i n v a l u e s o f u n c e r t a i n t i e s ? :
i . u s e t h e s m a l l e s t d i v i s i o n o n t h es c a l e , i f t h e r e i s a v i s i b l e g a p
b e t w e e n d i v i s i o n s , o r i f t h e
d i v i s i o n s o f t h e s c a l e a r e t o oc l o s e , o r
i i . b y u s i n g a s c a l e o f i n t e r p o l a t i o n
i f t h e g a p b e t w e e n d i v i s i o n s i sl a r g e .
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C O N C E P T O F
M E A S U R E M E N TW h a t a b o u t s i g n i f i c a n t f i g u r e s i n
a b s o l u t e u n c e r t a i n t i e s ? : R u l e # 1 :
R u l e # 2 :
Section 2.2, page 15, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
U n c e r t a i n t i e s s h o u l d a l m o s t a l w a y s
b e r o u n d e d t o 1 s i g . f i g .
T h e l a s t s i g . f i g . o f t h e b e s t e s t i m a t e
m u s t b e i n t h e s a m e d e c i m a l p o s i t i o n a s
t h e u n c e r t a i n t y
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C O N C E P T O F
M E A S U R E M E N TF o r e x a m p l e , v a l i d m e a s u r e m e n t si n c l u d e :
i . 9 . 8 2 0 . 0 2 m s - 2
i i . 6 0 5 0 3 0 m s - 1
i i i . 9 0 3 0 c m - 3
b u t :
i . 9 . 8 2 0 . 0 2 3 8 5 m s - 2 ( v i o l a t e s R u l e# 1 a n d e v e n R u l e # 2 )
i i . 6 0 5 1 . 7 8 3 0 m s- 1
( v i o l a t e s R u l e# 2 )
a r e n o t v a l i d m e a s u r e m e n t s
Section 2.2, page 15, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
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C O N C E P T O F
M E A S U R E M E N T
Section 2.2, page 16, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
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C O N C E P T O F
M E A S U R E M E N T
E x a m p l e o f m e a s u r i n g l e n g t h :
Figure 1.2, page 8, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
Q: What is the length of the pencil?
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C O N C E P T O F
M E A S U R E M E N T
Q: What is the length of the pencil?
indicates presence of uncertainty
37.0best estimate
1.0 mm
value of uncertainty
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C O N C E P T O F
M E A S U R E M E N T
E x a m p l e o f m e a s u r i n g v o l t a g e :
Figure 1.3, page 8, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
Q: What is the value of the reading?
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C O N C E P T O F
M E A S U R E M E N T
Reading on voltmeter: 5.5 1.0 V
Q: What is the value of the reading?
Use interpolation; obtaining values between scale values; to get 5.5 V
The uncertainty of 1.0 V is the size of the divisions.
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M E A S U R I N G L E N G T H S W e w i l l l o o k a t t w o o t h e r
a p p a r a t u s u s e d t o m e a s u r el e n g t h s :
i . t h e v e r n i e r s c a l ei i . t h e m i c r o m e t e r s c r e w g a u g e
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M E A S U R I N G L E N G T H S
A VERNIER SCALE
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
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M E A S U R I N G L E N G T H S E X A M P L E S :
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
I. LOOK FOR ZERO VALUE ON VERNIER SCALE
Here ZERO VALUE is between 3.0 and 4.0 mm on
main scale. Take lower value of 3.0 mm.
II. LOOK FOR DIGIT ON VERNIERSCALE WHERE VERNIER AND
MAIN SCALES COINCIDE
Here they coincide at 0.7 mm.
The vernier scale divides the
linear scale into smaller
subdivisions
III. ADD BOTH READINGS
Here we get 3.0 + 0.7 mm = 3.7 mm
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M E A S U R I N G L E N G T H S
Source:
https://changkatphysics.wikispaces.com/file/view/Worksheet+1.2_2011.pdf
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M E A S U R I N G L E N G T H S
Source:
https://changkatphysics.wikispaces.com/file/view/Worksheet+1.2_2011.pdf
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M E A S U R I N G L E N G T H S
Source:
http://home.vs.moe.edu.sg/imran/midyearworksheet.pdf
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M E A S U R I N G L E N G T H S
Source:
http://home.vs.moe.edu.sg/imran/midyearworksheet.pdf
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M E A S U R I N G L E N G T H S
Source:
http://home.vs.moe.edu.sg/imran/midyearworksheet.pdf
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M E A S U R I N G L E N G T H S
A MICROMETER SCREW GAUGE
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
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M E A S U R I N G L E N G T H S
III. Obtain value of measurement by adding main scale reading
with vernier scale reading. Here it is 7.72 mmSource:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
II. Read value of vernier
scale where scale
coincides with line
perpendicular to
main scale. Here it is
0.22 mm.
