From the discussion so far, it should be clear that to accomplish your goals in any of the various fields within the natural sciences and engineering, a thorough grounding in the laws of physics is necessary. The reason for this is simply that the laws of physics govern everything in the observable universe at all measurable scales of length, mass, and time. Now, that is easy enough to say, but to come to grips with what it really means, we need to get a little bit quantitative. So, before surveying the various scales that physics allows us to explore, let’s first look at the concept of “order of magnitude,” which we use to come to terms with the vast ranges of length, mass, and time that we consider in this text (Figure 2).
Order of magnitude
The order of magnitude of a number is the power of 10 that most closely approximates it. Thus, the order of magnitude refers to the scale (or size) of a value. Each power of 10 represents a different order of magnitude. For example, 101,102,103 and so forth, are all different orders of magnitude, as are 100=1,10−1,10−2, and 10−3. To find the order of magnitude of a number, take the base-10 logarithm of the number and round it to the nearest integer, then the order of magnitude of the number is simply the resulting power of 10. For example, the order of magnitude of 800 is 103 because log10800 ≈ 2.903, which rounds to 3. Similarly, the order of magnitude of 450 is 103 because log10450 ≈ 2.653 which rounds to 3 as well. Thus, we say the numbers 800 and 450 are of the same order of magnitude: 103. However, the order of magnitude of 250 is 102 because log10250 ≈ 2.397 which rounds to 2.
An equivalent but quicker way to find the order of magnitude of a number is first to write it in scientific notation and then check to see whether the first factor is greater than or less than \(\sqrt {10} = 10^{0.5} ≈ 3. \) The idea is that \(\sqrt {10} = 10^{0.5}\) is halfway between 1 = 100 and 10 = 101 on a log base-10 scale. Thus, if the first factor is less than \(\sqrt{10}\) then we round it down to 1 and the order of magnitude is simply whatever power of 10 is required to write the number in scientific notation. On the other hand, if the first factor is greater than \(\sqrt {10}\) then we round it up to 10 and the order of magnitude is one power of 10 higher than the power needed to write the number in scientific notation. For example, the number 800 can be written in scientific notation as 8 × 102 Because 8 is bigger than \(\sqrt {10} ≈ 3\) we say the order of magnitude of 800 is 102+1 = 103. The number 450 can be written as 4.5 × 102 so its order of magnitude is also 103 because 4.5 is greater than 3. However, 250 written in scientific notation is 2.5 × 102 and 2.5 is less than 3, so its order of magnitude is 102.
The order of magnitude of a number is designed to be a ballpark estimate for the scale (or size) of its value. It is simply a way of rounding numbers consistently to the nearest power of 10. This makes doing rough mental math with very big and very small numbers easier. For example, the diameter of a hydrogen atom is on the order of 10−10 m, whereas the diameter of the Sun is on the order of 109 m, so it would take roughly 109/10−10 = 1019 hydrogen atoms to stretch across the diameter of the Sun. This is much easier to do in your head than using the more precise values of 1.06 × 10−10 m for a hydrogen atom diameter and 1.39 × 109 m for the Sun’s diameter, to find that it would take 1.31 × 1019 hydrogen atoms to stretch across the Sun’s diameter. In addition to being easier, the rough estimate is also nearly as informative as the precise calculation.
Known ranges of length, mass, and time
The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times (given as orders of magnitude) in Figure 3. Examining this table will give you a feeling for the range of possible topics in physics and numerical values. A good way to appreciate the vastness of the ranges of values in Figure 3 is to try to answer some simple comparative questions, such as the following:
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How many hydrogen atoms does it take to stretch across the diameter of the Sun?
(Answer: 109 m/10–10 m = 1019 hydrogen atoms)
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How many protons are there in a bacterium?
(Answer: 10–15 kg/10–27 kg = 1012 protons)
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How many floating-point operations can a supercomputer do in 1 day?
(Answer: 105 s/10–17 s = 1022 floating-point operations)
In studying Figure 3, take some time to come up with similar questions that interest you and then try answering them. Doing this can breathe some life into almost any table of numbers.
Figure 3 This table shows the orders of magnitude of length, mass, and time.
