Notes
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Uncertainty and Significant Figures |
1. The general scheme of things
2. A law is an analytic statement, usutally with an empirically determined constant.
It is a concise but general statement about how nature behaves.
To be called a law, it must be
experimentaly valid over a wide range of observed phenomena.
3. Principle. Used for less general statements
than those in a law.
4. A theory may contain a set of laws, or a
theory may be implied from an empirically determined law.
5. Examples
Ø Theories
Astronomy: Big Bang
Biology: Evolution
Mathematics: Chaos
Ø
Laws
Kepler’s three laws of
planetary motion
Newton’s three laws of motion
Conservation laws: mass,
energy, momentum
Ø Principles
Heisenberg’ Uncertainty Principle
Archimedes’ Principle (buoyancy)
6. The "Scientific Method"
Ø From observations, determine the nature of the phenomenon that is of interest
Ø Develop one or more hypotheses, or educated guesses, to explain this phenomenon.
This hypotheses should be predictive - given a set of circumstances, the hypothesis should predict an outcome
Ø Devise experiments to test the hypotheses. All valid scientific hypotheses should be testable
Ø Analyze the experimental results and determine to what degree the results fit the predictions of the hypothesis
Ø Modify if necessary and repeat
Uncertainty and Significant Figures
7. No measurement is absolutely precise.
8. It is absurd to claim a precision greater than obtainable by either the measuring device or computation method used.
9. Uncertainty
Ø Estimated
Ø Percent
10. Significant figures
Ø The number of reliably known digits in a number. Do not count place holder 0's that merely show where the decimal goes
Example: .00002 has 1 significant figure.
Ø Multiplication and Division
The final result of a multiplication or division should have only as many digits as the number with the least number of significant
figures used in the calculation
Example: 11.3 x 6.8 = 76.8400 (on a calculator). The answer should be expressed as 77 (rounding up and using 2 significant
figures in 6.8
11. The International System of Units (le Système International d'unités), is the official name for the modern metric system. It is usually
referred to by its abbreviated name SI.
The metric system was conceived by a group of scientists (among them, Antoine-Laurent Lavoisier, who is known as the "father of
modern chemistry") who had been commissioned by Louis XVI of France to create a unified and rational system of measures.
After the French Revolution, the system was adopted by the new government.[7] On August 1, 1793, the National Convention adopted
the new decimal metre with a provisional length as well as the other decimal units with preliminary definitions and terms.
SI Base Units
| Name | Symbol | Quantify | Note |
| meter | m | length | 1 meter @ 39 inches |
| kilogram | kg | mass | 1 kg @ 2.2 pounds |
| second | s | time | |
| ampere | A | electric current | |
| kelvin | K | thermodynamic temperature | |
| candela | cd | luminous intensity | |
| mole | mol | amount of substance |
A prefix may be added to a unit to produce a multiple of the original unit. All multiples are integer powers of ten. For example, kilo- denotes a multiple of a thousand and milli- denotes a multiple of a thousandth; hence there are one thousand millimeters to the meter and one thousand meters to the kilometer. The prefixes are never combined: a millionth of a kilogram is a milligram not a microkilogram.
The text discusses units that are derived from these base units (base units are derived from a standard).
12. Order of magnitude calculations are useful for rapid estimating. For example, to determine if an answer or a claim is at least believable.
13. In performing rough calculations, estimates, or comparisons, we occasionally round off a number to zero significant figures -
which is the nearest power of 10. A number rounded to the nearest power of 10 is called an order of magnitude.
14. For example
Assume the average height of a human being is about 1.7 meters (about 5'7"). For the sake of simplicity, let's round off 1.7 meters
to the nearest power of 10, which is 100 m (or 1 m). We are not saying that the average height of a person is a mere 1 meter, but rather
the average height is closer to 1 meter (or 100 meters) than it is to 10 meters (or 101 meters).
Similarly, rounding the height of an ant, which is about 8 x 10-4 meters, to the nearest power of ten results in 10-3 meters.
Another way of saying this is that the order of magnitude of the height of an ant is 10-3 meters.
Now, if we compare the height of a human being (100 meters) with the height of an ant (10-3 meters), we come up with the ratio human
height/ant height = 100/10-3 = 100 - (-3) = 103 = 1000.
A human being is roughly 1000 times (or 103 times) taller than an ant.
In other words, a human being is 3 orders of magnitude (3 powers of 10) taller than an ant. The table below shows some
interesting comparisons.