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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Drag - Dimensional Analysis / Buckingham Pi

Dimensional Analysis / Buckingham Pi Theorem I'm working on dimensional analysis and I'm having trouble. Here's a problem from my book I'm working on for practice. I'm suppose to consider a small ...
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2answers
6k views

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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1answer
277 views

Why dimensionality of the Electric Charge varies with the spacetime dimensions?

The point is: We can find via dimensional analysis that the electric charge dimensionality varies with the dimension of space-time. $$[\text{charge}] = eV^{(3-D)/2}$$(You can see below the way I did ...
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3answers
158 views

Why are log scales so common?

I am currently reading a very nice book on scales in physics. There is a discussion on the different physical scales which are based on the effect of the corresponding phenomena, the given examples ...
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146 views

$c^4$ in Einstein field equations

I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $c^4$ in the denominator. the $8{\pi}G$ term can be obtained ...
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71 views

Relating period, volume, surface area and the velocity of sound by dimensional analysis

The question is:- There is a dimensional relation between period T, volume V, surface area A and the velocity of sound C. Assume that period increases with volume and decrease with increase in area. ...
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1answer
132 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
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214 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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5answers
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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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2answers
123 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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0answers
50 views

Exercises with solutions in dimensional analysis - reference request

I am currently trying to brush up on my skills in dimensional analysis, and computing with units. Is there a good source of worked examples, and exercises with solutions? I'd prefer to have solutions ...
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2answers
106 views

What does this equation mean? [closed]

So I have just entered 11th grade and started limits on my own but my Physics textbook has an equation which I don't understand, I suspect it uses integration which I haven't learned yet. So can ...
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19 views

Intuitive understanding of multiplied units [duplicate]

I intuitively grasp division of units as "per", or "the amount you get out compared to what you get in," but when units are multiplied together, as in Newton-meters or whatever, I'm not really capable ...
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2answers
76 views

Dimensional Analysis Question [closed]

First of, I would like to say that I have tried this question, and have my answer as well, just not sure such a method of obtaining the answer is valid or not, therefore trying to look for help here. ...
3
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1answer
64 views

Buckingham-$\Pi$ theorem application: the case of only 0 or 1 dimensionless groups?

In dimensional analysis, we might consider a problem like: $$ f(q_1, q_2, ..., q_{n-1}) = q_{n} $$ where $q_i$ are physical quantities of interest. In order to determine how many $\Pi$ groups can be ...
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1answer
94 views

Is the dimension “number of particles” a fundamental, or derived dimension (based on mass), or does it depend on the context, or is it dimensionless?

I consider "fundamental quantities" to be those that have dimensions that are are like length, mass, time, temperature, and so on. "Derived quantities" have dimensions that can be written in terms of ...
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0answers
50 views

What does mathematical equivalence means here?

On Motls blog, http://motls.blogspot.com/2012/06/on-importance-of-conformal-field.html, while I was trying to understand what dimensional transmutation means, he said: I said that by omitting the ...
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1answer
99 views

Square-Cube Law?

I've heard about something called the square-cube law. What is it? All I know of it is that it has something to do with mass of large objects and their gravitational influence.
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2answers
44 views

Natural unit conversion

I'm a bit confused about different notions of "natural units" that I encounter occasionally. I'm familiar with Planck units, and in particular I can understand the conversion between, say, metres and ...
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4answers
1k views

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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1answer
45 views

What are the units of the creation and annihilation operators?

The creation and annihilation operators - also known as ladder operators are; $ \hat{a}^\dagger$ and $\hat{a}$ respectively. Using the equation $\hat{H} = \hbarω\left(\hat{a}^\dagger \hat{a} + ...
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4answers
697 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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1answer
122 views

Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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1answer
69 views

Does the path integral measure have dimension?

For example, in the field functional integral: $$\int D\phi \ e^{S[\phi]} $$ Does the $D\phi$ here have dimensions?
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75 views

What does a unit like $C^{1/5}$ or $kg^{1/2}$ physically mean?

