Questions tagged [dimensional-analysis]

The study of the relationships between physical quantities by identifying their units of measure and fundamental dimensions. It is used to convert from one set of units to others such as from miles per hour to meters per second, or from calories per slice of cake to kilocalories per whole cake.

What questions should have this tag? Questions which are related to the conversion between different units of measurement or fundamental dimensions (for example, converting speed from miles per hour to meters per second).

What are the basic methods? The factor-label method for converting units involves first creating a series of ratios equaling 1 from known identities and using those to multiply your input, cancelling dimensional units which appear both in the numerator and denominator until the result is in the desired dimensions.

For example, $10 \frac{\text{miles}}{\text{hour}} = 10\frac{\text{miles}}{\text{hour}}\times 5280\frac{\text{feet}}{\text{miles}}\times \frac{1}{3600}\frac{\text{hour}}{\text{second}} = 14.\overline{6}\frac{\text{feet}}{\text{second}}$

Important links:

Dimensional Analysis (chem.tamu.edu)

Wikipedia: Dimensional Analysis

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Why do units (from physics) behave like numbers?

What are units (like meters $m$, seconds $s$, kilogram $kg$, …) from a mathematical point of view? I've made the observation that units "behave like numbers". For example, we can divide them (as in $m/s$, which is a unit of speed), and also square…
user377104
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What mathematical structure models arithmetic with physical units?

In physics we deal with quantities which have a magnitude and a unit type, such as 4m, 9.8 m/s², and so forth. We might represent these as elements of $\Bbb R\times \Bbb Q^n$ (where there are $n$ different fundamental units), with $(4,…
MJD
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What is $5^\circ \mathrm{F}$ minus $5^\circ \mathrm{F}$?

Is it $0^\circ \mathrm{F}$ or is it $0\,\mathrm{K}$ (Kelvin)? From an arithmetic standpoint, it seems like it should be $0^\circ \mathrm{F}$, but that seems inconsistent because the result represents a delta ($0 \,\Delta^\circ \mathrm{F}$ perhaps)…
Brent
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Why is an equation necessarily dimensionally correct?

I have just read a fascinating proof of the value of the integral $$ \int_{-\infty}^\infty e^{-ax^2} dx, $$ which proceeds by dimensional analysis, as follows: we know that we can write $$ \int_{-\infty}^\infty e^{-ax^2} dx = f(a) $$ for some $f$.…
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Why are radians dimensionless?

According to https://en.wikipedia.org/wiki/Dimensionless_quantity, "A dimensionless quantity is a quantity to which no physical dimension is applicable." The article then explains, a few sentences later, that time, as an example, is a quantity with…
Blitzquark
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Arbitrarily discarding/cancelling Radians units when plugging angular speed into linear speed formula?

Why is the radians implicitly cancelled? Somehow, the feet just trumps the numerator unit. For all other cases, you need to introduce the unit conversion fraction, and cancel explicitly. Is it because radians and angles have no relevance to linear…
JackOfAll
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How do fractional tensor products work?

In this blog post, Terry Tao discusses the $n$-fold tensor product of a one-dimensional vector space $V^L$ ($L$ is just a non-numeric label, not an exponent). He claims that With a bit of additional effort (and taking full advantage of the…
tparker
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Difference between units and dimensions

Though this question may seem related to Physics, I think that at the very root this is a mathematical question and so I have posted this on math.stackexchange. Background: Initially I thought that the terms-unit and dimension, refer to the same…
MrAP
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What really are units? And why is it valid to ignore them (once you have dimensional homogeneity), as is done in class?

All my life the approach has been as follows: In math class I learn the rules and almost always deal with purely numerical problems. In physics class I apply the things learned in math class but this time our quantities have units. Now, once the…
ben ari
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Why is an angle dimensionless?

I was trying out Dimensional Analysis on a few equations and realized that angles have no dimension. Otherwise equations such as $s=r\theta$ are not dimensionally consistent. Further, why don't trigonometric ratios have any dimension? PS: I couldn't…
Green Noob
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The Power of Dimensional Analysis

What are some illustrative examples of dimensional analysis at work? Especially, where can it be used to significantly reduce the difficultly of a computation?
user142299
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What is the difference between a space dimension and a time dimension?

The simplest case is obviously 4D spacetime composed of 3 space dimensions and 1 time dimension. In some talks i stumbled across physicists and mathematicians who talked about spaces in which you only work with space dimensions or only with time…
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What are the measurement units of Fisher information? (Dimensional Analysis)

I know that there is a strong relationship between Shannon entropy and thermodynamic entropy -- they even have the same units and differ only by a constant factor. This suggests that they both intrinsically describe the same fundamental…
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Physical dimensions in math

I was interested in the idea of of formalising the idea of physical dimensions with an algebraic structure containing "all physical quantities of any type". You'd need: Scalar multiplication over the reals (so you can get "2 kg" from "2 *…
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How to use dimensional analysis to find these integrals?

Use dimensional analysis to find $\displaystyle \int_0^∞ \mathrm{e}^{-ax} \,\mathrm{d}x$ and $\displaystyle \int\frac{\mathrm{d}x}{x^2 + a^2}$. A useful result is$$ \int \frac{\mathrm{d}x}{x^2 + 1} = \arctan x + C. $$ I am using the the book…
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