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Measurements

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Measurements

A very concrete methods of dealing with the description and understanding of nature

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Measurements

A very concrete method of dealing with the description and understanding of nature

Measurements give credibility to the interpretation of;

•Theories•Laws•Principles•Hypothesis

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Measurements

A very concrete way of dealing with the description and understanding of nature

Measurements give credibility to the interpretation of:

•Theories•Laws•Principles•Hypothesis

This credibility is directly related to the accuracy of the measurements

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Estimation (uncertainty)

• All measurements have some degree of estimation.

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It would be difficult to measure with an certainty beyond a millimeter

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The ruler has a limited amount of certainty. Thinner lines could increase the amount of certainty .

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All measuring devices have a certain amount of uncertainty

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The uncertainty of a measurement is determined by the

a. precision of the measurementand

b. accuracy of the measured value

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Precision verses Accuracy

• Precision in its strictest sense refers to the exactness to which the measuring instrument has been manufactured.

• If the same measurement is repeated multiple times with the same instrument, will the measurement be the same each time? (repeatability)

• smaller units would make the instruments more precise (exact_

• Accuracy is how close the measurement is to the true value

• Influenced by :1. person making the measurements 2. precision of the instrument

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Measurement of Uncertainty

• Estimated uncertainty is written with a ± sign; for example:

• Percent uncertainty is the ratio of the uncertainty to the measured value, multiplied by 100:

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Significant Figures

• The number of significant figures is the number of reliably known digits in a number. It is usually possible to tell the number of significant figures by the way the number is written:

• 23.21 cm has 4 significant figures

• 0.062 cm has 2 significant figures (the initial zeroes don’t count)

• 80 km is ambiguous – it could have 1 or 2 significant figures. If it has 3, it should be written 80.0 km. If it has 2 it should be written in scientific notation 8.0 x 101 km

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Significant Figures

• When multiplying or dividing numbers, the result has as many significant figures as the number used in the calculation with the fewest significant figures.

• Example: 11.3 cm x 6.8 cm = 76.84 cm2 77 cm2

• 11.3 cm / 77cm = 0.1467 0.15

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Adding and Subtracting Significant Figures

• When adding or subtracting quantities, leave the same number of decimal places (rounded) in the answer as there are in the quantity with the least number of decimal places.

• Examples

1) 2)

23.1 157

0.546 -5.5

1.45 151.5 152

25.096 25.1

1-4 Measurement and Uncertainty; Significant Figures

Calculators will not give you the right number of significant figures; they usually give too many but sometimes give too few (especially if there are trailing zeroes after a decimal point).

The top calculator shows the result of 2.0 / 3.0.

The bottom calculator shows the result of 2.5 x 3.2.

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Scientific Notation

• Scientific notation is the expression of a number in the “power of 10”36,900 3.69 x 104

allows a number to expressed in significant digits in the coefficient

eliminates the need to write multiple zeros

Know how to add, subtract, multiply, and divide numbers expressed in scientific notation

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Order of Magnitude Estimation

• Method of making an approximate value for a measurement the number is rounded to one (1) significant figure and its power of 10

3,675 m ----- 4 x 103 m5,000 m ----- 5 x 103 m

added together 9 x 103 m

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Order of Magnitude Estimation

• Reasons for 1) Rapid estimation 2) Accurate calculation is not worth the time3) Quick check of an accurate calculation to check

for large errorsa) Check the accuracy of the exponent

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Order of Magnitude Example

• Find the volume (V) of a lake o Lake has

oAverage depth of 65 mo Surface area of 52,500 m2

Volume = area x depth = (7 x 101m) x (5 x 104m2) = 3.5 x 106m3