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

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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 ...
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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 ...
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5answers
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What do units like joule * seconds imply?

I can easily understand what divisive units imply, but not what multiplicative units imply. What I mean is, when I read "$12 \:\mathrm{eggs/carton}$", I mentally convert it to, "There are 12 eggs ...
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11answers
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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 $L = 1 km$. $\lg L = \lg km$ It ...
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11answers
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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: ...
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2answers
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Why are expressions such as $\operatorname{ln}T$ used in thermodynamics where $T$ is not dimensionless?

In all thermodynamics texts that I have seen, expressions such as $\operatorname{ln}T$ and $\operatorname{ln}S$ are used, where $T$ is temperature and $S$ is entropy, and also with other thermodynamic ...
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1answer
554 views

Mass of empty AdS$_5$

Five dimensional empty AdS$_5$ space has mass $$ E = \frac{3 \pi \ell^2}{32 G}. $$ Is the above equation correct? Let's do some dimensional analysis to confirm. In natural units, in 5 dimensions ...
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How to get Planck length

I know that what Planck length equals to. The first question is, how do you get the formula $$\ell_P~=~\sqrt\frac{\hbar G}{c^3}$$ that describes the Planck length? The second question is, will any ...
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How can the speed of light be a dimensionless constant?

This is a quote from the book A first course in general relativity by Schutz: What we shall now do is adopt a new unit for time, the meter. One meter of time is the time it takes light to travel ...
14
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3answers
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How can I understand counterintuitive units like $\text{s}^2$?

One of the things I never understood, but was too afraid to ask is this: how should I think of things like $\frac{\text{kg}}{\text{s}^2}$. What exactly is a square second? Square foot makes sense to ...
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4answers
723 views

Why isn't it $E \approx 27.642 \times mc^2$?

Sorry for the strange question, but why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides? Why can so many ...
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6answers
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Why isn't temperature measured in units of energy?

Temperature is the average of the kinetic energies of all molecules of a body. Then, why do we consider it a different fundamental physical quantity altogether [K], and not an alternate form of ...
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1answer
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Fundamental question about the Buckingham $\pi$ theorem (dimensional analysis)

I have a rather fundamental question about the Buckingham $\pi$ theorem. They introduce it in my book about fluid mechanics as follows (I state the description of the theorem here, because I noticed ...
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8answers
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In dimensional analysis, why the dimensionless constant is usually of order 1?

Usually in all discussions and arguments of scaling or solving problems using dimensional analysis, the dimensionless constant is indeterminate but it is usually assumed that it is of order 1. What ...
13
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2answers
597 views

Is $0\,\mathrm{m}$ dimensionless?

Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$? I once was given an assignment where I was asked to deduce and write down some physical quantity. It ...
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6answers
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Can you find the length of a pencil without a ruler or clock?

Edit: Assume you do have access to all sorts of instruments, but they are all shrunk in proportion. My real question is: If you are shrunken (or expanded) by a constant factor and put in a room ...
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5answers
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Dimensional analysis restricted to rational exponents?

After some reading on dimensional analysis, it seems to me that only rational exponents are considered. To be more precise, it seems that dimensional values form a vector space over the rationals. My ...
11
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5answers
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units and nature

I am wondering whether the five$^1$ units of the natural unit system really is dictated by nature, or invented to satisfy the limited mind of man? Is the number of linearly independent units a ...
10
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4answers
714 views

Can dimension analysis be used in developing more advanced physics equations?

It is obvious that dimensional analysis can be used to derive many classical mechanics equations (excluding constants). As long as all the dependent quantities are known. My question is whether this ...
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Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
10
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4answers
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How to indicate that a unit is dimensionless [duplicate]

For my dissertation I am preparing a list of symbols used in the text, which basically is a table that consists of the symbol, a short explanation and the dimension it has as indicated below: ...
10
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2answers
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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 ...
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2answers
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Why are smaller animals stronger than larger ones, when considered relative to their body weight?

I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat. It has been suggested to me that this is due to ...
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2answers
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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 = ...
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How does the period of an hourglass depend on the grain size?

Suppose I have an hourglass that takes 1 full hour on average to drain. The grains of sand are, say, $1 \pm 0.1\ {\rm mm}$ in diameter. If I replace this with very finely-grained sand $0.1 \pm 0.01\ ...
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3answers
217 views

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 ...
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Simple Harmonic Motion - What are the units for $\omega_0$?

I'm trying to understand the units in: $$mx''+kx=0$$ And the general solution is $$x(t)=A \cos(\omega_0 t)+B \sin(\omega_0 t).$$ Let $\omega_0 =\sqrt{\frac{k}{m}}$ - the unit for the spring ...
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3answers
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What are the physical dimensions (units) of the elements in a Hilbert space of a QM system?

