Introduction
Measurements and observations form the base of our scientific knowledge of the world. Some quantities we measure can be calculated in terms of others, while some cannot; those that cannot are the most fundamental quantities. The International System of Units (SI) is based on seven fundamental units from which we can express all other units. Sometimes, we deal in practice with quantities much larger or much smaller than the defined unit. In such cases, it is convenient to use prefixes. Last but not least, we look at significant figures and guidelines for their use in computations and the scientific notation.
Fundamental and Derived Units
Physics is exciting when it can predict how nature will behave in one situation on the basis of data obtained in another situation. Therefore, a great deal of effort goes into making measurements as accurate and reproducible as possible in a given situation.
For example, the atomic mass unit u has a value of one-twelfth (1/12) the mass of the carbon isotope C12. As long as we stay within the boundaries of atomic physics, the picture is nice and clear: The hydrogen atom has (approximately) a one-unit mass, 1 u; the oxygen atom has a mass of 16 u; and so forth. And these data can be obtained from nice atomic collision measurements. The ugly part (due to our somewhat arbitrary definition of u) comes when we want to link this to another mass standard, such as the kilogram. We find the following:
1 u = 1.66067 · 110–27 kg
and all our nice atomic data transform into strings carrying four or five decimals. It can make us wonder why the kilogram is the only one of the seven fundamental units still defined in terms of a physical platinum-iridium cylinder artifact kept in Paris.
Often, it is possible to split the units of measurement into simpler units. For example, a unit of area, say, a square foot, may be actually the area of a square that is 1 foot long by 1 foot wide. In other words, we can reduce the area measurement to a length measurement.
But there are times when this trick is not possible anymore. For example, to measure the length, we need to actually perform a direct comparison to a convenient unit of length. We call length a fundamental quantity and the unit of length a fundamental unit. Other well-known fundamental units are the ones measuring time and mass. We should think of these base units as some of the building blocks that we use in our dimensional puzzle to get units for other important physical quantities such as speed, force, or energy.
Choosing Units
Choosing a unit for a particular quantity is a highly subjective exercise; choosing a good one will allow accurate measurements that are easily accessible and reproducible so that everybody understands when we document our observations.
International System of Units (SI)
Revolutionary times call for revolutionary units! The main ideas for an international system of units were born out of the French Revolution, when the metric system and the kilogram were introduced to measure lengths and weights. Another revolution, and a New World, had already adopted a decimal system—you guessed it, in the United States, where a dime has ten cents, a dollar has ten dimes, and so forth. But the United States fell short of putting Thomas Jefferson's decimal idea in practice outside the financial system.
Eventually, these revolutionary ideas prevailed and took over the world, when in 1960, the International System of Units (S1) was adopted by the 11th General Conference on Weights and Measurements.
This system defines seven fundamental quantities and their associated units: length and meter, mass and kilogram, time and second, electric current and ampere, temperature and degree Kelvin, luminous intensity and candela, and, last but not least, the amount of substance and the mole. All other units can be consistently derived from these seven fundamental units.
The metric system had a strong influence on the scientific community. Therefore, eventually the first coherent system of units and a precursor to SI showed up in the 1860s as the CGS system. It defined the centimeter, gram, and second (or CGS) as fundamental units for length, mass, and time. Another hundred years had to pass for us to fully realize we needed another four more fundamental units to get the whole picture, that is, so that we could derive any other measurement in terms of these seven fundamental quantities.
A coherent system of units enforces a certain set of rules for manipulating the results of the actual measurements. To get sensible results, only quantities with the same unit can be added or subtracted.
When listing data and performing calculations, do the following:
- Write down the units explicitly, convert everything to the same unit, and use the same prefixes throughout a calculation.
- Treat all units as algebraic quantities. In particular, when identical units are divided, they cancel each other out.
- Use conversion factors between different unit systems. For instance, the conversion factor of 1 lb = 0.4536 kg might be used to go back and forth between 51 and British units of mass.
- Check to see that your calculations are correct by verifying that the units combine algebraically to give the desired unit for the answer. Remember, only quantities with the same units can be added or subtracted.
Some usual practical conversion factors between different units in the SI and the u.s. or British system are as follows:
1 inch = 2.54 centimeters
1 yard = 3 feet = 36 inches = 91.44 centimeters
1 mile = 1.609 kilometers
1 pound (lb) = 0.4536 kilograms
1 quart = 946 milliliters = 0.946 liters
Example
You travel to Canada where the highway posted speed limit is 100 km/h. How fast are you allowed to go in mi/h?
Solution
Obviously, your car speedometer shows you values in miles per hour, so we have to figure out this km/h value. Reversing the relation 1 mi = 1.609 km, we get:
1 km = 0.621 mi
Therefore:
100 km/h = 100 . (l km/l h) = 100·
(0.621 mill h) = 62.1 mi/h
100 km/h = 62.1 mi/h
Unit Prefixes and Use
We already mentioned that S1 is a decimal-based system, where each unit may be divided or multiplied by 10 to form the next subunit or multiple units. Initially, it was decided that multiples would use root words coming from Greek: deka for 10, hecto for 100, kilo for 1,000. And submultiples would use Latin root words: deci for one-tenth, centi for one-hundredth, milli for one-thousandth, and so on. Eventually, our needs expanded so much that now there are 20 usual prefixes used to form multiples and submultiples of SI units, and the initial rules were relaxed somehow. Here are the 12 most usual of these prefixes.
