Fundamentals of Physics IDr. Jones
Lecture 1, September 24, 2013:
Scientific Measurement
What is measurement?
You do it constantly and for your entire life.
Humans are homeotherms--our bodies try to keep us at the same constant temperature. In order to do so, it needs some form of indication of the current
Thermoreceptors are nerves in the body that detect change in temperature. Their TRPM8 ion channel varies with temperature; the colder, the more ions it lets pass into the cell, the more electrical
What is the "state" of a "system"?
In Physics, a "physical system" is a piece of the
The "state" of a system are the numerical values of all of the
For example, temperature is a physical quantity that medical doctors use to help determine the state of your body. It is a quantity that describes heat content.
How do we measure?
We construct models of systems--in physics we use mathematics as our building block for such models. Each physical reality of a system, such as how fast it is going, its position in space, and so on, corresponds to a physical quantity such as
We use other physical systems, such as a clock, ruler, thermometer, and so on, to measure these quantities by creating a correspondence. For example, we define a meter stick, and compare everything else to that meter stick.
A quick refresher on Scientific Notation:
| Number | Scientific Notation | Product of | Places after 1st Digit |
| 1 | \(1.0\times 10^0\) | 1 | 0 places |
| 10 | \(1.0\times 10^1\) | 1\(\times\)10 | 1 places |
| 100 | \(1.0\times 10^2\) | 1\(\times\)10\(\times\)10 | 2 places |
| 1,000 | \(1.0\times 10^3\) | 1\(\times\)10\(\times\)10\(\times\)10 | 3 places |
| 10,000 | \(1.0\times 10^4\) | 1\(\times\)10\(\times\)10\(\times\)10\(\times\)10 | 4 places |
| 100,000 | \(1.0\times 10^5\) | 1\(\times\)10\(\times\)10\(\times\)10\(\times\)10\(\times\)10 | 5 places |
| 1,000,000 | \(1.0\times 10^6\) | 1\(\times\)10\(\times\)10\(\times\)10\(\times\)10\(\times\)10\(\times\)10 | 6 places |
SI Units
*Official in every nation except three.Only three nations to not adopt SI units: Burma, Liberia, and the US
Map colored based on year of adoption.
Table 1. SI base units |
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SI Derived Units
All other physical quantities can be described by some combination of the seven units. For example, speed is distance over time and is measured in terms of meters and seconds: \[ [v] = \frac{[m]}{[s]} \] Area is measured in terms of meters squared and has units \([m]^2\) and acceleration is measured in terms of
Precision verses accuracy
Accuracy of a measurement is how
Precision of a measurement tells us how close the results of
The International Organization for Standardization has modified these definitions as of 2008 with ISO 5725-1,
but they haven't quite settled in with general scientific community. They hold
trueness to be how close a measurement is to its true value, precision how close together repeated measurements are, and accuracy to be a combination of the two.
Identify the high accuracy / low precision version and the low accuracy high precision version according to the classic standards? How would you identify these according to ISO 5725?
Significant Figures
The number of sig figs that result from an experiment is the number of digits that do not change with
On the other hand, if I am using a more sophisticated instrument that is calibrated such that I know that all three decimal places can be trusted, then I would count five sig-figs, average and find the maximum difference from the average, and write my result as
Significant Figures
Science is about honesty and we would not say that the book is 31.345 cm long, we would say only that it is 31.345 +/- 0.011 cm long. The true answer is very likely to be somewhere inside the region from
Different branches of science have slight variations on these rules, what you've seen here is a general outline; you will learn how your field does precision/trueness/accuracy in greater detail in more specialized classed. The take-away point is that measurement depends very heavily upon
Reporting Measurement and Uncertainty
Note: If the equations appear jumbled, refresh your browser.
Suppose you own a thermometer which is certified to have a resolution of and a measuring device which is certified to have a resolution of
Counting sig figs: Atlantic-Pacific rule:
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If a decimal point is Present, ignore zeros on the Pacific (left) side. If the decimal point is Absent, ignore zeros on the Atlantic (right) side. Everything else is significant.
OR
Scientific Notation Rule: Convert the number into scientific notation. Any leading or trailing zeros the decimal point bumps past in the conversion will vanish. Everything else is significant.
Counting Sig Figs
| Number | Atlantic-Pacific rule | Scientific notation rule |
| 0.001010 | decimal point Present: ignore zeros on the Pacific side. 4 sig. digits. | In scientific notation: 1.010 × 10-3. 4 sig. digits. The decimal point moved past the three leading zeros; they vanished. |
| 0.30000 | decimal point Present: ignore zeros on the Pacific side. 5 sig. digits. | In scientific notation: 3.0000 × 10-1. 5 sig. digits. The decimal point bumped past the leading zero; it vanished. |
| 100.0000 | decimal point Present: ignore zeros on the Pacific side (none!) 7 sig. digits. | In scientific notation: 1.000000 × 102. The decimal point moved past two zeros, but they aren't trailing zeros; they're in the middle of the number. 7 sig. digits. |
| 12303000 | decimal point Absent: ignore zeros on the Atlantic side. 5 sig. digits. | In scientific notation: 1.2303 × 107. The decimal point moved past the trailing three zeros; they vanished. It moved past the zero between the threes, too, but that's not a trailing or leading zero; it stays. 5 sig. digits. |
References
Note:
Some scientist use the following notation:
330 m means 330 meters plus or minus a few tens of meters,
2 sig figs
330. m means 330 meters plus or minus a few meters,
3 sig figs
330.0 m means 330 meters plus or minus a few tenths of a meter,
4 sig figs
and so on...
Math with sig figs
Suppose we want to determine speed by measuring how far an object and
dividing that by our measurement for how long it took to get that far.
How to we preserve sig-figs?
Multiplying and Dividing: Take the numbers and perform the mathematical
operation; at the very end, the answer should have the same number of sig-figs
as the smallest number of sig-figs any of the input numbers had. Round
if necessary.
\[ 3.3 \times 2 = 6.6 \ \rightarrow \ 6 \]
Math with sig figs
Suppose we want to determine the total length of two wires measured
with two different measuring devices with different precisions.
How to we preserve sig-figs?
Adding and subtracting: Perform the mathematical calculation. The answer
should have the same number of places after the decimal as does the input
which has the smallest number of decimal places. Round if necessary.
\[5.27 + 3,000.1 = 3,005.37 \ \rightarrow \ 3,005.4 \]
Sig Fig practice
Note: If the equations appear jumbled, refresh your browser.
We divide the found distance by the measured time, entering the numbers in our calculator as given to find; we then round to the smallest number of sig-figs in the inputs (2 for the time measurement).