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Questions tagged [dimensional-analysis]

Dimensional analysis is the process of obtaining results by analysing the units and dimensions in questions, equations, and so on using The Principle of Homogeneity. Note: DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.

Dimensional analysis means to obtain results by analyzing the units in question, etc. DO NOT USE THIS TAG if your question is about dimensions as in degrees of freedom.

An example of dimensional analysis would be constructing the Planck time. The Planck time is to involve the Planck constant, the gravitational constant, and the speed of light. So, we let the Planck time be products of these constants raised to unknown exponents. Then by requiring that this Planck constant has units of time, we calculate these unknown exponents.

Other instances of the usefulness of dimensional analysis include:

  • Calculating other Planck units; e.g. Planck length, Planck mass, etc.
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Are units of angle really dimensionless?

I know mathematically the answer to this question is yes, and it's very obvious to see that the dimensions of a ratio cancel out, leaving behind a mathematically dimensionless quantity. However, I've been writing a c++ dimensional analysis library…
Nicolas Holthaus
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Is $\pi^2 \approx g$ a coincidence?

In spite of their different dimensions, the numerical values of $\pi^2$ and $g$ in SI units are surprisingly similar, $$\frac{\pi^2}{g}\approx 1.00642$$ After some searching, I thought that this fact isn't a coincidence, but an inevitable result of…
nalzok
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What justifies dimensional analysis?

Dimensional analysis, and the notion that quantities with different units cannot be equal, is often used to justify very specific arguments, for example, you might use it to argue that a particular formula cannot possibly be the correct expression…
Jack M
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Why is it "bad taste" to have a dimensional quantity in the argument of a logarithm or exponential function?

I've been told it is never seen in physics, and "bad taste" to have it in cases of being the argument of a logarithmic function or the function raised to $e$. I can't seem to understand why, although I suppose it would be weird to raise a…
sangstar
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Why are "degrees" and "bytes" not considered base units?

From Wikipedia: The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the candela for luminous intensity,…
Wais Kamal
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Is the Boltzmann constant really that important?

I read a book in which one chapter gave a speech about the fundamental constants of the Universe, and I remember it stated this: If the mass of an electron, the Planck constant, the speed of light, or the mass of a proton were even just slightly…
EXVII
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Should zero be followed by units?

Today at a teachers' seminar, one of the teachers asked for fun whether zero should be followed by units (e.g. 0 metres/second or 0 metre or 0 moles). This question became a hot topic, and some teachers were saying that, yes, it should be while…
Vidyanshu Mishra
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Dimensionless Constants in Physics

Forgive me if this topic is too much in the realm of philosophy. John Baez has an interesting perspective on the relative importance of dimensionless constants, which he calls fundamental like alpha, versus dimensioned constants like $G$ or $c$ […
Michael Luciuk
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Fundamental question about dimensional analysis

In dimensional analysis, it does not make sense to, for instance, add together two numbers with different units together. Nor does it make sense to exponentiate two numbers with different units (or for that matter, with units at all) together; these…
Jubilee
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What is the logarithm of a kilometer? Is it a dimensionless number?

In log-plots a quantity is plotted on a logarithmic scale. This got me thinking about what the logarithm of a unit actually is. Suppose I have something with length $L = 1 \:\mathrm{km}$. $\log L = \log \mathrm{km}$ It seems that the unit of $\log…
Statec
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Why are angles dimensionless and quantities such as length not?

So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths. Ok so far, so good. Then came the question: "In that sense even length is a ratio. Of length…
xrisk
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Square bracket notation for dimensions and units: usage and conventions

One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol varies from source to source; there are a few…
Emilio Pisanty
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What are the units or dimensions of the Dirac delta function?

In three dimensions, the Dirac delta function $\delta^3 (\textbf{r}) = \delta(x) \delta(y) \delta(z)$ is defined by the volume integral: $$\int_{\text{all space}} \delta^3 (\textbf{r}) \, dV = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}…
Andrew
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Why do we assume, in dimensional analysis, that the remaining constant is dimensionless?

Walter Lewin's first lecture (at 22:16) analyzes the time $t$ for an apple to fall to the ground, using dimensional analysis. His reasoning goes like this: It's natural to suppose that height of the apple to the ground ($h$), mass of the apple…
Fine Man
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Is it possible to speak about changes in a physical constant which is not dimensionless?

Every so often, one sees on this site* or in the news† or in journal articles‡ a statement of the form "we have measured a change in such-and-such fundamental constant" (or, perhaps more commonly, "we have constrained the rate of change of...").…
Emilio Pisanty
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