687: Dimensional Analysis
Title text: Or the pressure at the Earth's core will rise slightly.
| This explanation may be incomplete or incorrect: Randall's solutions for a different Prius are not explained well.|
If you can address this issue, please edit the page! Thanks.
This comic makes fun of how scientists (often physicists) use dimensional analysis to quickly check if a given formula can possibly relate to a physical system or if there were some (obvious) mathematical errors in its derivation. Dimensional analysis here refers to the check if both sides of the equation arrive at the same physical unit if the units of all variables get plugged into the equation. This usually requires knowledge of the system of units and the relation between different physical units.
The comic uses the following equation to make fun of it:
(Planck energy) / (Pressure at the core of the earth) * (Prius combined EPA gas mileage) / (minimum width of the English Channel) = π
The right hand side is dimensionless (The constant π = 3.14... by definition is 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: miles/gallon, SI units: meters/litres [m/l]
- 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.
In addition, if a better Prius were built, the equation would no longer be accurate, so one of the other quantities would have to be change to suit the equation. A higher gas mileage would require a wider English Channel or it would no longer be the same answer. The title text also refers to this, as a higher pressure at Earth's core would also cause the equation to equal pi.
For such relations it is also true that many of them can be made up by searching for matching values for variables to derive at the wanted number finally. E.g. if it is desired to arrive at e instead of π on the comic-equation, this could be done by using a different car model and/or a different length measurement and/or a different pressure (e.g. by choosing a different planet) and/or some other arbitrary energy.
Some numbers for this calculation
The Planck energy is the only nearly exact value we do have. And according to other Planck values it is small.
E_planck = 1.956 x 109 J = 1.956 x 109 Nm
Pressure at the core of the earth. Using a simple value like this:
P_core = 350 GPa = 3.5 x 1011 N/m²
Prius combined EPA gas mileage: Highway US units 50 mpg
50 mpg => 13.2 miles per litre => 21,000,000 meter per m³
Minimum width of the English Channel is about
34km or 34,000 meters.
Calculating from these values you will get π=3.45... what is pretty close to π=3.14... while using a Planck value. Just try the ePrius and you will come closer to that target.
- 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.
- Interestingly, this sort of dimensional analysis is formalized by the so called Buckingham π theorem, where each dimensionless grouping is called a "Pi". Thus formally each grouping can be denoted as equal to "Pi", although this is a rather obscure pun in the context of the comic.
- The fine-structure constant α =~ 1/137 is a common known analogue in physics.
- This is another comic in the infrequent My Hobby series.
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