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Measurement and Significant Digits
Measurement and Significant Digits
>>>>>>>>>>>>>>>>>>>>> object
---------|---------|---------|---------|
10
11
12
cm ruler
13
How do we record the length of this object?
Length of object = _________________ cm ?
Measurement and Significant Digits
>>>>>>>>>>>>>>>>>>>>> object
---------|---------|---------|---------|
10
11
12
cm ruler
13
How do we record the length of this object?
Length of object = 12.2 or 12.3 cm
Measurement and Significant Digits
>>>>>>>>>>>>>>>>>>>>> object
---------|---------|---------|---------|
10
11
12
cm ruler
13
How do we record the length of this object?
Length of object = 12.2 or 12.3 cm = 12.3 ± 0.1 cm
Measurement and Significant Digits
>>>>>>>>>>>>>>>>>>>>> object
---------|---------|---------|---------|
10
11
12
cm ruler
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.
Measurement and Significant Digits
>>>>>>>>>>>>>>>>>>>>> object
---------|---------|---------|---------|
10
11
12
cm ruler
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”
Significant Digits

when recording measurements, physicists only record
the digits that they know for sure plus only one
uncertain digit
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
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...
Rules for counting sigs
Rules for counting sigs
1)
0.00254 s
Rules for counting sigs
1)
0.00254 s
3 significant figures or 3 digit accuracy
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.
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
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
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.
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
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
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.
A “Tricky” Counting Sigs Rule
A “Tricky” Counting Sigs Rule
4.
8000 m/s
A “Tricky” Counting Sigs Rule
4.
8000 m/s
Not sure how many sigs: Ambiguous
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.
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
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
1 significant figure
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
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
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
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
Accuracy vs Precision
Accuracy
Precision
Accuracy vs Precision
Accuracy

tells us how close a
measurement is to the
actual or accepted value
Precision
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
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
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
More on Accuracy vs Precision
Accuracy

Reflected in the number
of significant digits
Precision
More on Accuracy vs Precision
Accuracy

Reflected in the number
of significant digits
Precision

Reflected in the number
of decimal places
Accuracy and Precision: A Golf Analogy
Accuracy and Precision: A Golf Analogy
* * * *
*
* * *
*
*
*
hole
*
*
Red golfer =
Blue golfer =
Green golfer =
* *@* *
* * *
*
*
*
*
*
Accuracy and Precision: A Golf Analogy
* * * *
*
* * *
*
*
*
hole
*
*
* *@* *
*
*
*
*
* * *
Red golfer = good precision and poor accuracy
Blue golfer =
Green golfer =
*
Accuracy and Precision: A Golf Analogy
* * * *
*
* * *
*
*
*
hole
*
*
* *@* *
*
*
*
*
* * *
Red golfer = good precision and poor accuracy
Blue golfer = poor precision and poor accuracy
Green golfer =
*
Accuracy and Precision: A Golf Analogy
* * * *
*
* * *
*
*
*
*
hole
*
* *@* *
*
*
*
*
* * *
Red golfer = good precision and poor accuracy
Blue golfer = poor precision and poor accuracy
Green golfer = good precision and good accuracy
*
Formula Numbers
Formula Numbers

are found in mathematics and physics equations and
formulas
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
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
d=2r
C=2πr
Eff%=Wout/WinX 100
T=2π√ (l/g)
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.
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
A=L X W
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
A=L X W
=(2.148m)(3.09m)
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
A=L X W
=(2.148m)(3.09m)
=6.63732 m2
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
A=L X W
=(2.148m)(3.09m)
=6.63732 m2 = =6.64 m2
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
A=L 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.
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
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)
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)
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 )
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)
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
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.
☺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.
☺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
☺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