Physical quantities and their
measurements
•
•
•
•
•
•
Measurement of length and volume
Instruments of measurement
Least count and Precision of an instrument
Significant figures
Uncertainty in a measurement
Accuracy and errors – Systematic and
random
Measuring Volume of a liquid
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
• Beaker
graduated cylinder
buret
Beaker – least count and significant
figures
• The smallest division is 10 mL, so we can read
the volume to 1/10 of 10 mL or 1 mL. The
volume we read from the beaker has a reading
error of 1 mL.
• The volume in this beaker is 47 1 mL. You
might have read 46 mL; your friend might read
the volume as 48 mL. All the answers are
correct within the reading error of 1 mL.
• So, How many significant figures does our
volume of 47 1 mL have? Answer - 2! The "4"
we know for sure plus the "7" we had to estimate.
Graduated cylinder
• First, note that the surface of the liquid is curved. This is
called the meniscus. This phenomenon is caused by the
fact that water molecules are more attracted to glass
than to each other (adhesive forces are stronger than
cohesive forces). When we read the volume, we read it
at the BOTTOM of the meniscus.
• The smallest division of this graduated cylinder is 1 mL.
Therefore, our reading error will be 0.1 mL or 1/10 of the
smallest division. An appropriate reading of the volume
is 36.5 0.1 mL. An equally precise value would be 36.6
mL or 36.4 mL.
• How many significant figures does our answer have? 3!
The "3" and the "6" we know for sure and the "5" we had
to estimate a little.
Buret
• The smallest division in this buret is 0.1
mL. Therefore, our reading error is 0.01
mL. A good volume reading is 20.38 0.01
mL. An equally precise answer would be
20.39 mL or 20.37 mL.
• How many significant figures does our
answer have? 4! The "2", "0", and "3" we
definitely know and the "8" we had to
estimate.
Conclusion – sf and precision
• The number of significant figures is directly
linked to a measurement. If a person needed
only a rough estimate of volume, the beaker
volume is satisfactory (2 significant figures),
otherwise one should use the graduated cylinder
(3 significant figures) or better yet, the buret (4
significant figures).
• So, does the concept of significant figures deal
with precision or accuracy? Hopefully, you can
see that it really deals with precision only.
Rulers
• Meter rule in inch, centimetre and
millimetre
• Half metre rule
• Precision is how fine of a measurement
that the measuring instrument is marked
off for.
Rulers in inches
Standard English Ruler
Standard ruler Metric
Least Count
• The least count is the smallest subdivision marked on a
measuring instrument.
• Determine the least count of your measuring stick.
• Record the numerical value of the least count and the unit of
measurement.
• Example: A meter stick is divided into 100 equal divisions and
numbered. Each of these numbered divisions is called 1 cm.
('one centimeter' means 'one one-hundredth' of a meter). Each
centimeter is further divided into 10 equal divisions. This is the
smallest subdivision on the meter stick.
Rounding off and significant figures
•
http://www.chem.tamu.edu/class/fyp/mathrev/mr-sigfg.html
Other instruments to measure
length
• Micrometer screw gauge
• Vernier calipers
Rules for Working with
Significant Figures:
• Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the decimal point is
specified.
Hint: Change the number to scientific notation. It is
easier to see.
• Addition or Subtraction:
The last digit retained is set by the first doubtful digit.
• Multiplication or Division:
The answer contains no more significant figures than the
least accurately known number.
Exact and inexact numbers
• exact:
– example: There are exactly 12 eggs in a
dozen.
– example: Most people have exactly 10 fingers
and 10 toes.
inexact numbers:
– example: any measurement.
If I quickly measure the width of a piece of
notebook paper,
– I might get 220 mm (2 significant figures).
– If I am more precise, I might get 216 mm (3
significant figures).
– An even more precise measurement would be
215.6 mm (4 significant figures).
Significant figures
•
•
•
•
•
0.00682
1.072
300
300.
300.0
3
4
1
3
4
6.82 x 10-3
1.072 (x 100)
3 x 102
3.00 x 102
3.000 x 102
• 453
3 significant
• .5057
4
• 5.00
• 0.007
3
1
figures
Significant digits
• 0.00341........
• 3 sig. digs.
• 1.0040........
• .5 sig. digs.
• 0.00005........
• 1 sig. dig.
• 65000..........
• 2 sig. digs.
• 40300..........
• 3 sig. digs.
• 200300.........
• 4 sig. digs.
Rules for Working with
Significant Figures:1
• Leading zeros are never significant.
Imbedded zeros are always significant.
Trailing zeros are significant only if the
decimal point is specified.
Hint: Change the number to scientific
notation. It is easier to see.
Rules for Working with Significant
Figures: 2
• Addition or Subtraction:
The last digit retained is set by the first
doubtful digit.
• Multiplication or Division:
The answer contains no more significant
figures than the least accurately known
number.
Rounding off
Addition or Subtraction:
•
The last digit retained is set by the first doubtful
digit
Addition or Subtraction:
The last digit retained is set by the
first doubtful digit
Multiplication or Division:The answer
contains no more significant figures
than the least accurately know number.
Multiplication or Division:
The answer contains no more significant
figures than the least accurately known
number.
