Instructions
This carefully engineered gauge is designed to be placed on the side of a standard US pint glass in order to determine the amount of beer that is missing from the glass. The precisely designed gauge fits along the side of the pint glass and the easy-to-read beer level indicators are used to determine the amount of beer that is GONE (see Figures 1 and 2).
Figure 1 Figure 2
The operational instructions are as follows:
The Theory
The standard US pint glass is conical in shape; essentially an inverted truncated cone around six inches tall and tapering by about an inch in diameter over the height of the glass.
Mathematically speaking the type of structure is referred to as a conical frustum, which is created by slicing the top off of a cone (where the slice is made parallel to the base). A frustum is defined as the portion of a solid which lies between two parallel planes cutting the solid. To determine the volume of this conical frustum (i.e., a pint glass), we could have looked in a set of mathematical tables. But what would be the fun of that? If we derive it ourselves, the beer tastes better at the end.
Thus, we got our international science team (these types of guys roam the halls at will at my real job and are looking for anything to derive like a beer connoisseur might look for his next microbrew) together along with a few bombers (the Holy Trinity) and a couple of six packs of Belgian ale (Abbey and Trippel) and started our derivation.
I do not care what they say, I like to drink and DERIVE.
The volume of interest is shown in Figure 3. In this figure, L is the inside height of the pint glass, D_{bot} is the inside diameter of the bottom of the pint glass, and D_{top} is the inside diameter of the top of the pint glass.
Figure 3: The geometry of the standard US pint glass.
We start by writing the volume as
where V is the total volume of the pint glass and dV is the differential volume element. We next realize that since the pint glass is cylindrical in shape we can use the cylindrical coordinate system to derive the needed volume. Hence, in cylindrical coordinates, the differential volume element is illustrated in Figure 4 and is expressed as
and the volume is expressed as
Figure 4: The differential volume element.
Now we must determine the limits of integration. The limits of dz and dphi are very straight forward. The limits for dz are 0 to L, and the limits for phi are 0 to 2pi. We have to be careful with the limits on rho. As the height of the beer in the glass increases, the radius of the beer surface increases. The radius for a given height “z” is given by
Thus, the limit of the rho integral will be 0 to rho_{z}. With the limits of integration determined, the volume is given
The dphi integral is elementary to evaluate, which can be done first to give
Since rho_{z} is a function of z, we have to be careful about the order of integration of the remaining integral. We evaluate drho first to give
This integral is also very straight forward to evaluate and after all the dust clears we have the following for the volume of the pint glass
The volume for any height (between 0 and L) of beer in the glass is given
where H_{top} is the distance from the surface of the beer to the top of the pint glass, and D_{H} in the diameter of the beer surface at the H_{top} location.
We can now use this equation to plot a graph that shows the amount of beer in a standard pint glass as a function of the distance H_{top} (the distance of the beer surface from the top of the pint glass). This plot is shown below. Also shown in this figure is the amount of beer missing as a function of H_{top}.
Figure 5: Amount of beer in a standard US pint glass as a function of H_{top} (the distance of the beer surface from the top of the pint glass).
The Important RESULTS!
This figure shows that if a beer is poured to a distance just less then 1/2 inch from the top, less than 14 oz are in the glass, and 13% of the beer is GONE. A distance just less then 1 inch from the top of the glass indicates that 25% of the beer is missing, and less than 12 oz are in the glass. At a distance of 2 inches from the top, only 8 oz of beer are in the glass and 50% of the beer is GONE; which is indeed a sad state of affairs!
Keep your bartender honest and Order your very own "Beer Gauge" today (Order Information).
email: thebeergauge@yahoo.com
Brought to you by: Three Phase Designs, LLC.