687: Dimensional Analysis - Revision history

FaviFake: adding trivia, credit to 108.162.245.126 20:47, 15 May 2025 (UTC)

2025-06-05T16:11:22Z

adding trivia, credit to 108.162.245.126 20:47, 15 May 2025 (UTC)

172.70.207.19: /* Dimensional analysis */ lol

2025-01-01T03:26:20Z

‎Dimensional analysis: lol

172.69.234.160: /* Explanation */

2024-12-12T20:31:17Z

‎Explanation

Mumiemonstret: Added my favourite example of dimension-analysis-derived insight

2024-12-12T11:03:08Z

Added my favourite example of dimension-analysis-derived insight

162.158.137.212: Undo revision 356913 by RandallIsAZionist (talk) rv vandalism

2024-11-14T23:56:48Z

Undo revision 356913 by RandallIsAZionist (talk) rv vandalism

RandallIsAZionist at 23:43, 14 November 2024

2024-11-14T23:43:14Z

Uranium233 at 11:51, 3 November 2023

2023-11-03T11:51:51Z

FaviFake: Fixed redirect

2023-08-20T10:29:33Z

Fixed redirect

FaviFake: Fixed old http:// link

2023-05-23T20:32:12Z

Fixed old http:// link

JLZ0kTC5 at 11:01, 20 November 2022

2022-11-20T11:01:38Z

@@ Line 64: / Line 64: @@
 :Student: But what if they build a better Prius?
 :Cueball: ''Then England will drift out to sea.''
 {{comic discussion}}

@@ Line 30: / Line 30: @@
 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 energy (joule) as well as all the unit of volume (cubic 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. For instance the inverse of gas mileage, fuel consumption per distance, is dimensionally a surface. If a car consumes 10 liters per 100 km, i.e. 10 dm<sup>3</sup> / 100 km, this value has a dimension of m<sup>3</sup> / m = m<sup>2</sup>. This surface corresponds to the area of an imagined cylinder (or other prism) of fuel along the road that the car consumes as it rides along, 0.1 mm<sup>2</sup> in the example. However, in reality such an equation 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.
+Another aspect of the comic is that sometimes dimension analysis of equations that were not derived but rather "made up" can provide insight. For instance the inverse of gas mileage, fuel consumption per distance, is dimensionally a surface. If a car consumes 10 liters per 100 km, i.e. 10 dm<sup>3</sup> / 100 km, this value has a dimension of m<sup>3</sup> / m = m<sup>2</sup>. This surface corresponds to the area of an imagined cylinder (or other prism) of fuel along the road that the car consumes as it rides along, 0.1 mm<sup>2</sup> in the example. However, in reality such an equation 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.{{Citation needed}}
 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.

@@ Line 30: / Line 30: @@
 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 energy (joule) as well as all the unit of volume (cubic 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. For instance the inverse of gas mileage, fuel consumption per distance, is dimensionally a surface. If a car consumes 10 liters per 100 km, i.e. 10 dm<sup>3</sup> / 100 km, this value has a dimension of m<sup>3</sup> / m = m<sup>2</sup>. This surface corresponds to the area of an imagined cylinder of fuel along the road that the car consumes as it rides along, 0.1 mm<sup>2</sup> in the example. However, in reality such an equation 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.
+Another aspect of the comic is that sometimes dimension analysis of equations that were not derived but rather "made up" can provide insight. For instance the inverse of gas mileage, fuel consumption per distance, is dimensionally a surface. If a car consumes 10 liters per 100 km, i.e. 10 dm<sup>3</sup> / 100 km, this value has a dimension of m<sup>3</sup> / m = m<sup>2</sup>. This surface corresponds to the area of an imagined cylinder (or other prism) of fuel along the road that the car consumes as it rides along, 0.1 mm<sup>2</sup> in the example. However, in reality such an equation 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.

@@ Line 30: / Line 30: @@
 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 energy (joule) as well as all the unit of volume (cubic 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 equation 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.
+Another aspect of the comic is that sometimes dimension analysis of equations that were not derived but rather "made up" can provide insight. For instance the inverse of gas mileage, fuel consumption per distance, is dimensionally a surface. If a car consumes 10 liters per 100 km, i.e. 10 dm<sup>3</sup> / 100 km, this value has a dimension of m<sup>3</sup> / m = m<sup>2</sup>. This surface corresponds to the area of an imagined cylinder of fuel along the road that the car consumes as it rides along, 0.1 mm<sup>2</sup> in the example. However, in reality such an equation 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.

