[math-fun] "Baby" dimensional analysis
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units. (For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.) The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this. Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful? Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?" (My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.) Thanks, Jim Propp
Kepler's [third?] law, which states that the squared period is proportional to the cubed orbital radius, is derivable from dimensional analysis. Of course, the dimensional analysis 'proof' of Kepler's third law is somewhat dissatisfying to me because it doesn't say *which* radius should be cubed -- this matters if different planets have different eccentricities. [It happens that it's the semi-major axis, but this is by no means obvious. Does anyone know an *elegant* justification for this being the case (i.e. not involving churning through lots of coordinate calculations)?] Best wishes, Adam P. Goucher
Sent: Tuesday, January 16, 2018 at 7:03 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] "Baby" dimensional analysis
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
if I recall correctly, for 1/r^2 gravity there is a conserved quantity other than the energy and the angular momentum — this is why this system is “integrable". this additional quantity puts a bunch of elliptical orbits into an equivalence class with the same period. but I don’t remember how it works… can someone tell us? - Cris
On Jan 16, 2018, at 1:02 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Kepler's [third?] law, which states that the squared period is proportional to the cubed orbital radius, is derivable from dimensional analysis.
Of course, the dimensional analysis 'proof' of Kepler's third law is somewhat dissatisfying to me because it doesn't say *which* radius should be cubed -- this matters if different planets have different eccentricities.
[It happens that it's the semi-major axis, but this is by no means obvious. Does anyone know an *elegant* justification for this being the case (i.e. not involving churning through lots of coordinate calculations)?]
Best wishes,
Adam P. Goucher
Sent: Tuesday, January 16, 2018 at 7:03 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] "Baby" dimensional analysis
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
https://en.wikipedia.org/wiki/Laplace–Runge–Lenz_vector
On Jan 16, 2018, at 2:36 PM, Cris Moore <moore@santafe.edu> wrote:
if I recall correctly, for 1/r^2 gravity there is a conserved quantity other than the energy and the angular momentum — this is why this system is “integrable". this additional quantity puts a bunch of elliptical orbits into an equivalence class with the same period. but I don’t remember how it works… can someone tell us?
- Cris
There is an extra conserved quantity, the Runge-Lenz vector. Look up the excellent Wiki page on this topic. -- Gene On Tuesday, January 16, 2018, 2:37:11 PM PST, Cris Moore <moore@santafe.edu> wrote: if I recall correctly, for 1/r^2 gravity there is a conserved quantity other than the energy and the angular momentum — this is why this system is “integrable". this additional quantity puts a bunch of elliptical orbits into an equivalence class with the same period. but I don’t remember how it works… can someone tell us? - Cris
On Jan 16, 2018, at 1:02 PM, Adam P. Goucher <apgoucher@gmx.com> wrote:
Kepler's [third?] law, which states that the squared period is proportional to the cubed orbital radius, is derivable from dimensional analysis.
Of course, the dimensional analysis 'proof' of Kepler's third law is somewhat dissatisfying to me because it doesn't say *which* radius should be cubed -- this matters if different planets have different eccentricities.
[It happens that it's the semi-major axis, but this is by no means obvious. Does anyone know an *elegant* justification for this being the case (i.e. not involving churning through lots of coordinate calculations)?]
Best wishes,
Adam P. Goucher
Two of my favorites: . to raise the note of a stringed instrument by an octave, while keeping the length and density of the string the same, you need to quadruple the tension (this was known to Galileo’s father) . the period of a pendulum scales as \sqrt(L/g) where L is its length and g the acceleration of gravity — and the mass at the end doesn’t (can’t) matter. - Cris
On Jan 16, 2018, at 12:03 PM, James Propp <jamespropp@gmail.com> wrote:
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I saw the Reynolds number introduced as purely dimensional analysis, ie: turbulence is probably related to viscosity, density, relative velocity, and a size parameter. Building a dimensionless number (regardless of how) out of those turns out to be useful. On Tue, Jan 16, 2018 at 3:07 PM, Cris Moore <moore@santafe.edu> wrote:
Two of my favorites:
. to raise the note of a stringed instrument by an octave, while keeping the length and density of the string the same, you need to quadruple the tension (this was known to Galileo’s father)
. the period of a pendulum scales as \sqrt(L/g) where L is its length and g the acceleration of gravity — and the mass at the end doesn’t (can’t) matter.
- Cris
On Jan 16, 2018, at 12:03 PM, James Propp <jamespropp@gmail.com> wrote:
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Right. We then basically ask whether these dimensionless numbers are >> 1 or << 1, to tell which physical regime (e.g. smooth vs. turbulent flow) we’re in. Cris
On Jan 16, 2018, at 3:27 PM, James Davis <lorentztrans@gmail.com> wrote:
I saw the Reynolds number introduced as purely dimensional analysis, ie: turbulence is probably related to viscosity, density, relative velocity, and a size parameter. Building a dimensionless number (regardless of how) out of those turns out to be useful.
On Tue, Jan 16, 2018 at 3:07 PM, Cris Moore <moore@santafe.edu> wrote:
Two of my favorites:
. to raise the note of a stringed instrument by an octave, while keeping the length and density of the string the same, you need to quadruple the tension (this was known to Galileo’s father)
. the period of a pendulum scales as \sqrt(L/g) where L is its length and g the acceleration of gravity — and the mass at the end doesn’t (can’t) matter.
