Measurement and Significant Digits

>>>>>>>>>>>>>>>>>>>>> object | | | | cm ruler How do we record the length of this object? Length of object = _________________ cm ?

Measurement and Significant Digits >>>>>>>>>>>>>>>>>>>>> object | | | | cm ruler How do we record the length of this object? Length of object = 12.2 or 12.3 cm

Measurement and Significant Digits >>>>>>>>>>>>>>>>>>>>> object | | | | cm ruler 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 | | | | cm ruler 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 | | | | cm ruler 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

1) s

Rules for counting sigs 1) s 3 significant figures or 3 digit accuracy

Rules for counting sigs 1) 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) 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) kg

Rules for counting sigs 1) 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) kg 5 significant digits or 5 digit accuracy

Rules for counting sigs 1) 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) kg 5 significant digits or 5 digit accuracy Zeros between non-zero digits do count.

Rules for counting sigs 1) 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) kg 5 significant digits or 5 digit accuracy Zeros between non-zero digits do count. 3) N

Rules for counting sigs 1) 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) kg 5 significant digits or 5 digit accuracy Zeros between non-zero digits do count. 3) N4 digit accuracy or 4 sig figs

Rules for counting sigs 1) 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) kg 5 significant digits or 5 digit accuracy Zeros between non-zero digits do count. 3) N4 digit accuracy or 4 sig figs Trailing zeros to the right of the decimal do count.

A “Tricky” Counting Sigs Rule

m/s

A “Tricky” Counting Sigs Rule m/s Not sure how many sigs: Ambiguous

A “Tricky” Counting Sigs Rule 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 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 X 10 3 m/s

A “Tricky” Counting Sigs Rule 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 X 10 3 m/s

A “Tricky” Counting Sigs Rule 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 X 10 3 m/s

A “Tricky” Counting Sigs Rule 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 X 10 3 m/s

A “Tricky” Counting Sigs Rule 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 X 10 3 m/s4 sig figs or 4 digit accuracy

A “Tricky” Counting Sigs Rule 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 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

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

* * * * * * * * * *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

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=2rC=2πrT=2π√ (l/g) Eff%=W out /W in X 100

Weakest Link Rule for Multiplying and Dividing Measured Quantities Example:A rectangular deck is 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 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 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 m long and 3.09 m wide. Find the area of the rectangular deck A=L X W =(2.148m)(3.09m) = m 2

Weakest Link Rule for Multiplying and Dividing Measured Quantities Example:A rectangular deck is m long and 3.09 m wide. Find the area of the rectangular deck A=L X W =(2.148m)(3.09m) = m 2 = =6.64 m 2

Weakest Link Rule for Multiplying and Dividing Measured Quantities Example:A rectangular deck is m long and 3.09 m wide. Find the area of the rectangular deck A=L X W =(2.148m)(3.09m) = 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.

Weakest Link Rule for Adding and Subtracting Measured Quantities Example:A rectangular deck is 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 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 m long and 3.09 m wide. Find the perimeter of the rectangular deck P = 2(L + W) = 2(2.148 m m)

Weakest Link Rule for Adding and Subtracting Measured Quantities Example:A rectangular deck is m long and 3.09 m wide. Find the perimeter of the rectangular deck P = 2(L + W) = 2(2.148 m m) = 2(5.238 m )

Weakest Link Rule for Adding and Subtracting Measured Quantities Example:A rectangular deck is m long and 3.09 m wide. Find the perimeter of the rectangular deck P = 2(L + W) = 2(2.148 m m) = 2(5.238 m ) = 2(5.24 m)

Weakest Link Rule for Adding and Subtracting Measured Quantities Example:A rectangular deck is m long and 3.09 m wide. Find the perimeter of the rectangular deck P = 2(L + W) = 2(2.148 m 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 m long and 3.09 m wide. Find the perimeter of the rectangular deck P = 2(L + W) = 2(2.148 m 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 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 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.

☺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 cm = 3.06 X m X m = X m = 4.52 X m

☺Review Question b)The constant G = 6.67 X 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 )(15.2 g)(4.1 kg)/(4.52 X m) 2 = (6.67 X )(15.2 x kg)(4.1 kg)/(4.52 X m) 2 = X N = 2.0 X N