In QM, the state vector $|\psi\rangle$ seem to have various dimensions under different representations: (only in space of continuous dimension) $$\langle x|\psi\rangle = [\frac{1}{\sqrt{Length}}]$$ ...
8
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4answers
353 views

Physical representation of volume to surface area

I was looking at this XKCD what-if question (the gas mileage part), and started to wonder about the concept of unit cancellation. If we have a shape and try to figure out the ratio between the volume ...
8
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2answers
350 views

Why does the dimension of the electric charge depend on the number of spacetime dimensions?

We can find via dimensional analysis that the dimension of the electric charge varies with the dimension of space-time $(D+1)$: $$[\text{charge}] = (\text{eV})^{(3-D)/2}.$$ It is dimensionless if ...
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7answers
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Why are radians more natural than any other angle unit?

I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for ...
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Is the number 1 a unit?

In dimensionless analysis, coefficients of quantities which have the same unit for numerator and denominator are said to be dimensionless. I feel the word dimensionless is actually wrong and should be ...
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2answers
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What does the Reynolds Number of a flow represent physically?

What does the Reynolds Number of a flow represent physically? I am having trouble understanding the meaning and the utility of the Reynolds number for a certain flow, could someone please tell me how ...
7
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1answer
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Why should it be allowed to set the einbein to unity?

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by $$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$ where ...
7
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3answers
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In terms of scale, where does the concept of Reynold's number cease to have meaning?

The Reynolds number is classically described in terms of pipe geometries but its use has also been usefully extended to other more complex surface geometries to predict transitional flow behavior. But ...
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1answer
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Why can you remove the gravitational constant from a computer game simulation?

I've seen in a few gravity simulation games (ie. bouncing balls) the equation: force = G * m1 * m2 / distance^2 shortened to this by removing the gravitational ...
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1answer
396 views

What are units actually?

This question is about the concept of units in physics. Firstly - do units have a formal mathematical definition? How are they different from pure numbers? Are pure numbers defined to be ratios of ...
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3answers
996 views

Tips on teaching Dimensional Analysis? [closed]

What's a good way to explain dimensional analysis to a student? Here's a simple question which this method would be useful: Let's say a truck is moving with a speed of 18 m/s to a new speed of 13 ...
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3answers
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Units of a discrete Fourier transform

Normally a Fourier transform (FT) of a function of one variable is defined as $$f_k=\int^\infty_{-\infty}f(x)\exp\left(-2\pi i k x\right) dx.$$ This means that $f_k$ gets the units of $f$ times the ...
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Can Planck's constant be derived from Maxwell's equations?

Can mathematics (including statistics, dynamical systems,...) combined with classical electromagnetism (using only the constants appearing in chargefree Maxwell equations) be used to derive the Planck ...
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2answers
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What are the units of the quantities in the Einstein field equation?

The Einstein field equations (EFE) may be written in the form: $$R_{\mu\nu}-\frac {1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\frac {8\pi G}{c^4}T_{\mu\nu}$$ where the units of the gravitational constant $G$ ...
5
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2answers
271 views

Units inside a logarithm

I have troubles understanding a seemingly simple integral in a physical context. Take a look at $\int_{V_1}^{V_2} \frac{\mathrm{d}V}{V}$ which appears in isothermal expansions (V being the volume of a ...
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1answer
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Deriving or justifying fundamental constants

Is there a fundamental way to look at the universal constants ? can their orders of magnitude be explained from a general points of view like stability, causality, information theory, uncertainty? ...
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2answers
135 views

Is the “dimension” in dimensional analysis the same as the “dimension” in “three spatial dimensions”?

When we talk about the dimension of a quantity (e.g. the dimension of acceleration is$[ L \ T ^ {-2}]$) are we talking about the same "dimension" as when we talk about three dimensional space? Are ...
5
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1answer
167 views

Data requirement to determine proportionality

A common result of theoretical analysis in physics is some sort of relation derived from physical parameters and typically expressed in the form of a non-dimensional parameter. These scale relations ...
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2answers
131 views

Do bras and kets have dimensions?

I'm trying to understand more intuitively what bras and kets are, but some aspects of them remain a mystery to me. We usually think of $\psi (x)$ as having dimension of $[1/\sqrt{L}]$ so that ...
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4answers
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What happens to the units when squaring a variable?

What happens to the units of a squared variable? For example, if I squared velocity, would the units, metres per second (${\rm m}/{\rm s}$), change as well?
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Why does Coulomb's constant have units?

I think of Coulomb's constant as a conversion factor (not sure if this is correct). Kind of like how you would do calculations in kg and then times it by the conversion constant to convert your answer ...
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Does the unit of a quantity change if you take square root of it?

For example, I have a mass, m = 0.1kg and I square root it, giving me m = 0.316 (3s.f.) does the unit still stay as kg, or does it change?
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$\hbar$, the angular momentum and the action

Is there anything interesting to say about the fact that $\hbar$, the angular momentum and the action have the same units or is it a pure coincidence?