Fermi Questions
Larry Weinstein, Column Editor
Old Dominion University, Norfolk, VA 23529;
weinstein@odu.edu.
Solutions for Fermi Questions, December 2011
w Question 1: Dry your hands!
What are the relative costs of drying your hands on a
cloth towel, a paper towel, and with a hot-air hand dryer?
(Thanks to Chuck Adler of St. Mary’s College of Maryland
for suggesting the question.)
Answer: At home almost everybody dries their hands
on cloth towels, but commercial bathrooms offer either
paper towels, hot-air hand dryers, or both. Let’s estimate
the costs of each.
A bathroom cloth hand towel costs around $5 (more
than $1 and less than $20) and lasts about 10 years (more
than one year and less than a century). It is typically used
about five times per day and washed about once every 20
days (less than once a day and more than once a year).
The cost of washing a load of clothes is about $3 (more
than $1 and less than $10) and one load can contain
about 30 hand towels (more than 10 and less than 100).
Thus, it costs about $0.10 to wash a single hand towel (as
part of a full washing machine load). The average cost to
dry your hands once with a cloth towel is the sum of the
costs to buy and wash the towel (suitably amortized):
C=
$5
$0.10
+
(10 yrs)(400 days/yr)(5 dry/day) (20 days)(5 dry/day)
= $10 −3 / dry
or about 0.1 cents per hand drying. Note that this does
not include the labor involved in washing the towels.
This is remarkably inexpensive, but impractical for public
bathrooms where the spread of germs is a serious concern.
Now let’s consider paper towels. A roll of kitchen paper
towels costs about $1 (more than $0.1 and less than $10)
and contains about 100 towels (more than 10 and less
than 1000). It takes two or three paper towels to thoroughly dry your hands, so the cost of drying your hands
with paper towels is about $0.03 (ignoring disposal and
cleanup costs). Paper towels are thus 30 times more
expensive than cloth towels.
Hot-air hand dryers will have a much higher initial cost
and a much lower operating cost. A hot-air hand dryer
will cost about $300 (more than $100 and less than
$1000) plus another $100 or so to install. In order to
estimate the operating cost, we need to estimate the time
it takes to dry your hands, the electrical power used, and
the cost of the electrical energy. An old-fashioned hand
dryer takes about a full (interminable) minute to dry
your hands. It uses about as much electrical power as
a typical 1 kW home hair dryer. At $0.10 per kiloWatthour, this will cost
C = ($0.10 / kW-hr)(1 kW)(1/ 60 hr)
= $2 × 10 −3
or about 0.2 cents per hand drying. This is 10 times
cheaper than paper towels.
The high-velocity hand dryers can dry your hands five
times faster, reducing the costs to $4310­-4 per hand drying.
Let’s consider a busy highway rest-stop bathroom. If a
customer enters once every 10 seconds, then there will be
53103 customers in a 12-hour day and 2310­6 customers
per year. This bathroom will need more than six regular
hand dryers and more than one high-velocity hand dryer.
If these hand dryers last for three years (more than one,
less than 10), then the regular hand dryers will dry 106
hands and the high-velocity hand dryer will dry several
times that. Even if the total cost (including installation
and repairs) of each hand dryer is $103, that is only $10-3
or 0.1 cents of capital costs per hand drying.
Thus, the total cost to the bathroom owner for paper
towels will be
Cpaper = (2 × 10 6 cust/yr)($3 × 10 −2 / cust)
= $6 × 10 4 / yr
and the cost of the hand dryers will be
Chot-air = ( 2 × 10 6 cust/yr)($2 × 10 −3 / cust)
= $4 × 103 / yr.
That is a difference of $60 thousand dollars per year, a
not-insignificant amount.
Note that we only considered the cost to the bathroom
owner. The cost to the user is also significant. It takes
only a few seconds to dry your hands with paper towels.
Many people are unwilling to wait an entire minute to
dry their hands with a regular hand dryer and either
don’t wash their hands at all or don’t dry them. This is
because time is money. The average yearly per-capita
gross domestic product (GDP) of the U.S. is about
The Physics Teacher ◆ Vol. 49, December 2011
$5310­4, which translates to $25 per hour for a typical 2000-hour work-year. This means that one minute
of your time is worth about $0.50. This far outweighs
the costs of either the paper towels or the electricity
involved. The total cost of the time spent drying hands
by all of the bathroom customers in one year will be
Ctime = (2 × 10 6 cust/yr)($0.50 / cust)
= $10
6
or one million dollars per year.
Thus, regular hot-air hand dryers cost the users far
more than they save the bathroom owner.
Copyright 2011, Lawrence Weinstein.
w Question 2: Mercury and the Sun
How close can Mercury orbit the Sun without being
ripped apart by tidal forces?
Answer: Mercury is the closest planet to the Sun, orbiting at about half the distance of the Earth to the Sun
(0.5 Astronomical Units or 831010 m). The closer a
planet orbits the Sun, the greater the tidal forces on the
planet. When the tidal forces exceed the gravitational
attraction of the planet, the planet will be ripped apart.
More specifically, the tidal acceleration is the difference
between the gravitational acceleration due to the Sun
at the center of the planet and that at the surface of the
planet
∆g( d , r ) =
GM
GM
−
d2
( d − r )2
where d is the distance from the Sun to the planet and r
is the radius of the planet. When this exceeds the gravitational acceleration at the surface of the planet, then
the planet is ripped part.
This means that we need to estimate the mass of the
Sun, M", the gravitational constant, G, the radius of
Mercury, and the surface gravity of Mercury. If we know
that G = 7310-11N-m2/kg2, then we can calculate the
mass of the Sun from the Earth’s orbital period and
radius. Alternatively, if we remember that M"= 23
1030 kg (or that the mass of the Sun is 106 times greater
than the mass of the Earth), then we can derive G.
Mercury is a small rocky planet. It is smaller than the
Earth (ME = 631024 kg, RE = 63106 m) and larger than
the Moon (MMoon = 10-2 ME, RMoon = 106 m) so we
will take the geometric means to get MM =
631023 kg and RM = 23106 m. These values are well
within a factor of two of the actual values. Similarly, we
can take the geometric mean of the Earth’s and Moon’s
surface gravities (gE = 10 m/s2, gMoon = 2 m/s2) to get
Mercury’s surface gravity of gM = 4 m/s2.
Now we have all the information we need. Making the
approximation that the radius of Mercury is much less
than its distance to the Sun (r << d), we can rewrite the
first equation as
∆g( d , r ) =
so that
dmin
 2 GM r 
=

 g
1/ 3
M
 2(7 × 10 −11 N-m 2 /kg 2 )(2 × 1030 kg)(2 × 106 m) 
=

4 m/s2

1/ 3
= (10 20 m 3 )1/ 3 = 10 7 m
This is much much closer than Mercury’s current orbital distance of about 1011 m.
Note the increasing tidal forces at smaller orbital distances probably disrupted planet formation at those
locations, rather than pulling apart already existing
planets.
GM
GM
2GMr
,
−
≈
2
2
d
(d − r )
d3
The Physics Teacher ◆ Vol. 49, December 2011
Copyright 2011, Lawrence Weinstein.