Significant figures and electronic
calculators
• . For example, dividing 5.0 by 1.67 on a
calculator may give the following answer:
• 5.0 / 1.67= 2.9940119
• The correct answer, 3.0, has only two
significant figures, as in the least accurate
number (5.0) in the problem. All other
digits displayed by the calculator are
insignificant.
Rounding off
• Let's round off 64,492 to:
(a) 1 significant figure
which is 60,000
(b) 2 significant figures which is 64,000
(c) 3 significant figures which is 64,500
(d) 4 significant figures which is 64,490
(e) 5 significant figures which is 64,492
rounding off to given significant
figures
• When rounding off numbers to a certain
number of significant figures, do so to the
nearest value.
– example: Round to 3 significant figures:
2.3467 x 104 (Answer: 2.35 x 104)
– example: Round to 2 significant figures: 1.612
x 103 (Answer: 1.6 x 103)
What happens if there is a 5?
There is an arbitrary rule:
– If the number before the 5 is odd, round up.
– If the number before the 5 is even, let it be.
The justification for this is that in the course of a
series of many calculations, any rounding errors will
be averaged out.
– example: Round to 2 significant figures: 2.35 x 102
(Answer: 2.4 x 102)
– example: Round to 2 significant figures: 2.45 x 102
(Answer: 2.4 x 102)
– Of course, if we round to 2 significant figures: 2.451 x
102, the answer is definitely 2.5 x 102 since 2.451 x
102 is closer to 2.5 x 102 than 2.4 x 102.
QUIZ:
• Question 1
• Give the correct number of significant
figures for 4500, 4500., 0.0032, 0.04050
• Question 2
• Give the answer to the correct number of
significant figures:
4503 + 34.90 + 550 = ?
Questions
• Question 3
• Give the answer to the correct number of
significant figures: 1.367 - 1.34 = ?
• Question 4
• Give the answer to the correct number of
significant figures:
(1.3 x 103)(5.724 x 104) = ?
Questions
• Question 5
• Give the answer to the correct number of
significant figures:
• (6305)/(0.010) = ?
Example Significant figures
•
Number
• 560,000
Exponential expression
Significant figures
5.6 X 105
two
• (The zeros show only the location of the decimal point.)
• 560,000. 5.60000 X 105
six
(The decimal point in the original number shows that all the
zeros are significant.)
• 30,290
3.029 X 104
four
• (The first zero is between two digits and is significant. The last
shows only the location of the decimal point.)
• 0.0160
•
1.60 X 10-2
three
(The first two zeros show the location of the decimal; they are not
significant. The last one does not show the location of the decimal
point; it reports a measurement and therefore is significant.)
Errors
Random Errors
Systematic Errors
Changes conditions or surroundings of
equipment
Zero error on instrumentation
Non-perfect observations by
experimentalist
Wrong calibration of instrumentation
Readability of equipment
Incorrect measurement method for every
measurement
Random Errors: Errors that can not be predicted.
Systematic Errors: Errors which are the same for each measurement
Precision and accuracy
•
http://www.studyphysics.ca/newnotes/20/unit01_kinematicsdynamics/chp02_intro/images/precise.GIF
Good Precision ->
all the hits are close
to each other
Poor Accuracy ->
the hits are not near
their intended target
Poor Precision ->
the hits are not
near each other
Poor Accuracy ->
the hits are not
near their
intended target
Good Precision -> all the hits
are close to each other
Good Accuracy -> all the hits
are near their intended
target
Precision:
Small random error
Accurate:
Small systematic error
Absolute uncertainty
• Size of an error and its units
30m tape measure has an error of ±0.5cm
So the Absolute error is
30±0.005 m
Fractional uncertainty
•
Absolute uncertainty / measurement
30m tape measure has an error of ±0.5cm
Fractional Uncertainty = 0.005/30
Percentage uncertainty
•
Fractional uncertainty x 100%
Therefore
0.005/30 x 100% = 0.017%
Addition and Subtraction
5.9 ±0.6m + 7.9 ±0.8m = 13.8 ±1.4m
(add absolute errors)
6.9 ±0.6m - 3.9 ±0.8m = 3.0 ±1.4m
(add absolute errors)
Task
1.)Use your ruler to measure the length of
your desk. State the uncertainty in your
measurement.
Multiplication and Division
5.6 ±0.5m x 2.6 ±0.5m = 15 ±??m
0.5 / 5.6 = 0.0893
0.5 / 2.6 = 0.192
Sum of relative errors = 0.281
Absolute error = 0.281 x 15 = 4.2m
FINAL ANSWER = 15 ±4 m
Example Question
The length of a piece of paper is measured
as 297± 1mm. Its width is measured as
209± 1mm
a)What is the fractional uncertainty in its
length
b)What is the percentage uncertainty in its
length?
c)What is the area of one side of the piece
of paper? State your answer with its
uncertainty
2.) What is the area of your
desk?
• Remember
A/A =
L/L + W/W
Example Question 2
A Voltmeter has a reading of 2.00±0.05 V,
a miliammeter reads 3.3±0.1 mA
Estimate the Resistance and state your
uncertainties.
Question 3
• What do the terms systematic and random
errors mean?