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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How to determine that the renormalization constant $Z_3$ must depend only on $g$ and $\Lambda/m$

In Le Bellac's book, Quantum and Statistical Field Theory, the renormalization constant $Z_3$ is introduced with the equation $$ \Gamma^{(2)}_R(k^2, m^2, g) = Z_3 \Gamma^{(2)}(k^2, m_0^2, g_0; ...
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561 views

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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720 views

Does the Lennard-Jones force equation give its answer in Newtons?

I'm trying to do the dimensional analysis of the Lennard-Jones force to work out what units are being used in my MD simulation. The lennard Jones force is given as the negative derivative of the ...
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1answer
200 views

Functional derivative and units

The both sides of below equation don't give the same units, e.g. $$ \frac{\delta}{\delta \phi (\tau)}\int_a^b \phi (\tau') d\tau'=1\;. $$ where $a<\tau<b$. To show this assume that the field ...
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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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1answer
141 views

Unit analysis of a Fourier transform

This is an extension of a Phys.SE question I asked earlier, Fourier transform of two pulses of light. I start with 2 Gaussians, represented by the equation $e^{-(t/a)^2}\times e^{2\pi i d ...
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2answers
193 views

Understanding units manipulation (speed of falling coconut after 20m)

When I was on holidays, I was told a story about how someone passing under a palmtree and almost got a coconut fall on his head. Given that these palmtrees where about $20m$ high, we wondered at what ...
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126 views

Dimensional analysis of magnetic energy: dimensions of µ0 and H

When calculating the energy difference between the normal and the superconducting state in a superconductor at zero magnetic field, the result is as follows: Now I'm quite confident of this result, ...
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193 views

Dimensions in lagrangian potential

According to Mankowski flat space dimensions We can write, $$L= \int \text{dt} \text d^d{x} \left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
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1answer
82 views

Non-dimensionalization for spatially varying material parameters

For a homogeneous material of length $L$, we can write the heat equation as $$\rho c\frac{dT}{dt}=k\frac{du^2}{dx^2}, \text{ } x\in (O,L)$$ where $T$ is the temperature, $\rho$ is the thermal ...
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Scaling in Vlasov equations

This question is in reference to the paper, http://arxiv.org/abs/1301.7182 What exactly is the argument being made on page 6 and 7? One deduces that the function $\Delta$ has to be such that, ...
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297 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 ...
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1answer
102 views

The units of gain and number of atoms in population inversion in a laser

I am following my university course notes on amplification in laser media, and have come across expressions for the gain of a medium, but the notes are not exactly rigorous... The expression given for ...
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1answer
251 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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1answer
134 views

Why Planck scale is so important?

I know that Planck scale is the scale where both, gravity and quantum effects are relevant simultaneously. Are there more reasons?
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2k views

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 ...
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64 views

Suggestions contents for soliton theory [closed]

I have been studying solitons theory to make a note on dimension analysis for solitons At first I have derived one space dimensional kink solution for soliton theory. I want to go to higher ...
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3answers
73 views

Interaction photons-matter and dimensional analysis

I know that when photons pass through matter, the law that describes the intensity in function of the thickness is: $$I(x)=I_0 e^{-\mu x}$$ where $\mu = \rho \frac{N_a}{A} \sigma$ and ...
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5answers
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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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132 views

Units for physical constants

Someone told me that units for $G$ and $\epsilon_0$ (gravitational constant and Coulomb's constant) are placed there simply to make equations work dimensionally and that there is no real physical ...
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2answers
106 views

Dimensional analysis - the argument of functions

Suppose we have some function of time and space (1-D for simplicity) $G(x, t)$ which, by considering some equation relating $G$ to other quantities we know to be dimensionless. We now conclude that ...
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161 views

Curious relation between the dependance in ℏ of Planck units and units dimensions

Looking at Planck units, there seems to be a curious rule between the dependance in $\hbar$ of a Planck unit and the unit dimensions of the corresponding physical quantity. Let the dimensions of the ...
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Temperature in CGS (Gaussian) units

I've been struggling with conversion from Gaussian to SI units for sometime, trying to figure out how derived units in CGS (current, charge etc) relate to the SI units. But I couldn't find any ...
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Kingston corners [closed]

It is approximately 275km from WVWHS to Kingston corners. If your car gets 30.0 miles per gallon and gas costs 3.85 per gallon, how many cents of gas is burned from going from school to Kingston ...
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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 ...
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Where might hertz per dioptre actually be useful?

