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Measurement and Significant Digits
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>>>>>>>>>>>>>>>>>>>>> object ---------|---------|---------|---------| cm ruler 10 11 12 13 How do we record the length of this object? Length of object = _________________ cm ?
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Measurement and Significant Digits >>>>>>>>>>>>>>>>>>>>> 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
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Measurement and Significant Digits >>>>>>>>>>>>>>>>>>>>> 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 >>>>>>>>>>>>>>>>>>>>> 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 >>>>>>>>>>>>>>>>>>>>> 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...
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Rules for counting sigs
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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
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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
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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.
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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. 3) 35.00 N
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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. 3) 35.00 N4 digit accuracy or 4 sig figs
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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. 3) 35.00 N4 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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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 10 3 m/s 8.0 X 10 3 m/s 8.00 X 10 3 m/s 8.000 X 10 3 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 10 3 m/s1 significant figure 8.0 X 10 3 m/s 8.00 X 10 3 m/s 8.000 X 10 3 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 10 3 m/s1 significant figure 8.0 X 10 3 m/s 2 significant digits 8.00 X 10 3 m/s 8.000 X 10 3 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 10 3 m/s1 significant figure 8.0 X 10 3 m/s 2 significant digits 8.00 X 10 3 m/s3 sigs 8.000 X 10 3 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 10 3 m/s1 significant figure 8.0 X 10 3 m/s 2 significant digits 8.00 X 10 3 m/s3 sigs 8.000 X 10 3 m/s4 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 10 3 m/s1 significant figure 8.0 X 10 3 m/s 2 significant digits 8.00 X 10 3 m/s3 sigs 8.000 X 10 3 m/s4 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
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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 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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* * * * * * * * * *hole* * * * * * *@* ** * * * * * Red golfer = Blue golfer = Green golfer =
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Accuracy and Precision: A Golf Analogy * * * * * * * * * *hole* * * * * * *@* ** * * * * * Red golfer = good precision and poor accuracy Blue golfer = Green golfer =
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Accuracy and Precision: A Golf Analogy * * * * * * * * * *hole* * * * * * *@* ** * * * * * 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* * * * * * *@* ** * * * * * 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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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 d=2rC=2πrT=2π√ (l/g) Eff%=W out /W in X 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 A=L 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 A=L 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 A=L X W =(2.148m)(3.09m) =6.63732 m 2
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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 A=L X W =(2.148m)(3.09m) =6.63732 m 2 = =6.64 m 2
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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 A=L X W =(2.148m)(3.09m) =6.63732 m 2 = =6.64 m 2 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)
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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
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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 r 1 = 3.06 mm and r 2 = 4.21 cm. Each sphere has a mass m 1 = 15.2 g and m 2 = 4.1 kg. a) If d = r 1 + r 2, 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 = Gm 1 m 2 /d 2. 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 r 1 = 3.06 mm and r 2 = 4.21 cm. Each sphere has a mass m 1 = 15.2 g and m 2 = 4.1 kg. a) If d = r 1 + r 2, 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 = Gm 1 m 2 /d 2. Given that all measured quantities must be in MKS units, find F in Newtons. F = Gm 1 m 2 /d 2 = (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
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