I. Read value from main scale. Here
it is 7.5 mm.each line = 1.0 mm
each line = 0.5 mm
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M E A S U R I N G L E N G T H S
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
What is the value of this measurement?
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M E A S U R I N G L E N G T H S
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
What is the value of this measurement? 3.56 mm
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M E A S U R I N G L E N G T H S
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
What is the value of this measurement?
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M E A S U R I N G L E N G T H S
Source:
http://teachers.cie.org.uk/community/resources/pt_view_resource?id=1474
What is the value of this measurement? 0.29 mm
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M E A S U R I N G W E I G H T S A N D
M A S S E S A p p a r a t u s t h a t c a n b e u s e d
t o m e a s u r e w e i g h t / m a s s o f a n o b j e c t :
i . S p r i n g b a l a n c ei i . L e v e r b a l a n c e
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M E A S U R I N G W E I G H T S A N D
M A S S E S
Source:
http://images.tutorvista.com/content/forces/spring-balance.gif
The diagram on the left shows a spring
balance.
The reading of the pointer gives the
value of the gravitational force that is
exerted on the object.
To obtain the mass ofthe object, we
must divide the reading of the scale bythe local gravitational acceleration,
often taken as 9.8 m s-2
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M E A S U R I N G W E I G H T S A N D
M A S S E S
Source:
http://etc.usf.edu/clipart/67200/67265/67265_proportions.htm
The diagram on the left shows
a primitive lever balance.
The lever balance functionsusing the principle of
moments.
Weight, W to be determined is
hung on the right side offulcrum, F.
The mass, P is moved until the
the scale is horizontal.
rd
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M E A S U R I N G A N G L E S
Source:
http://www.mathsisfun.com/geometry/images/protractor-2-angles.gif
The diagram on the left
shows a protractor.
Protractors can be usedto measure angles
between 0 to 180 .
The absolute uncertainty
of a protractor = 1.0 . You will use protractors
later during the practical
sessions.
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M E A S U R I N G T I M E T i m e i s u s u a l l y m e a s u r e d u s i n g :
W e w i l l l o o k a t t h e s e i n a l i t t l e
b i t m o r e d e t a i l i n t h e n e x t f e ws l i d e s .
i . a c l o c k ,i i . a s t o p w a t c h ,
i i i . a c a l i b r a t e d t i m e b a s e
s e t t i n g o f a C R O .
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M E A S U R I N G T I M E
C l o c k / s t o p w a t c h :
C l o c k s a n d s t o p w a t c h e s a r et i m e m e a s u r i n g d e v i c e s .
C l o c k s a n d s t o p w a t c h e s c a n b ee i t h e r a n a l o g u e o r d i g i t a l .
T h e u n c e r t a i n t i e s o f d i g i t a l
s t o p w a t c h / c l o c k a r e . .
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U S I N G A C R OC a t h o d e R a y O s c i l l o s c o p e ( C R O ) :
V e r y g o o d e x p l a n a t i o n s o f t h eC R O u s e i s p r o v i d e d i n t h e
l i n k s b e l o w :
I. http://www.antonine-
education.co.uk/Pages/Physics_1/Electricity
/EL_10/Electricity_10.htmII. http://www.doctronics.co.uk/scope.htm
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U S I N G A C R O
Source:
http://www.antonine-education.co.uk/Pages/Physics_1/Electricity/EL_10/Electricity_10.htm
The diagram below shows a sinusoidal trace on the screen of a CRO
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OT H E R M E A S U R E M E N T S O t h e r m e a s u r e m e n t s d o n e i n
t h e l a b i n c l u d e :i . m e a s u r i n g t e m p e r a t u r e s
u s i n g t h e r m o m e t e r s ,
i i . m e a s u r i n g c u r r e n t s u s i n g
a m m e t e r s / g a l v a n o m e t e r s ,
i i i . m e a s u r i n g p o t e n t i a ld i f f e r e n c e u s i n g v o l t m e t e r s .
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E X A M P L E SMay/Jun 2009, Paper 1, question 3.
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E X A M P L E SMay/Jun 2010, Paper 11, question 4.
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E X A M P L E SMay/Jun 2010, Paper 11, question 7.
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E X A M P L E SOct/Nov 2010, Paper 12, question 4.
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H O M E W O R K Q U E S T I O N SHomework:
1. Question 4, Paper 1, May/Jun 2009.2. Question 4, Paper 11, Oct/Nov 2009.
3. Question 1, Paper 11, Oct/Nov 2010.
4. Question 5, Paper 11, Oct/Nov 2010.
5. Question 3, Paper 23, Oct/Nov 2010.
6. Question 4, Paper 11, May/Jun 2011.7. Question 5, Paper 12, May/Jun 2011.
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C A L I B R AT I O N C U R V E S Calibration curves are functions of an
instrument's responses to a range of factor
levels. (*) A factor level could be current, potential
difference, or some other measurable quantity.
