687: Dimensional Analysis
Title text: Or the pressure at the Earth's core will rise slightly.
Cueball has a hobby — showing correct calculations according to the dimensional analysis — but with ridiculous correlations of uncorrelated events and measurements. Here Cueball is teaching a class and uses this trick to convince his students that the Toyota Prius combined EPA gas mileage is somehow connected to the constant π via the Planck energy, the pressure at the earth's core and the width of the English Channel.
Scientists — often physicists — use dimensional analysis to quickly check if a given formula can possibly relate to a physical system, because if you end up with an equation claiming that joules are meters, something is clearly wrong. Dimensional analysis here refers to the check if both sides of the equation arrive at the same physical unit when the units of all variables get plugged into the equation. This requires knowledge of the system of units and the relation between different physical units.
Cueball uses the following equation to make a mockery of the practice:
(Planck energy) / (Pressure at the core of the earth) * (Prius combined EPA gas mileage) / (minimum width of the English Channel) = π
 Dimensional analysis
The right hand side is dimensionless, it's the constant π = 3.14... which is defined by the relation of two lengths, the circumference and the diameter of a circle. The left hand side requires to plug in the dimensions of the named physical quantities:
- Planck energy: given in Joules [J]
- Pressure at the core of the earth: Given in Pascals [Pa]
- Prius combined EPA gas mileage: gallons/mile, cubic meters/meter [l/m]
- Fuel efficiency has two formats that are commonly used: length/volume and volume/length. The former must be used here in order to get the units to cancel correctly.
- minimum width of the English channel: meters [m]
When plugged into the left hand side this amounts to:
[(J / Pa) * ((m/l) / m)] = [(Nm / (N/m²)) * ((m/m³) / m)] = 1
Using the following unit relations (this does not reduce units to the seven SI units, but does use some derived units):
- 1 Joule = 1 Newton-meter [J] = [Nm]
- 1 Pascal = 1 Newton per square-meter [Pa] = [N/m²]
- 1 cubic-metre = 1000 litres [m³] = 1000 [l]
Note that for dimensional analysis constant factors are not taken into account. Here square brackets are used to denote dimensional analysis. In the above equation the unit of force (newton) as well as all the units of length (meter) cancel out each other.
Another aspect of the comic is, that sometimes dimension analysis of equations that were not derived but rather "made up" can provide insight. However, in reality such an equations would have to be somehow "motivated", which is more of an art than science and requires great experience in the field the equation should relate to. The presented equation combines values that have no immediate causal relation with each other, so it does not make sense. Furthermore, since the values have absolutely no causal relation to each other, the ratios presented are simple coincidence; despite Cueball's claim, building a better Prius would not cause any changes to the English Channel.
The title text also refers to this, as a higher pressure at Earth's core could also balance the equation, keeping the result constant equal to π. The Planck energy is an absolute, however, so it is not mentioned as a way to balance the next version of Prius.
 Some numbers for this calculation
The Planck energy is the only nearly exact value we do have. Compared to other Planck values it is very large (macroscopic).
E_planck = 1.956 x 109 J = 1.956 x 109 Nm
Pressure at the core of the earth ranges from 330 to 360 gigapascals. Using a simple value like this:
P_core = 350 GPa = 3.5 x 1011 N/m²
Prius combined EPA gas mileage: For the third generation (from 2010) the City mileage is 51 mpg and the Highway mileage is 48 mpg. But it is the combined EPA gas mileage which is used in the equation and that is 50 mpg.
50 mpg => 13.2 miles per litre => 21,000,000 meter per m³
Minimum width of the English Channel is about
33.1 km or 33,100 meters.
Calculating from these values you will get π = 3.54... that is pretty close to π = 3.14... while using a Planck value. According to Cueball this will be within the experimental error (the combined error for all four numbers - none are exact numbers). For instance if you tried the ePrius you would probably get closer to that target — as the mileage in real life usually is somewhat lower than the value given — and that would reduce the result.
This definition of pi uses, among other values, the width of the English channel. Incidentally, an approximation of pi can be found using natural water bodies: The sinuosity of a freely meandering river approaches pi. The sinuosity is the ratio of the true length (as the water flows) and the straight line, from river source to river delta.
- My Hobby:
- Abusing dimensional analysis
- [On a blackboard.]
- (Planck energy/Pressure at the Earth's core) x (Prius combined EPA gas mileage/Minimum width of the English Channel) = π
- [Cueball indicates this equation with a pointer in front of a class.]
- Cueball: It's correct to within experimental error, and the units check out. It must be a fundamental law.
- Student: But what if they build a better Prius?
- Cueball: Then England will drift out to sea.
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