Aside from providing the solution to an amusing puzzle, this page is primarily an
exercise in the use of HTML5 MathML
for math expressions. MathML is supported by most modern web browsers, but the quality
of the rendering may vary.
(version 31.0, at the time of this writing) does a rather good job,
probably the best of most current browsers.
When a can of beer is full the center of gravity is at the center of the can.
As the beer is consumed the center of gravity moves down.
When the can is finally empty the center of gravity is back at the center.
At what point does the center of gravity reach its lowest point and start turning up again?
We will assume that the can is a uniform cylinder and that we are only concerned
with its center of gravity when it is sitting upright. We will assume that the
gravitational field is constant over the height of the can, so we may consider
the center of mass to be equivalent to the center of gravity.
If we define x, where 0 ≤ x ≤ 1, as the proportion of beer remaining in the can, and
we define y, where 0 ≤ y ≤ 1, as the height of the center of mass of the can and the beer
together, as a proportion of the height of the center of mass of the can alone, then
The mass times the distance of the center of mass (from the base of the can, in this case)
is sometimes called the "mass moment".
The mass moment of the beer and the can together must be equal to
the mass moment of the beer plus the mass moment of the can:
That is another way of saying that the height of the center of mass of the beer and can together
is the mass-weighted average of the height of the center of mass of the beer and the
height of the center of mass of the can.
Solving for y,
That function tells us how the center of mass moves as the beer is consumed.
The shape of the curve depends on the ratio k of the initial mass of the beer
to the mass of the can.
y reaches a minimum when its derivative goes to zero. Differentiating,
It can be seen that the derivative is zero when
The positive solution for that polynomial is the proportion of beer left when the
center of mass is at its lowest level. Using the
Which simplifies to
For a typical 12-ounce aluminum beer can, the mass of the beer is about 10 times
the mass of the can (k = 10), so the center of mass is at its lowest when there is slightly
less than one-quarter of the beer remaining in the can:
We can find the minimum without using calculus by making the observation that
the center of mass will be going down as long as we are removing beer that is
above it and the center of mass will start going up when we begin removing beer
below it. In other words, the center of mass will be at its lowest when it
coincides with the surface of the beer.
At that point,
so if we substitute
for y in the mass moment equation,
and simplify, we get the same polynomial as before, without having to use