- Cris
On Jan 16, 2018, at 12:03 PM, James Propp <jamespropp@gmail.com> wrote:
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Here's a nice one. What is the energy flux = power per unit area F (W m^-2 = kg s^-3) carried by a gravitational wave of strain h? Strain is dimensionless, and the only physical constants available are speed of light c (m s^-1) and Newton's G (kg^-1 m^3 s^-2), since this is not a quantum phenomenon. Since the power flux should vary as h^2, we can write F = c^x G^y h^2. Expanding out, kg: 1 = -ym: 0 = x + 3ys: -3 = -x - 2y But these are inconsistent. So something is missing. Let's put in the frequency f (s^-1), and write F = c^x G^y f^z h^2. kg: 1 = -ym: 0 = x + 3ys: -3 = -x - 2y - z Then y = -1, x = 3, z=2, and F = (c^3 / G) (f h)^2. This makes sense. Strain is the deviation of the metric from flat space, but the gravitational energy should vary as the square of the connection coefficients, which are first derivatives of the metric. There is some proportionality constant which cannot be found from dimensional analysis. Is all this correct? Indeed it is. According to https://arxiv.org/pdf/1209.0667.pdf eq. (25) for a linearly polarized wave, the constant is π/8. -- Gene
Math-fun seems to have garbled up the line feeds in this message, so I'll try again. On Tuesday, January 16, 2018, 3:05:36 PM PST, Eugene Salamin <gene_salamin@yahoo.com> wrote: Here's a nice one. What is the energy flux = power per unit area F (W m^-2 = kg s^-3) carried by a gravitational wave of strain h? Strain is dimensionless, and the only physical constants available are speed of light c (m s^-1) and Newton's G (kg^-1 m^3 s^-2), since this is not a quantum phenomenon. Since the power flux should vary as h^2, we can write F = c^x G^y h^2. Expanding out, kg: 1 = -y m: 0 = x + 3y s: -3 = -x - 2y But these are inconsistent. So something is missing. Let's put in the frequency f (s^-1), and write F = c^x G^y f^z h^2. kg: 1 = -y m: 0 = x + 3y s: -3 = -x - 2y - z Then y = -1, x = 3, z=2, and F = (c^3 / G) (f h)^2. This makes sense. Strain is the deviation of the metric from flat space, but the gravitational energy should vary as the square of the connection coefficients, which are first derivatives of the metric. There is some proportionality constant which cannot be found from dimensional analysis. Is all this correct? Indeed it is. According to https://arxiv.org/pdf/1209.0667.pdf eq. (25) for a linearly polarized wave, the constant is π/8. -- Gene
On Tue, Jan 16, 2018 at 3:07 PM, Cris Moore <moore@santafe.edu> wrote:
. the period of a pendulum scales as \sqrt(L/g) where L is its length and g the acceleration of gravity — and the mass at the end doesn’t (can’t) matter.
In high school physics, our teacher had us time the periods of pendula of various lengths (measured in meters) and masses (measured in kilograms). Numerically, we got 2 sqrt(L) seconds, where the units, of course, make no sense. Inserting g to get the units right, we got 2 k sqrt(L/g), where k was the unitless square root of 9.8---which we then realized was pi to roughly one part in 300. -- Mike Stay - metaweta@gmail.com http://www.math.ucr.edu/~mike http://reperiendi.wordpress.com
Hi Jim, On that question, I once spent a year thinking about dimensional analysis and its implications (both for scalars and in the context of linear algebra) and came up with some non-standard conclusions. If you are interested, check out my book "Multidimensional Analysis" (Springer-Verlag, 1995). George http://georgehart.com On 1/16/2018 2:03 PM, James Propp wrote:
In high school I was taught a baby version of dimensional analysis in which the quantity you're trying to compute is either directly or inversely proportional to each of the quantities you're given, and the only pitfall is that you might divide when you should multiply or vice versa. In such a case, you keep yourself on the right path by including units along with the numbers, and checking that your big product-quotient expression has the right units.
(For instance, you don't multiply distance by velocity if you want an answer that has units of time. m meters times v meters-per-second has units of meters-squared-per-second, while m meters DIVIDED BY v meters-per-second has units of seconds.)
The Wikipedia page https://en.wikipedia.org/wiki/Dimensional_analysis does a pretty good job of explaining this.
Anyway, I recall from high school chemistry that there were some pretty complex problems involving reaction rates of different chemical species where the keeping-track-of-units trick was really helpful because the complexity of the problem overloaded my short-term memory. Can anyone think of a problem (maybe chemical, maybe not) where keeping track of units, and trying to get them to cancel out, is really helpful?
Of course, this is not to deprecate the sense-making approach that determines whether a number should be put "up top" or "down below" by asking "If this number were larger, would my final answer get bigger or smaller?"
(My February "Mathematical Enchantments" essay will be about dimensional analysis, and I want to discuss these sorts of "baby" applications before I get into deeper ones where the exponents aren't all +1 and -1.)
Thanks,
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (8)
-
Adam P. Goucher -
Cris Moore -
Eugene Salamin -
George Hart -
James Davis -
James Propp -
Mike Stay -
Veit Elser