Measurement and Problem Solving

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Measurement and Science
He has it down to an exact science
What the heck does that mean?
Science is not about being for sure.
Science is about exploring options and always
being open to other interpretations.
There is no such thing as an exact science!
There is only one thing for certain in science?
Nothing is for certain!
2
Science cannot exist without quantifiable
comparisons
He is tall--- compared to who or what?
He is 7 ft in height-- quantifiable
Tall, short, fat, skinny, long, short, hot,
cold.
. These are relative terms and not quantities!
Comparisons are meaningless in science unless
compared to a standard value.
And those standard values need to be the same for
everyone in order to be widely useful!
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Fundamental Units of Measure-directly comparable
to a standard
These are the only units used in Mechanics
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Fundamental Standards for Units in Mechanics
meter (m) ? now based on wavelengths of light
second (s) ? based upon vibrations of a Cesium
atom
kilogram (kg) ? still a non-reproducible
standard- chunk of metal in France
all other units in Mechanics are combinations
(derived) of these three fundamental values!
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Derived Units -derived from fundamental units

• Velocity (m/s)- meter/second
• Acceleration (m/s2)- meter per second squared
• Force (N)- Newton
• Energy (J)- Joule
• Power (w)- Watt

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In making measurements, it is important to have a
standard for comparison, and to make those
measurements with as much precision as possible.
20
30
Definitely more than 20 and less than 30 units!
23, 24, 25?
Definitely more than 24 and less than 25!
24.2 units, 24.3?
Either reading is considered correct.
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20
30
To call this measurement 20 units would be poor
when it is obviously more!
To call this measurement 30 units would be
equally lousy because it is obviously less!
24 or 25 would be a better effort and be
considered more accurate, even though they
contain estimated values.
There is no such thing as an exact measurement--
All measurements are inherently estimates!
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All measurements contain some degree of
uncertainty depending upon the device.
An accurate measurement will contain all the
known values of the measurement plus one (and
only one) estimated value.
Why only one? More than one estimated value
becomes wild guesses and have no meaning.
An estimate is not a guess-- it is an attempt to
approximate and make a reading more precise.
All known values of a measurement plus one
estimate are called significant figures (digits).
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Significant Figures/Digits

• A method of maintaining accuracy and precision in
  measurements and calculations.
• All known values of a measurement or calculation
  PLUS one and only one estimated value.
• In measurements, SD are totally determined by the
  device being used.
• In calculations, SD in the answer are determined
  by a basic rule

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In a given value, what is a SD?

• All non-zero numbers are SDs
• 12.35cm (4) 4.26 m (3)
• Zeros between other SD count
• 102 s (3) 5.007 (4)
• Zeros ending decimals count
• 12.30 (4) .3400 (4)
• Zeros marked with a bar count
• 100 (3) 12,000 (4)

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When is 0 not a SD?
When it merely shows where the decimal is

• ending whole numbers (with no bar)

12,000 m (2) 305,000 m (3)

• starting a pure decimal

.0035 cm (2) 0.000240 km (3)

• part of the magnitude of scientific notation

3.50 X 105 J (3)
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In calculations, only measured (or values
calculated from measurements) count for SD!The
following would have no SD

• the accepted value used in finding experimental
  error and deviation
• standard/accepted values such as the acceleration
  due to gravity
• constant, non-measured values, such as Newtons
  Universal Gravitational Constant, Pi (p), etc.
• Counted values

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General Rules for Calculating in SD
Find the area in SD
You might assume the answer to be 339.1 m2
because it is a tenth times a tenth
125.6 mx 2.7 m
8 7 9 22 5 1 2 0 3 3 9. 1 2
Estimated Values
In SD, only one estimated value is kept!!
Therefore, the correct, precise and accurate
answer is
340 m2
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Rule for Calculating in SD

• In a calculation done in SD, the answer can never
  be more precise than the least precise part of
  the problem!

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Rules for Adding and Subtracting in SD

• Your answer will have its last SD in the same
  decimal place as the least precise part of the
  problem!
• 11.2 cm 8.66 cm 2.345 cm
• last SD in the tenths column
• 45.3578 L - 23.26 L
• last SD in the hundredths column

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Rules for Multiplying and Dividing in SD
Keep the same number of SD in your answer as the
smallest ( SD) part of your problem!
(12.6 cm)(11.22 cm)(5.8 cm)
3SD4SD2SD? 2 SD in your answer

• (55.6g)
• (11.34cm)(18.345cm)(3.4cm)

2 SD in your answer
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How many significant figures in the following
measurements
1)   400     2)    200.0 3)  0.0001 4)
 218 5)  320    6)    0.00530 7)    22
568 8) 4755.50 

• Complete these addition problems.
• 6.201 cm 7.4 cm 0.68 cm 12.0 cm
• 1884 kg 0.94 kg 1.0 kg 9.778 kg
• c) 16. 156 g 28.2 g 0.0058 g 9.44 g

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• Complete these subtraction problems.
• 10.8 g 8.264 g
• 2104.1 m 463.09 m
• c) 16.50 mL 8.0 mL

• Complete these multiplication problems.
• 10.19 m x 0.013 m
• 3.2145 km x 4.23 km
• (7.50 x 106 m)(2.2 x 10-3 m)

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Complete these division problems. a) 80.23 m
2.4 s   b) 4.301 kg 1.9 cm3
6.6 x 108 m 2.31 x 10-2 s
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Accuracy and Precision in Labwork
Bad accuracy, good precision
Better accuracy, poor precision
Bad accuracy and precision
Good and good
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Accuracy and Precision

• A way of indicating the the degree of uncertainty
  in measurements

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Accuracy?Error

• Refers to how close a measured value comes to the
  accepted value for a quantity
• Absolute error- actual difference
• Ea O - A O?Observed in lab
  (data)
• A?Accepted
  answer
• Relative Error- comparative miss
• Er Ea/A 100

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Precision?Deviation

• Refers to how well several measurements agree
  with each other- about the same average answer
  each trial
• Absolute Deviation- difference each trial is from
  the average answer
• Da O - M M?mean (average of data)
• Relative Deviation- percentage
• Dr Da (average)/M ? 100
• ?Only 1 value for Dr! ?

24
A student performs a lab in which he tries to
find the acceleration due to gravity. His data
produces the following values 9.5 m/s2 , 8.9
m/s2, 9.9 m/s2, and 9.1 m/s2. Find his accuracy
and his precision if the accepted value is 9.8
m/s2.
Ea O - A
Er Ea/A x 100
9.5 - 9.8 m/s2 .3 m/s2
.3/9.8 x 100 3
8.9 - 9.8 m/s2 .9 m/s2
.9/9.8 x 100 9
9.9 - 9.8 m/s2 .1 m/s2
.1/9.8 x 100 1
9.1 - 9.8 m/s2 .7 m/s2
.7/9.8 x 100 7
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M (9.5 8.9 9.9 9.1) m/s2
4
9.4 m/s2
Avg. Da
Da O - M 9.5 - 9.4 .1 m/s2
(.1.5.5.3)m/s2 4
Da O - M 8.9 - 9.4 .5 m/s2
.4 m/s2
Da O - M 9.9 - 9.4 .5 m/s2
Dr Avg Da X 100 M
Da O - M 9.1 - 9.4 .3 m/s2
.4 / 9.4 X 100 4
26
Find the accuracy and precision of the following
lab done to find the density of a sample of lead
(accepted D 11.6 g/cm3)
Trial 1 12.3 g/cm3 Trial 2 11.0 g/cm3 Trial
3 10.4 g/cm3 Trial 4 12.8 g/cm3 Trial 5 13.1
g/cm3