Measurement and Significant Digits

1
Measurement and Significant Digits
2
Measurement and Significant Digits

• gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt object
• ------------------------------------
  cm ruler
• 10 11 12 13
• How do we record the length of this object?
• Length of object _________________ cm ?

3
Measurement and Significant Digits

• gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt object
• ------------------------------------
  cm ruler
• 10 11 12 13
• How do we record the length of this object?
• Length of object 12.2 or 12.3 cm

4
Measurement and Significant Digits

• gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt object
• ------------------------------------
  cm ruler
• 10 11 12 13
• How do we record the length of this object?
• Length of object 12.2 or 12.3 cm 12.3 0.1
  cm

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Measurement and Significant Digits

• gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt object
• ------------------------------------
  cm ruler
• 10 11 12 13
• How do we record the length of this object?
• Length of object 12.2 or 12.3 cm 12.3 0.1
  cm
• Recorded measured quantities include only
  digits known for certain plus only one estimated
  or uncertain digit.

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Measurement and Significant Digits

• gtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgtgt object
• ------------------------------------
  cm ruler
• 10 11 12 13
• How do we record the length of this object?
• Length of object 12.2 or 12.3 cm 12.3 0.1
  cm
• Recorded measured quantities include only digits
  known for certain plus only one estimated or
  uncertain digit.
• These digits are called Significant Digits
  (Figures) or simply sigs or sig figs

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Significant Digits

• when recording measurements, physicists only
  record the digits that they know for sure plus
  only one uncertain digit

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Significant Digits

• when recording measurements, physicists only
  record the digits that they know for sure plus
  only one uncertain digit
• reflect the accuracy of a measurement

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Significant Digits

• when recording measurements, physicists only
  record the digits that they know for sure plus
  only one uncertain digit
• reflect the accuracy of a measurement
• Depends on many factors apparatus used, skill of
  experimenter, number of measurements...

10
Rules for counting sigs
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Rules for counting sigs

• 1) 0.00254 s

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Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy

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Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.

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Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg

15
Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg
• 5 significant digits or 5 digit
  accuracy

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Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg
• 5 significant digits or 5 digit
  accuracy
• Zeros between non-zero digits do count.

17
Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg
• 5 significant digits or 5 digit
  accuracy
• Zeros between non-zero digits do count.
• 3) 35.00 N

18
Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg
• 5 significant digits or 5 digit
  accuracy
• Zeros between non-zero digits do count.
• 3) 35.00 N 4 digit accuracy or 4 sig
  figs

19
Rules for counting sigs

• 1) 0.00254 s
• 3 significant figures or 3 digit
  accuracy
• Leading zeros don't count. Start counting sigs
  with the first non-zero digit going left to
  right.
• 2) 1004.6 kg
• 5 significant digits or 5 digit
  accuracy
• Zeros between non-zero digits do count.
• 3) 35.00 N 4 digit accuracy or 4 sig figs
• Trailing zeros to the right of the decimal do
  count.

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A Tricky Counting Sigs Rule
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A Tricky Counting Sigs Rule

• 4. 8000 m/s

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs Ambiguous

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s
• 8.0 X 103 m/s
• 8.00 X 103 m/s
• 8.000 X 103 m/s

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s 1 significant figure
• 8.0 X 103 m/s
• 8.00 X 103 m/s
• 8.000 X 103 m/s

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s 1 significant figure
• 8.0 X 103 m/s 2 significant
  digits
• 8.00 X 103 m/s
• 8.000 X 103 m/s

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s 1 significant figure
• 8.0 X 103 m/s 2 significant
  digits
• 8.00 X 103 m/s 3 sigs
• 8.000 X 103 m/s

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s 1 significant figure
• 8.0 X 103 m/s 2 significant
  digits
• 8.00 X 103 m/s 3 sigs
• 8.000 X 103 m/s 4 sig figs or 4 digit
  accuracy

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A Tricky Counting Sigs Rule

• 4. 8000 m/s Not sure how many sigs
  Ambiguous
• Must write quantities with trailing
  zeros to the left of the decimal in scientific
  notation.
• 8 X 103 m/s 1 significant figure
• 8.0 X 103 m/s 2 significant
  digits
• 8.00 X 103 m/s 3 sigs
• 8.000 X 103 m/s 4 sig figs or 4 digit
  accuracy
• In grade 12, assume given data with trailing
  zeros to the left of the decimal are
  significant...not true in general

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Accuracy vs Precision

• Accuracy

• Precision

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Accuracy vs Precision

• Accuracy
• tells us how close a measurement is to the actual
  or accepted value

• Precision

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Accuracy vs Precision

• Accuracy
• tells us how close a measurement is to the actual
  or accepted value

• Precision
• tells us how close repeated measurements of a
  quantity are to each other

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Accuracy vs Precision

• Accuracy
• tells us how close a measurement is to the actual
  or accepted value
• Depends on many factors experiment design,
  apparatus used, skill of experimenter, number of
  measurements...

