Download - Measurement and Significant Digits. >>>>>>>>>>>>>>>>>>>>> object ---------|---------|---------|---------| cm ruler 10 11 12 13 How do we record the length.

Measurement and Significant Digits


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


>>>>>>>>>>>>>>>>>>>>> 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


>>>>>>>>>>>>>>>>>>>>> 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


>>>>>>>>>>>>>>>>>>>>> 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.


>>>>>>>>>>>>>>>>>>>>> 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


reflect the accuracy of a measurement

Significant Digits


reflect the accuracy of a measurement Depends on many factors: apparatus used, skill of

experimenter, number of measurements...

Rules for counting sigs


1) 0.00254 s


1) 0.00254 s 3 significant figures or 3 digit accuracy


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.



first non-zero digit going left to right.2) 1004.6 kg



first non-zero digit going left to right.2) 1004.6 kg 5 significant digits or 5 digit accuracy



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.



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



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



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


4. 8000 m/s


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


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.



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



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



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



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



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



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 tells us how close a

measurement is to the actual or accepted value

Precision




Precision tells us how close

repeated measurements of a quantity are to each other




Depends on many factors: experiment design, apparatus used, skill of experimenter, number of measurements...






Depends on many factors: experiment design, apparatus used, skill of experimenter, number of measurements...



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 =


* * * * * * * * * * hole * * * * * * *@* * * * * * * *Red golfer = good precision and poor accuracy Blue golfer =Green golfer =


* * * * * * * * * * hole * * * * * * *@* * * * * * * *Red golfer = good precision and poor accuracy Blue golfer = poor precision and poor accuracyGreen golfer =


* * * * * * * * * * 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 not measured quantities and therefore are considered as “exact” numbers with an infinite number of significant digits

Formula Numbers


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.


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



A=L X W =(2.148m)(3.09m)



A=L X W =(2.148m)(3.09m) =6.63732 m2



A=L X W =(2.148m)(3.09m) =6.63732 m2 = =6.64 m2



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



P = 2(L + W)



P = 2(L + W) = 2(2.148 m +3.09 m)



P = 2(L + W) = 2(2.148 m +3.09 m) = 2(5.238 m )



P = 2(L + W) = 2(2.148 m +3.09 m) = 2(5.238 m ) = 2(5.24 m)



P = 2(L + W) = 2(2.148 m +3.09 m) = 2(5.238 m ) = 2(5.24 m) =10.5 m



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