A few rules on the use of prefixes follow:
- A prefix should always be followed by the unit that it divides or multiplies.
- Prefixes cannot be combined: We use pm (picometer) and not μμm (micro-micro-meter).
- The only SI unit that has a prefix already in its name is the kilogram. In this case, the prefix names are used with the name gram and the symbol g. Another exception of the previously mentioned rule exists: It is also usual to say ktan (kiloton) instead of megagram.
The use of prefixes allows us to conveniently scale up or scale down our units to the task at hand and for simplifying the calculations involving our measurements. It is more convenient to use centimeters when measuring the circumference of your waist or the length of your arm (if you are a tailor, for example) and to use kilometers when measuring the distance between two cities than the other way around.
Significant Figures
We explore and know our world by making a subjective use of our five senses (vision, touch, hearing, smell, and taste). This is one of the reasons that there is no such thing as an exact or absolute measurement. In our measurements, we are sometimes held back by the quality of the tools we use, by our own skills, or by the methods we use to gather our data. To overcome some of these limitations, a simple solution exists: Repeat the measurement several times, and then somehow figure out which of the measurements we get as our results are significant. For a single direct measurement, the rule of thumb is that we can accurately (precisely) estimate by up to half of the smallest division on the scale of our instrument, and we can take a best guess up to one-tenth of this smallest division. For example, using a vernier caliper that can measure exactly to one-tenth of a millimeter to find the diameter of a rod, the bestcase es'jmate of our precision would be 0.01 mm. We choose to accept a result of 3.23 mm for the diameter. But given our tool, if someone tells us he or she measured the rod diameter to be 3.22778 mm, we would have to be very skeptical about the last three digits. We say in this case our measurement has only three significant figures, because only these first three figures are known with a certain degree of confidence. And certainly, the last digit is the least accurate; this is where our confidence is somewhat lower than for the other digits.
How Many Digits?
As a rule: The last digit we keep (the 6 in bold) is rounded up to the next value (in this case, it would be 7) if the first of the insignificant figures being dropped is 5 or greater; if it's less than 5, we leave it unchanged (which is the case here).
Example 1
You measure the mass of a small crucible on a precision electronic balance that is rated to measure down to one-tenth of a milligram. You get the following results when you perform this measurement three times: 3.0757 g, 3.0754 g, and 3.0758 g. What are the significant figures in your results?
Solution 1
From these data, it is clear that even if you claim to have a very precise instrument, something happens during the measuring process that limits your precision to only milligrams. Maybe you lean on the bench where the balance sits, or air currents affect your experiment, or some hardware problem exists with your balance. Averaging your measurements, you can say that your crucible has 3.0756333 g. But wait: Your balance can measure only up to a tenth of a milligram; in other words, we can have only five significant figures in our measurement. We have to somehow limit our answer to only five digits.
Therefore, we quote our result for the crucible weight as 3.0756 g. From these figures, we know for sure the first four digits are more accurate than the last one, which is just an estimate—a best effort to guess the exact value.
To get more accurate data, we could do the following:
- Try to be more careful when taking the actual data, and see if this helps.
- Repeat the experiment more than ten times, and use mathematics (statistics) to get a best-guess average of the result (which is beyond the scope of this text).
- Try both of the previous suggestions.
- Upgrade your hardware. Buy a more expensive balance that has an even better precision.
An important rule to be observed when we use experimental data to further calculate other quantities is that the worst precision of our measurements propagates to the result: The result has as many significant
Example 2
What is the area of a room that is 9.66 feet by 5.2 feet?
Solution 2
The area is the product of the two measured dimensions. If we keep the unit feet and look only for the significant figures after multiplication, we find the area is 50.232 square feet. However, because one of the factors has only two significant figures, we have to limit our result to only two figures as well. Therefore, we say that the area of the room is 50 square feet.
Introduce Rule
Rule for addition and subtraction: the result of addition and subtraction has the same number of decimal places as the least number of decimal places of the original numbers.
30.25 + 2.01089 = 32.26 (only second decimial place is considered)
Rule for multiplication and division: the result of multiplication or division has the same number of significant figures the least number of significant figures in any of the original numbers.
30.25 × 2.01089 = 60.83 (only four significant figures are considered)
Scientific Notation
One way to express more elegantly very small (atomic size) or very large (astronomic size) numbers is to use a notation that allows us to multiply a standard number (say, between 1 and 10) with a multiple. This format is called scientific notation.
The usefulness can be seen also in expressing results with the correct number of significant figures, as discussed in the previous section.
Scientific Notation
x . 10y (x is a number between 1 and 10, and y can be any power)
Example
A long rod has a radius of 5.2 mm and a length of 1.20 m. Express the volume in scientific notation.
Solution
We first convert the data to the same unit of length.
r=5.2 mm =5.2 mm ·10–3 m/1 mm =5.2 ·10–3 m
L = 1.20 m
V=A· L=?
V = 1T · r2 . L = π · (5.2 · 10–3 m)2 · 1.20 m = 32 ·10– 6m3
And to finish the problem, we must express this result in scientific notation:
V= 32 · 10–6 m3 = 3.2 · 10–5m3
V=3.2 · 10–5m3
Practice problems of this concept can be found at: Units and Prefixes Practice Questions
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From Physics Success in 20 Minutes A Day. Copyright © 2006 by LearningExpress, LLC. All Rights Reserved.
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