A calibration curve is used to give readings ofvalues for analogue meters that have:
i. non linear scales, or
ii. are not properly calibrated, or
iii. are faulty.
(*) Source:
http://www.aerosols.eas.gatech.edu/EAS%20Graduate%20Lab/Class%20Notes%20Uncertainty.p
df
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C A L I B R AT I O N C U R V E S The gradient of a calibration curve
gives the sensitivity of the measuringdevice.
The sensitivity of a measuring devicegives amount of change in
measurement output for a given
measurement input change.
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C A L I B R AT I O N C U R V E S A more sensitive measurement tool
will give a larger range of response to asmaller input change.
e.g. the needle of a sensitive analoguemeter will deflect more for a smaller
change in input.
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E X A M P L E SOct/Nov 2010, Paper 11, question 4.
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E X A M P L E SMay/June 2008, Paper 1, question 5.
R A N D O M v s S Y S T E M AT I C
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R A N D O M v s . S Y S T E M AT I C
E R R O R S
MEASUREMENT ERRORS
RANDOMERRORS
SYSTEMATICERRORS
R A N D O M v s S Y S T E M AT I C
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R A N D O M v s . S Y S T E M AT I C
E R R O R S
Figure 4.1, page 95, Chapter 4: Statistical Analysis of Random Uncertainties, AN INTRODUCTION TO ERROR
ANALYSIS: THE STUDY OF UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University
Science Books, California 1997.
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R A N D O M v s . S Y S T E M A T I C E R R O R S
RANDOM ERROR SYSTEMATIC ERROR
Definition: Experimental
uncertainties that can be revealed by
repeating the measurement arecalled random errors.
Definition: Experimental
uncertainties that cannot be
revealed by repeating themeasurement are called
systematic errors.
Examples of causes of this type of
error:
i. misreading of data/value on part
of data collector.
ii. variations in thickness of a wirewhen measuring its thickness.
Examples of causes of this type
of error:
i. zero error, a type of
calibration error.
ii. poorly calibrated equipment.
* Definition from page 94, Chapter 4: Statistical Analysis of Random Uncertainties, AN INTRODUCTION TO
ERROR ANALYSIS: THE STUDY OF UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR;
University Science Books, California 1997.** Source for examples:http://www.ibid.com.au/Uploads/English/Documents/Subject/Physics%20Ch%201.pdf
R A N D O M S Y S T E M A T I C E R R O R S
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R A N D O M v s . S Y S T E M A T I C E R R O R S
RANDOM ERROR SYSTEMATIC ERROR
Examples of causes of this type of
error (contd):
iii. reaction time errors, if reactiontimes are inconsistent.
iv. usage of less sensitive equipment
when more sensitive equipment is
available.v. effect of air currents when
reading mass.
vi. parallax error on part of observer,if inconsistent.
Examples of causes of this type
of error (contd):
iii. reaction time errors,provided consistent reaction
times.
iv. instrumental parallax error.
v. observers parallax error, ifparallax error consistent.
* Definition from page 94, Chapter 4: Statistical Analysis of Random Uncertainties, AN INTRODUCTION TO
ERROR ANALYSIS: THE STUDY OF UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR;
University Science Books, California 1997.** Source for examples:http://www.ibid.com.au/Uploads/English/Documents/Subject/Physics%20Ch%201.pdf
R A N D O M v s S Y S T E M AT I C
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R A N D O M v s . S Y S T E M AT I C
E R R O R SRANDOM ERROR SYSTEMATIC ERROR
Effects the standarddeviation (the spread of
results) of an experiment.
Each result from experiments
with higher random error
will be spread further apart
from the mean.
Effects the mean of theexperimental results.
Results of experiments with
more significant systematic
error will have a mean
further away from actual
value.
Effect of error on average of best estimate can be reduced
by repeating the experiment.
Effect of error on average of best estimate cannot be
reduced by numerous trials
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E X A M P L E SOct/Nov 2010, Paper 11, question 6.
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P R E C I S I O N v s . A C C U R A C Y
PRECISION ACCURACY
For measurements:
Relates to the spread; i.e. differenceof the results from the mean value
of all the measurements.
For a particular equipment: Relates to how much resolution a
measuring tool gives.
E.g. a micrometer screw gauge is
more precise in measurement than
a vernier scale because its
resolution is to the nearest 0.01
mm.
For measurements:
Relates to how close the reading isto the actual value.
For a particular equipment: Relates to how close to the actual
value a measurement tool can
measure.
Greatly depends on the calibration
of the equipment prior to
performing experiment.
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P R E C I S I O N v s . A C C U R A C Y
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E X A M P L E SMay/June 2008, Paper 1, question 5.
