Energy and Human Ambitions on a Finite Planet, 2021a

A.10 Units Manipulation 372
we sought. 34 In this case, we end up with 1.04 × 10 20 J/year, which is
what we were after.
We just carried out unit conversions (in time) in Example A.10.2, when
we multiplied by constructs like 60 s/1 min. The key to unit conversions
is to arrange a fraction expressing the same physical thing in both the numerator
<strong>and</strong> denominator, just using different units. So we’re looking for
equivalent measures. Most of the time, one of them will be 1, numerically,
as in the following example.
34: So units, h<strong>and</strong>led carefully, can provide
important clues as to how to get the problem
right.
Example A.10.3 We might want to convert the 1.04 × 10 20 J/year from
Example A.10.2 into quadrillion Btu per year. We know that 1 Btu is
1,055 J, <strong>and</strong> that a quadrillion is 10 15 . So we arrange the following:
1.04 × 10 20 J
year · 1 Btu
1, 055 J · 1 quadrillion ≈
10 15 100 quadrillion Btu/year.
Finally, units can help guide correct usage of factors in a problem. In
Example A.10.3, what if we did not know whether to divide or multiply
by 1,055? The fact that we wanted to eliminate Joules told us we needed
the Joules in the denominator, <strong>and</strong> so the relation 1 Btu = 1,055 J told us
the 1,055 travels with Joules <strong>and</strong> must be in the denominator.
But what if we are faced with a problem whose application is not as
apparent? 35 Let’s explore how this might go in a less familiar setting.
Example A.10.4 For some inexplicable reason, you put a brick in the
refrigerator, which is 20 ◦ C colder than the brick is, initially, <strong>and</strong> want
to find out how long it will take for the brick to cool off. The brick has
a mass of 2.5 kg, will dump its heat into the refrigerator at a rate of
So
you feel overwhelmed by lack of familiarity, right? Good, because the
units are here to help.
35: ...orwedon’tknowtheformula, which
is no bad thing, as we then have the chance
to construct it from what we know, like a
real expert!
10 W (10 J/s), <strong>and</strong> has a specific heat capacity of 1,000 J/kg/ ◦ C. 36 36: This construction means that kg <strong>and</strong> ◦ C
You want an answer in time units, <strong>and</strong> see one instance in the 10 J/s
rate at which heat leaves the brick. To get seconds “up top,” you want
to make sure the 10 W value is in the denominator. This puts Joules in
the denominator, <strong>and</strong> we don’t want it to survive to the final answer.
We notice Joules in the specific heat capacity thing in the numerator,
so that thing must go in the numerator. Let’s take stock of where that
leaves us.
1 1, 000 J ·
10 J/s kg ·◦ C 100
s
kg ·◦ C
It looks like if we multiply by the mass in kg <strong>and</strong> multiply by the
temperature difference, we’re home free. Doing so results in 5,000
seconds. Whenever possible, try to extract the most context/intuition
out of an answer as you can. Does 5,000 seconds mean a lot to you?
Divide by 60 (or multiply by 1 min
60 s
) to get 83 minutes. Better. Or another
factor of 60 <strong>and</strong> we’re at 1.4 hours. That seems like the most natural
are both in the denominator together.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.