Editing 687: Dimensional Analysis
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==Explanation== | ==Explanation== | ||
− | Cueball has a [[ | + | Cueball has a [[My Hobby|hobby]] — showing correct calculations according to the {{w|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 {{w|Toyota Prius}} combined {{w|United States Environmental Protection Agency|EPA}} gas mileage is somehow connected to the constant {{W|Pi}} via the {{w|Planck energy}}, the pressure at the {{w|Inner_core|earth's core}} and the width of the {{w|English Channel}}. |
− | + | 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 | ||
===Dimensional analysis=== | ===Dimensional analysis=== | ||
− | The right hand side is dimensionless, it's the constant | + | 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] | *Planck energy: given in Joules [J] | ||
− | *Pressure at the core of the | + | *Pressure at the core of the earth: Given in Pascals [Pa] |
− | + | *Prius combined EPA gas mileage: miles/gallon, SI units: meters/litres [m/l] | |
− | *Prius combined EPA gas mileage: miles | + | *minimum width of the English channel: meters [m] |
− | * | ||
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− | When plugged into the left | + | 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] | ||
− | 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 | + | 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, with a higher gas mileage, the equation would only be accurate if one of the other quantities changed. A higher gas mileage could be balanced if the English Channel became wider thus explaining why Cueball states that England will drift to sea away from France. | ||
+ | |||
+ | 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 Pi. 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=== | ===Some numbers for this calculation=== | ||
The {{w|Planck energy}} is the only nearly exact value we do have. Compared to other Planck values it is very large (macroscopic). | The {{w|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 10<sup>9</sup> J | + | E_planck = 1.956 x 10<sup>9</sup> J = 1.956 x 10<sup>9</sup> Nm |
− | Pressure at the {{w|Inner_core#Temperature_and_pressure|core}} of the | + | Pressure at the {{w|Inner_core#Temperature_and_pressure|core}} of the earth ranges from 330 to 360 gigapascals. |
Using a simple value like this: | Using a simple value like this: | ||
− | P_core = 350 GPa = 3.5 x 10<sup>11</sup> | + | P_core = 350 GPa = 3.5 x 10<sup>11</sup> N/m² |
Prius combined {{w|Toyota_Prius#Fuel_economy_and_emissions|EPA gas mileage}}: | Prius combined {{w|Toyota_Prius#Fuel_economy_and_emissions|EPA gas mileage}}: | ||
− | For the third generation (from 2010) the [ | + | For the third generation (from 2010) the City mileage is 51 mpg and the Highway mileage is 48 mpg. But it is the [http://www.fueleconomy.gov/feg/Find.do?action=sbs&id=26425 combined EPA gas mileage] which is used in the equation and that is 50 mpg. |
− | 50 mpg => 21,000,000 | + | 50 mpg => 13.2 miles per litre => 21,000,000 meter per m³ |
Minimum width of the {{w|English Channel}} is about | Minimum width of the {{w|English Channel}} is about | ||
− | 33.1 | + | 33.1 km or 33,100 meters. |
− | Calculating from these values you will get | + | 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. |
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==Transcript== | ==Transcript== | ||
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:Student: But what if they build a better Prius? | :Student: But what if they build a better Prius? | ||
:Cueball: ''Then England will drift out to sea.'' | :Cueball: ''Then England will drift out to sea.'' | ||
+ | |||
+ | ==Trivia== | ||
+ | *Interestingly, this sort of dimensional analysis is formalized by the so called {{w|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 {{w|fine-structure constant}} α =~ 1/137 is a common known analogue in {{w|physics}}. | ||
+ | *This is another comic in the infrequent [[My Hobby]] series. | ||
{{comic discussion}} | {{comic discussion}} | ||
− | + | [[Category:My Hobby]] | |
[[Category:Comics featuring Cueball]] | [[Category:Comics featuring Cueball]] | ||
[[Category:Comics featuring Ponytail]] | [[Category:Comics featuring Ponytail]] | ||
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