H O M E W O R K
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H O M E W O R K
1. Oct/Nov 2008, Paper 1, question 4.
2. Oct/Nov 2009, Paper 11, Question 3.3. May/Jun 2010, Paper 22, question 1.
4. May/Jun 2011, Paper 23, question 1.5. Oct/Nov 2011, Paper 12, question 5.
H A N D L I N G U N C E R TA I N T I E S
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H A N D L I N G U N C E R TA I N T I E S
UNCERTAINTIES
ABSOLUTEe.g. 9.8 0.2 m s-2FRACTIONAL/PERCENTAGE
e.g. 9.8 .
.
e.g. 9.8 m s-2 2%
H A N D L I N G U N C E R TA I N T I E S
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H A N D L I N G U N C E R TA I N T I E S
CONVERTING UNCERTAINTIES
between absolute and
fractional/percentage
%
between absolute and
percentage
%
H A N D L I N G U N C E R TA I N T I E S
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H A N D L I N G U N C E R TA I N T I E S
Quick Check 2.4, Section 2.7, page 29, AN INTRODUCTION TO ERROR ANALYSIS: THE
STUDY OF UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR;
University Science Books, California 1997.
H A N D L I N G U N C E R TA I N T I E S
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H A N D L I N G U N C E R TA I N T I E S
Quick Check 2.4, Section 2.7, page 29, AN INTRODUCTION TO ERROR ANALYSIS: THE
STUDY OF UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR;
University Science Books, California 1997.
a. 55 cm/s 4%,
b. -20 cm/s 10%,
c. 4.58 0.09 J
P R O P A G A T I N G
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U N C E R T A I N T I E SUncertainties propagate when
measurements with uncertainties are:I. added, or
II. subtracted, or
III. multiplied, or
IV. divided.
P R O P A G A T I N G
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U N C E R T A I N T I E S What happens when measurements
with uncertainties areadded/subtracted?:
the measurements areadded/subtracted.
their respective absoluteuncertainties are added.
P R O P A G A T I N G
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U N C E R T A I N T I E S
Example from page 50, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
P R O P A G A T I N G
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U N C E R T A I N T I E S
Example from page 50, AN INTRODUCTION TO ERROR ANALYSIS: THE STUDY OF
UNCERTAINTIES IN PHYSICAL MEASUREMENTS; JOHN R. TAYLOR; University Science
Books, California 1997.
P R O P A G A T I N G
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U N C E R T A I N T I E S What happens when measurements
with uncertainties aremultiplied/divided?:
the measurements aremultiplied/divided.
their respective percentage orfractional uncertainties are added.
E X A M P L E S
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May/June 2008, Paper 1, question 4.
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U N C E R T A I N T I E STWO SPECIAL CASES:
i. Constants
If a constant/exact value appears in an equation, themagnitude of the constant is multiplied by the absolute
uncertainties of the other measured variables that the
constant is multiplied by in the original equation.
e.g. Given the radius of a circle, 1.20 0.02cm, find
its circumference, with its uncertainty:
2 2
2 1.20 2 0.02
7.5 0.1cm
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U N C E R T A I N T I E STWO SPECIAL CASES:
ii. Powers
If a term is raised to a power, the magnitude of the poweris multiplied by the percentage uncertainties of the term
that is raised.
e.g. Given the length of a cube, 2.00 0.02cm, findits volume, with its uncertainty:
3 100%
2.00 3 0.02
2.00 100%
8. 00 3% 8.0 0.2
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Oct/Nov 2009, Paper 22, question 1.
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Oct/Nov 2009, Paper 22, question 1 (contd).
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Oct/Nov 2009, Paper 22, question 1 (contd).
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Oct/Nov 2009, Paper 22, question 1 (contd).
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Oct/Nov 2009, Paper 22, question 1 (contd).
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May/Jun 2010, Paper 23, question 1.
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May/Jun 2010, Paper 23, question 1 (contd).
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May/Jun 2010, Paper 23, question 1 (contd).
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Oct/Nov 2011, Paper 11, question 5.
H O M E W O R K
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1. Oct/Nov 2008, Paper 1, question 4.
2. May/Jun 2009, Paper 21, question 1.3. Oct/Nov 2009, Paper 21, question 1.
4. May/Jun 2010, Paper 11, question 6.
5. May/Jun 2010, Paper 22, question 1.
6. Oct/Nov 2010, Paper 12, question 5.
7. May/June 2011, Paper 11, question 5.8. May/June 2011, Paper 12, question 4.
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9. May/Jun 2011, Paper 21, question 1.
10.May/Jun 2011, Paper 23, question 1.11.Oct/Nov 2011, Paper 12, question 4.
12.Oct/Nov 2011, Paper 12, question 5.