Energy and Human Ambitions on a Finite Planet, 2021a

A.10 Units Manipulation 371
More complicated arrangements follow the same rules. For example, the
force of drag 30 on an object moving at speed v through a medium of
density ρ is F drag 1 2 c DAρv 2 , where A is the frontal (cross-sectional)
area of the object <strong>and</strong> c D is the dimensionless drag coefficient. 31 The
dimensions of area are m 2 ; density is kg/m 3 (mass per volume), <strong>and</strong>
velocity is m/s (distance over time). The whole arrangement therefore
has dimensions:
30: This choice is intentionally unfamiliar
<strong>and</strong> complicated–looking to demonstrate
that units can help bring a sense of order
<strong>and</strong> correctness even in alien contexts.
31: The drag coefficient, c D , is usually in
the range 0.3–1.
m 2 · kg ( ) m 2
m · m2 · kg · m 2
3 s m 3 · s 2
kg · m4
m 3 · s 2
kg · m
s 2 .
The end result matches the definition of Newtons, <strong>and</strong> can be verified by
the (possibly familiar) F ma form of Newton’s Second Law, 32 whereby
we have mass in kg times acceleration in m/s 2 making kg · m/s 2 .
32: Force equals mass times acceleration
When performing a chain of multiplications or divisions, we can carry
the units around <strong>and</strong> multiply, divide, or (hopefully) cancel them as we
go.
Example A.10.2 Let’s say we want to know how much energy, in
Joules, the U.S. uses in a year based on knowledge that the average
citizen accounts for 10,000 W (a Watt is a Joule per second) <strong>and</strong> the
U.S. has 330 million people. First, let’s work with the 10,000 W per
person metric:
10 4 J/s
person
10 4 J
person · s ,
where we have just moved the seconds into the denominator to
multiply “person” (order doesn’t matter). Most of the problem is in
going from seconds to years. It would like like this:
10 4 J
person · s ·
60 s 60 min 24 hour
· ·
1 min 1 hour 1day
· 365 day
1 year
Notice that each of the factors we multiply, even though they carry a
non-unity numeric value, are essentially identities that describe equal
intervals on top <strong>and</strong> bottom, in differing units. 33 So we are effectively
multiplying by 1 repeatedly in a unit conversion process.
33: E.g., 24 hours <strong>and</strong> 1 day describe the
same time interval.
Also note that the chain we construct allows a boatload of cancellations,
as almost all units present appear in both the numerator <strong>and</strong>
denominator once. The only ones that do not are J in the numerator
<strong>and</strong> year <strong>and</strong> person in the denominator. When we carry out the
multiplication above <strong>and</strong> cancel units, we find that we are left with:
3.15 × 10 11 J
year · person .
Oops, the units are helping us here by reminding us that we need
to multiply by the population (3.3 × 10 8 persons) to get the answer
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.