• Precision
• tells us how close repeated measurements of a
  quantity are to each other

34
Accuracy vs Precision

• Accuracy
• tells us how close a measurement is to the actual
  or accepted value
• Depends on many factors experiment design,
  apparatus used, skill of experimenter, number of
  measurements...

• Precision
• tells us how close repeated measurements of a
  quantity are to each other
• Depends on how finely divided or closely spaced
  the measuring instrument is...mm ruler is more
  precise than cm ruler

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More on Accuracy vs Precision

• Accuracy
• Reflected in the number of significant digits

• Precision

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More on Accuracy vs Precision

• Accuracy
• Reflected in the number of significant digits

• Precision
• Reflected in the number of decimal places

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Accuracy and Precision A Golf Analogy

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Accuracy and Precision A Golf Analogy

• 
  
• hole
• _at_
  
• Red golfer
• Blue golfer
• Green golfer

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Accuracy and Precision A Golf Analogy

• 
  
• hole
• _at_
  
• Red golfer good precision and poor accuracy
• Blue golfer
• Green golfer

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Accuracy and Precision A Golf Analogy

• 
  
• hole
• _at_
  
• Red golfer good precision and poor accuracy
• Blue golfer poor precision and poor accuracy
• Green golfer

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Accuracy and Precision A Golf Analogy

• 
  
• hole
• _at_
  
• Red golfer good precision and poor accuracy
• Blue golfer poor precision and poor accuracy
• Green golfer good precision and good accuracy

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Formula Numbers
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Formula Numbers

• are found in mathematics and physics equations
  and formulas

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Formula Numbers

• are found in mathematics and physics equations
  and formulas
• are not measured quantities and therefore are
  considered as exact numbers with an infinite
  number of significant digits

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Formula Numbers

• are found in mathematics and physics equations
  and formulas
• are not measured quantities and therefore are
  considered as exact numbers with an infinite
  number of significant digits
• Examples red symbols are formula numbers
• d2r C2pr T2pv (l/g)
• EffWout/WinX 100

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck.

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck
• AL X W

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck
• AL X W
• (2.148m)(3.09m)

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck
• AL X W
• (2.148m)(3.09m)
• 6.63732 m2

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck
• AL X W
• (2.148m)(3.09m)
• 6.63732 m2 6.64 m2

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Weakest Link Rule for Multiplying and Dividing
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the area of the rectangular
  deck
• AL X W
• (2.148m)(3.09m)
• 6.63732 m2 6.64 m2
• Rule When multiplying or dividing or square
  rooting, round the final answer to the same
  number of sigs as the least accurate measured
  quantity in the calculation.

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)
• 2(2.148 m 3.09 m)

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)
• 2(2.148 m 3.09 m)
• 2(5.238 m )

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)
• 2(2.148 m 3.09 m)
• 2(5.238 m ) 2(5.24 m)

57
Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)
• 2(2.148 m 3.09 m)
• 2(5.238 m ) 2(5.24 m) 10.5 m

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Weakest Link Rule for Adding and Subtracting
Measured Quantities

• Example A rectangular deck is 2.148 m long and
  3.09 m wide. Find the perimeter of the
  rectangular deck
• P 2(L W)
• 2(2.148 m 3.09 m)
• 2(5.238 m ) 2(5.24 m) 10.5 m
• Rule When adding or subtracting, round the
  final answer to the same number of decimal places
  as the least precise measured quantity in the
  calculation.

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?Review Question

• Two spheres touching each other have radii given
  by symbols r1 3.06 mm and r2 4.21 cm. Each
  sphere has a mass m1 15.2 g and m2 4.1 kg.
• a) If d r1 r2 , find d in meters
• b) The constant G 6.67 X 10-11 and the force of
  gravity between the spheres in Newtons is given
  by F Gm1m2/d2 . Given that all measured
  quantities must be in MKS units, find F in
  Newtons.

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?Review Question

• Two spheres touching each other have radii given
  by symbols r1 3.06 mm and r2 4.21 cm. Each
  sphere has a mass m1 15.2 g and m2 4.1 kg.
• a) If d r1 r2 , find d in meters
• 3.06 mm 4.21 cm
• 3.06 X 10-3 m 4.21 X 10-2 m
• 4.516 X 10-2 m 4.52 X 10-2 m

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?Review Question

• b) The constant G 6.67 X 10-11 and the force
  of gravity between the spheres in Newtons is
  given by F Gm1m2/d2 . Given that
  all measured quantities must be in MKS units,
  find F in Newtons.
• F Gm1m2/d2
• (6.67 X 10-11)(15.2 g)(4.1 kg)/(4.52 X 10-2
  m)2
• (6.67 X 10-11)(15.2 x 10-3 kg)(4.1
  kg)/(4.52 X 10-2 m)2
• 2.0345876 X 10-9 N 2.0 X 10-9 N