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5: Dimensions

  • Page ID
    24113
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    In 1906, Los Angeles received 540 millimeters of precipitation (rain, snow, sleet, and hail).

    Is this rainfall large or small?

    On the one hand, 540 is a large number, so the rainfall is large. On the other hand, the rainfall is also 0.00054 kilometers, and 0.00054 is a tiny number, so the rainfall is small. These arguments contradict each other, so at least one must be wrong. Here, both are nonsense.

    A valid argument comes from a meaningful comparison—for example, comparing 540 millimeters per year with worldwide average rainfall—which we estimated in Section 3.4.3 as 1 meter per year. In comparison to this rainfall, Los Angeles in 1906 was dry. Another meaningful comparison is with the average rainfall in Los Angeles, which is roughly 350 millimeters per year. In comparison, 1906 was a wet year in Los Angeles.

    In the nonsense arguments, changing the units of length changed the result of the comparison. In contrast, the meaningful comparisons are independent of the system of units: No matter what units we select for length and time, the ratio of rainfalls does not change. In the language of symmetry, which we met in Chapter 3, changing units is the symmetry operation, and meaningful comparisons are the invariants. They are invariant because they have no dimensions. When there is change, look for what does not change: Make only dimensionless comparisons.

    This criterion is necessary for avoiding nonsense; however, it is not sufficient. To illustrate the difficulty, let’s compare rainfall with the orbital speed of the Earth. Both quantities have dimensions of speed, so their ratio is invariant under a change of units. However, judging the wetness or dryness of Los Angeles by comparing its rainfall to the Earth’s orbital speed produces nonsense.

    Here is the moral of the preceding comparisons. A quantity with dimensions is, by itself, meaningless. It acquires meaning only when compared with a relevant quantity that has the same dimensions. This principle underlies our next tool: dimensional analysis.

    Exercise \(\PageIndex{1}\): Book boxes are heavy

    In Problem 1.1, you estimated the mass of a small moving-box packed with books. Modify your calculation to use a medium moving-box, with a volume of roughly 0.1 cubic meters. Can you think of a (meaningful!) comparison to convince someone that the resulting mass is large?

    Exercise \(\PageIndex{2}\): Making energy consumption meaningful

    The United States’ annual energy consumption is roughly 1020 joules. Suggest two comparisons to make this quantity meaningful. (Look up any quantities that you need to make the estimate, except the energy consumption itself!)

    Exercise \(\PageIndex{3}\): Making solar power meaningful

    In Problem 3.31, you should have found that the solar power falling on Earth is roughly 1017 watts. Suggest a comparison to make this quantity meaningful.

    Exercise \(\PageIndex{4}\): Energy consumption by the brain

    a. The human brain consumes about 20 watts, and has a mass of 1–2 kilograms. a. Make the power more meaningful by estimating the brain’s fraction of the body’s power consumption:

    \[\frac{\textrm{brain power}}{\textrm{basal metabolism}}.\]

    b. Make this fraction even more meaningful by estimating the ratio

    \[\frac{\textrm{the brain's fraction of the body's power consumption}}{\textrm{the brain's fraction of the body's mass}}.\]

    Exercise \(\PageIndex{5}\): Making oil imports meaningful

    In Section 1.4, we estimated that the United States imports roughly 3×109 barrels of oil per year. This quantity needs a comparison to make it meaningful. As one possibility, estimate the ratio

    \[\frac{\textrm{cost of the imported oil}} {\textrm{US military spending to "defend" oil-rich regions}}.\]

    If this ratio is less than 1, suggest why the US government does not cancel that part of the military budget and use the savings to provide US consumers with free imported oil.

    Exercise \(\PageIndex{6}\): Making the energy in a 9-volt battery meaningful

    Using your estimate in Problem 1.11 for the energy in a 9–volt battery, estimate

    \[\frac{\textrm{energy content of the battery}}{\textrm{cost of the battery}}.\]

    Compare that quotient to the same quotient for electricity from the wall socket.


    This page titled 5: Dimensions is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Sanjoy Mahajan (MIT OpenCourseWare) .

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