measurement and significant digits....

Measurement and Significant Digits

Measurement and Significant Digits

>>>>>>>>>>>>>>>>>>>>> object ---------|---------|---------|---------| cm ruler 10 11 12 13 How do we record the length of this object?Length of object = _________________ cm ?

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

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

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 cmRecorded measured quantities include only digits

known for certain plus only one estimated or uncertain digit.

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 cmRecorded 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 accuracyLeading 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 accuracyLeading 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 accuracyLeading 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 accuracyLeading 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 accuracyZeros between non-zero digits do count.

Rules for counting sigs

1) 0.00254 s 3 significant figures or 3 digit accuracyLeading 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 accuracyZeros between non-zero digits do count.3) 35.00 N

Rules for counting sigs

1) 0.00254 s 3 significant figures or 3 digit accuracyLeading 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 accuracyZeros 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 accuracyLeading 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 accuracyZeros between non-zero digits do count.3) 35.00 N 4 digit accuracy or 4 sig figsTrailing 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 1 significant figure 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 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 accuracyIn 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 accuracyGreen golfer =

Accuracy and Precision: A Golf Analogy

* * * * * * * * * * hole * * * * * * *@* * * * * * * *Red golfer = good precision and poor accuracy Blue golfer = poor precision and poor accuracyGreen 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 numbersd=2r C=2πr T=2π√ (l/g)

Eff%=Wout/WinX 100

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 mRule: 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/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 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/d

2 . 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