I'm more of a math guy than a physics guy so bear with me.... In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal ...
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1answer
58 views

Dimensionless numbers in relativistic theory

Dimensionless numbers allow physicists and engineers to extend the physical modeling landscape by reducing otherwise complex mathematics to a simple proportional relationship. For example by assuming ...
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5answers
206 views

What is the significance of the “squared” in $E =mc^2$? [duplicate]

If $c$ is just an arbitrary constant, why don't we say $E=mc$ and define the value of $c$ to be $\sqrt{299 792 458} \approx 17314$ meters per second? Or, why not use $E=mc^3$?
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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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1answer
48 views

Converting $F^2$ to $C^2$

I'm trying to convert Fahrenheit squared $F^2$ into Celsius squared $C^2$. I know how to convert a value $x$ in $F$ into $C$ with: $\frac{5}{9}(x - 32)$ I also know how to convert a value $x$ in ...
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0answers
39 views

Connection between the cosmological constant $\Lambda$ and the cutoff scale $\Lambda$

I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT. Cosmology: The cosmological constant enters the Einstein equations. In the special case of the ...
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2answers
173 views

Is a vector and a unit vector dimensionless

Lets say I have a position vector $\vec r$. Is it dimensionless or does it have a dimension of length i.e $[L]$. Also does the unit vector $\hat r$ have a dimension?
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36 views

Dimensional analysis explanation and teacher issues! [closed]

This is going to sound stupid but anyways. I am currently in a physics class and my teacher likes us to use dimensional analysis which I do not understand how to use or what to do with it! So firstly ...
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Dimensional analysis, valid reductions of dimensions, and their physical interpretation

So I have been thinking about dimensional analysis and I have been thinking about quantities with components that have negative and positive exponents in the same expression. Two examples: ...
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55 views

Black Hole Entropy Calculation

I was watching "Leonard Susskind on The World As Hologram" ( youtube ). At one point he describes the way Bekenstein calculates the entropy of a black hole. Paraphrasing: Take a minimally sized black ...
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1answer
47 views

Confusion in understanding wave number

The wave number is the number of complete wave cycles in a meter. So, $$K = \frac{1m}{\lambda}$$ and also, $$K = \frac{2\pi}{\lambda}$$ so according to both of the above equation how is $$2\pi ...
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1answer
235 views

making an equation dimensionless

I have a balance of energy equation as following (for a spherical particle that colliding with a spherical fluid droplet) Left hand side is for before collision and RHS for after that: ...
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1answer
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Formula for Electrical Arc Length

I was playing with some High-Voltage the other day, when a question popped into my head. Can you calculate length of an electrical arc? It probably would be proportional to :- 1. Voltage of the source ...
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2answers
114 views

Units in gravitational $N$ body simulations

I am trying to write a code in Python to simulate $N$ bodies interacting through gravity. In particular I am trying to see whether a system of particles with random initial positions and zero velocity ...
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1answer
36 views

Setting constants equal to 1 conditions

I have the following expression for the entropy of an ideal gas in a microcanonical ensemble, $$S=Nk_B\ln \left[ \frac{Ve}{N}\left(\frac{4\pi m e E}{3Nh_0^2}\right)\right] $$ Ideally I would like to ...
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2answers
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Dimensional analysis, dimensionless quantities and ratios

What justifies the "canceling out" of the same units? I have difficulty understanding the point of dimensionless quantities. Usually, when you have a concept like mass over volume, which is density, ...
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2answers
391 views

Checking units for equation with degree symbol

Using the following equation: $$ U = \left(\frac{B \times L \times \sin(\theta)}{C}\right)^{1/3} $$ I can calcukate the velocity of a flow traveling down a slope. I would like to check that the ...
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260 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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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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Deriving (dimensionless) physical constants from theory

The Wikipedia entry on Physical Constants says: With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants ...
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287 views

Center Of Mass Troubles

I understand the concept of Center Of Mass(com), but I am having a difficult time interpreting the equation of the simplified case of one-dimension. The book I am reading defines the position of the ...
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4answers
579 views

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: ...
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4answers
225 views

Problems with units of entropy in statistical thermodynamics

The statistical thermodynamics definition of entropy: $S = kN \ln (W)$ troubles me a lot with the problem of dimenstions. where $S$ is entropy; $k$, the Boltzmann constant; $N$ the number of particles ...
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1answer
60 views

I read, some time ago about a dimensionless constant in physics [closed]

and, my terminology is probably off, but, I think I can explain with an example. Take a Newton, which can be described as a KG*Meter/Second^2 - which frankly, written that way, looks confusing to me, ...
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Non-dimensionalizing the “bead on a rotating hoop, with viscous damping” problem

This is not a homework question. Rather, this is an exercise I have taken up on myself. In particular, I am trying to find an algorithmic way to non-dimensionalize known equations, using the ...
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Do the non-dimensionalizing equations that result from the Buckingham-$\pi$ algorithm necessarily have a unique solution?

Consider the Buckingham-$\pi$ algorithm: Let us say that we have $n+1$ relevant variables: $\{Q_0, ..., Q_n\}$. Let us say that we can define their dimensions in terms of $k$ basic dimensions. So, ...