We have heard
the famous two expressions “the glass is half-full” and the “glass is
half-empty”; the former is from the view of an optimist while the latter is
from the view of a pessimist. In life, I
have been both the optimist and the pessimist.
Let’s
consider when a glass is actually
half empty, oh sorry, half full. This is
achieved when the volume is half full when the volume of the liquid equals half
of the volume of the glass. Simple enough.
Let q be the
flow rate of the liquid being poured in the glass, and assume it is a constant
to keep it simple. (Yes we can a
variable flow rate q(t) – this would be appropriate when pouring viscous
liquids or considering problems when we are drinking water or our favorite
spirit or pop).
With V being
the volume of the liquid:
dV/dt = q
dV = q dt
Taking the
integral of both sides yield:
V = q*t + V0
Where V0 is
the initial volume of the liquid. Let’s
assume that we start with an empty glass.
Then the liquid of the liquid inside of the glass is:
V = q*t
Let C be the
volume (or the capacity) of the glass.
The glass is half full when V = C/2 or
t = C/(2 * q)
We can use
this equation for various shapes of drinking glasses. Case in point:
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For a
cylindrical glass: volume = π * r^2 * h
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For a
cylindrical glass: C = π * r^2 * h
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For a parabolic
glass: volume = π * r^2 * h/2
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For a parabolic
glass: C = π * r^2 * h/2
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For a
cup: volume = π * h/3 * (a^2 + a*b + b^2)
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For a
cup: C = π * h/3 * (a^2 + a*b + b^2)
(r, a, and b
are radii – half of the distance of the diameter.)
Take the
example that we have a cup with dimensions a = 1.375 in, b = 1.75 in, and h = 3.5
in. We are going to pour iced tea in it
at a flow rate of q = 8.3 in^3/sec.
C = π * 3.5/3
* (1.375^2 + 1.375 * 1.75 + 1.75^2) ≈ 26.93752 in^3
and t ≈
26.97352/(2 * 8.3) ≈ 1.62491 sec
P.S. The flow
rate I gave came from a short experiment of actually pouring Arnold Palmer Ice
Tea into the glass and timing how long it takes to pour the ice tea, which it
took about 3.25 seconds to fill it.
Doing the same experiment with pouring water from a bottle, which took
me 7 seconds (a flow rate of approximately 3.9 in^3/sec) – but I think held the
bottle closer to the cup.
Pessimists
and optimists (and everyone in between) – go out and have an awesome
weekend. Cheers!
Eddie
This blog is
property of Edward Shore. 2014.