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#IB-1573

The Scale of Physics

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).

 

Figure a shows a high resolution scanning electron microscope image of gold film. Figure b shows a magnified image of phytoplankton and ice crystals. Figure c shows a photograph of two galaxies.

Figure 2 (a) Using a scanning tunneling microscope, scientists can see the individual atoms (diameters around 10–10 m) that compose this sheet of gold. (b) Tiny phytoplankton swim among crystals of ice in the Antarctic Sea. They range from a few micrometers (1 μm is 10–6 m) to as much as 2 mm (1 mm is 10–3 m) in length. (c) These two colliding galaxies, known as NGC 4676A (right) and NGC 4676B (left), are nicknamed “The Mice” because of the tail of gas emanating from each one. They are located 300 million light-years from Earth in the constellation Coma Berenices. Eventually, these two galaxies will merge into one. (credit a: modification of work by "Erwinrossen"/Wikimedia Commons; credit b: modification of work by Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections; credit c: modification of work by NASA, H. Ford (JHU), G. Illingworth (UCSC/LO), M. Clampin (STScI), G. Hartig (STScI), the ACS Science Team, and ESA)

 

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,  and so forth, are all different orders of magnitude, as are  and  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 which rounds to 3. Similarly, the order of magnitude of 450 is 103 because  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  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 √10=100.5≈3. The idea is that √10=100.5 is halfway between  and  on a log base-10 scale. Thus, if the first factor is less than √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 √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 √10≈3 we say the order of magnitude of 800 is  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 

 

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  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 for a hydrogen atom diameter and for the Sun’s diameter, to find that it would take  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:

  • How many hydrogen atoms does it take to stretch across the diameter of the Sun?
    (Answer: 109 m/10–10 m = 1019 hydrogen atoms)

  • How many protons are there in a bacterium?
    (Answer: 10–15 kg/10–27 kg = 1012 protons)

  • 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.

 

This table of orders of magnitude of length, mass and time has three columns and thirteen rows. The first row is a header row and it labels each column, “length in meters (m),” “Masses in kilograms (kg),” and “time in seconds (s).” Under the “length in meters” column are the following entries: 10 to the minus 15 meters equals diameter of proton; 10 to the minus 14 meters equals diameter of large nucleus; 10 to the minus 10 meters equals diameter of hydrogen atom; 10 to the minus 7 meters equals diameter of typical virus; 10 to the minus 2 meters equals pinky fingernail width; 10 to the 0 meters equals height of 4 year old child, and a drawing of a child measuring himself against a meter stick is included; 10 to the 2 meters equals length of football field; 10 to the 7 meters equals diameter of earth; 10 to the 13 meters equals diameter of solar system; 10 to the 16 meters equals distance light travels in a year (one light year); 10 to the 21 meters equals milky way diameter; 10 to the 26 meters equals distance to edge of observable universe. Under the “Masses in kilograms” column are the following entries: 10 to the -30 kilograms equals mass of electron; 10 to the -27 kilograms equals mass of proton; 10 to the -15 kilograms equals mass of bacterium; 10 to the -5 kilograms equals mass of mosquito; 10 to the -2 kilograms equals mass of hummingbird; 10 to the 0 kilograms equals mass of liter of water, and a drawing of a balance scale with a liter on one side and a 1 kilogram mass on the other is shown; 10 to the 2 kilograms equals mass of person; 10 to the 19 kilograms equals mass of atmosphere; 10 to the 22 kilograms equals mass of moon; 10 to the 25 kilograms equals mass of earth; 10 to the 30 kilograms equals mass of sun; 10 to the 53 kilograms equals upper limit on mass of known universe. Under the “Time in seconds” column are the following entries: 10 to the -22 seconds equals mean lifetime of very unstable nucleus; 10 to the -17 seconds equals time for a single floating point operation in a supercomputer; 10 to the -15 seconds equals time for one oscillation of visible light; 10 to the -13 seconds equals time for one vibration of an atom in a solid; 10 to the -3 seconds equals duration of a nerve impulse; 10 to the 0 equals time for one heartbeat, and a drawing of the heart with a plot of three pulses is shown. The peak of the first pulse is labeled P. The next pulse is larger amplitude and shorter duration. The start of the second pulse is labeled Q, its peak is labeled R, and its end is labeled S. The peak of the third pulse is labeled T. The entries in the column continue as follows: 10 to the 5 seconds equals one day; 10 to the 7 seconds equals one year; 10 to the 9 seconds equals human lifetime; 10 to the 11 seconds equals recorded human history; 10 to the 17 seconds equals age of earth; 10 to the 18 seconds equals age of universe;

Figure 3 This table shows the orders of magnitude of length, mass, and time.