@@ Line 10: / Line 10: @@
 Cueball has a [[:Category:My Hobby|hobby]]: showing correct calculations according to the {{w|dimensional analysis}}, but with ridiculous correlations of uncorrelated events and measurements. Here Cueball is giving a talk and uses this trick to ''convince'' his listeners that the {{w|Toyota Prius}} combined {{w|United States Environmental Protection Agency|EPA}} gas mileage is somehow connected to the constant ''{{W|Pi|π}}'' via the {{w|Planck energy}}, the pressure at the {{w|Inner_core|Earth's core}} and the width of the {{w|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 Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong RandallIsAZionist Ygs have it all wrong 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+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:
 <math>\frac{\text{Planck energy}}{\text{Pressure at the core of the Earth}} \times  \frac{\text{Prius combined EPA gas mileage}}{\text{Minimum width of the English Channel}} = \pi</math>

@@ Line 43: / Line 43: @@
 Prius combined {{w|Toyota_Prius#Fuel_economy_and_emissions|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 [https://www.fueleconomy.gov/feg/Find.do?action=sbs&id=26425 combined EPA gas mileage] which is used in the equation and that is 50 miles per gallon.
+For the third generation (from 2010) the [https://www.fueleconomy.gov/feg/Find.do?action=sbs&id=26425 combined EPA gas mileage] is 50 miles per gallon.
 mpg => 21,000,000 m/m³

@@ Line 8: / Line 8: @@
 ==Explanation==
-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 giving a talk and uses this trick to ''convince'' his listeners that the {{w|Toyota Prius}} combined {{w|United States Environmental Protection Agency|EPA}} gas mileage is somehow connected to the constant ''{{W|Pi|π}}'' via the {{w|Planck energy}}, the pressure at the {{w|Inner_core|Earth's core}} and the width of the {{w|English Channel}}.
+Cueball has a [[:Category:My Hobby|hobby]]: showing correct calculations according to the {{w|dimensional analysis}}, but with ridiculous correlations of uncorrelated events and measurements. Here Cueball is giving a talk and uses this trick to ''convince'' his listeners that the {{w|Toyota Prius}} combined {{w|United States Environmental Protection Agency|EPA}} gas mileage is somehow connected to the constant ''{{W|Pi|π}}'' via the {{w|Planck energy}}, the pressure at the {{w|Inner_core|Earth's core}} and the width of the {{w|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.
+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:
 <math>\frac{\text{Planck energy}}{\text{Pressure at the core of the Earth}} \times  \frac{\text{Prius combined EPA gas mileage}}{\text{Minimum width of the English Channel}} = \pi</math>
@@ Line 26: / Line 24: @@
 *Minimum width of the English Channel: given in meters [m]
-When plugged into the left hand side, this amounts to:
+When plugged into the left-hand side, this amounts to:
 <math>\frac{\text{J}}{\frac{\text{J}}{\text{m}^3}} \times \frac{\frac{\text{m}}{\text{m}^3}}{\text{m}} = \text{m}^3 \times \frac{1}{\text{m}^3} = 1</math>
@@ Line 32: / Line 30: @@
 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 energy (joule) as well as all the unit of volume (cubic 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.
+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 equation 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.
@@ Line 68: / Line 66: @@
 {{comic discussion}}
 [[Category:Comics featuring Cueball]]
 [[Category:Comics featuring Ponytail]]

@@ Line 45: / Line 45: @@
 Prius combined {{w|Toyota_Prius#Fuel_economy_and_emissions|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 [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 miles per gallon.
+For the third generation (from 2010) the City mileage is 51 mpg and the Highway mileage is 48 mpg. But it is the [https://www.fueleconomy.gov/feg/Find.do?action=sbs&id=26425 combined EPA gas mileage] which is used in the equation and that is 50 miles per gallon.
 mpg => 21,000,000 m/m³
@@ Line 53: / Line 53: @@
 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 may get closer to pi — as the distance per fuel quantity in real life usage usually is somewhat lower than the standardized test value advertised — and that would reduce the result.
-[http://www.wolframalpha.com/input/?i=%28Planck+Energy%29%2F%28Pressure+at+the+Earth%27s+core%29*%282010+Prius+combined+EPA+Gas+Mileage%29%2F%2833.1+km%29 Wolfram|Alpha can find most of the statistics and do the calculations.]
+[https://www.wolframalpha.com/input/?i=%28Planck+Energy%29%2F%28Pressure+at+the+Earth%27s+core%29*%282010+Prius+combined+EPA+Gas+Mileage%29%2F%2833.1+km%29 Wolfram|Alpha can find most of the statistics and do the calculations.]
 ==Transcript==

@@ Line 51: / Line 51: @@
 .1&nbsp;km = 33,100&nbsp;m
-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 may get closer to pi — as the distance per fuel quantity in real life usage usually is somewhat lower than the standardised test value advertised — and that would reduce the result.
+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 may get closer to pi — as the distance per fuel quantity in real life usage usually is somewhat lower than the standardized test value advertised — and that would reduce the result.
 [http://www.wolframalpha.com/input/?i=%28Planck+Energy%29%2F%28Pressure+at+the+Earth%27s+core%29*%282010+Prius+combined+EPA+Gas+Mileage%29%2F%2833.1+km%29 Wolfram|Alpha can find most of the statistics and do the calculations.]