I once came across the strange, artificial unit "hertz per dioptre", which is dimensionally equivalent to "metres per second". Could this unit, by some stretch of the imagination, be used in some ...
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Can mass dimension of a field be viewed as another 'quantum number'?

While studying SUSY in 4D, I noticed the dynamical chiral superfield has dimension [GeV], whereas the dynamical vector superfield (for gauge theories) is unitless. Because I was introduced to the ...
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converting power spectrum to photon flux density

So I'm having trouble converting units and was hoping somebody could point out where I've gone wrong... It seems I'm missing something fundamental. a Power Spectrum has units $kW/m^2-\mu m$ for the ...
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3answers
374 views

What physical quantity has SI unit $\mathrm{kg}/\mathrm{m}$?

What physical quantity has SI unit $\mathrm{kg}/\mathrm{m}$? For example, the physical quantity with SI unit $\mathrm{kg}\cdot\mathrm{m}/\mathrm{s}^2$ is force $F$ and the physical quantity with SI ...
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371 views

Showing that position times momentum and energy times time have the same dimensions

I've been asked to show that both the position-momentum uncertainty principle and the energy-time uncertainty principle have the same units. I've never see a question of this type, so am I allowed to ...
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409 views

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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131 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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172 views

Gaussian integration and dimension argument

I made a mistake recently regarding the Gaussian density, by putting the determinant of the variance to the power $\frac{d}{2}$. Would the following argumentation be valid to highlight it should be to ...
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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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410 views

Why is the candela dimension J, not W?

According to the table at the bottom of the Wikipedia page for the candela, the dimension for candelas is J (joules). Why is this not W (watts)? The luminous intensity for light of a particular ...
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197 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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94 views

Introducing dimensionality of the $+$ and $-$ signs

Is it possible to introduce some dimensionalty as $\text{kg}$ for mass or $\text{m}/\text{s}^2$ for mathematical signs: plus - $+$ and minus - $-$. The main reason of this is to avoid some common ...
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150 views

Notation for two variables with same dimensions [duplicate]

What symbol represents "has the dimensions of", as in "x has the dimensions of d"? Does such a symbol exist?
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461 views

Planck time, distance, mass? Why do we take those values?

Say we want to make an educated guess for critical values of time, distance and mass, where quantum gravity effects are supposed to be non-negligible. These values are given the prefix "Planck-". Now, ...
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130 views

Dimension analysis of de Broglie equations

One form of one of the de Broglie's equations is this: $\lambda = \frac{2\pi\hbar}{p}$ Units: $\lambda = [m]$ $\hbar = [Js]$ $p = [\frac{kg m}{s}]$ $J=[Nm]$ How can one show with dimension ...
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How could the unit of a constant be unit of tension $N^{-1}$?

From my pervious Question:What are the units of the quantities in the Einstein field equation? i noticed that the unit of this constant $\frac {G}{c^4}$ is the unit of tenstion $$\frac ...
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842 views

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 $G$ ...
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767 views

Tips on teaching Dimensional Analysis?

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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357 views

Is dimensional analysis used outside fluid mechanics and transport phenomena?

Most dimensionless numbers (at least the ones easily found) used for dimensional analysis are about fluid dynamics, or transport phenomena, convection and heat transfer - arguably also sort of fluid ...
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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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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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326 views

Question about units in Lagrangian dynamics (inertia matrix)

I have a 3 degree of freedom system and my equation of motion is like this: $$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$ $M(q)$: inertia matrix $C(q,q_d)$: Coriolis-centrifugal matrix $G(q)$: potential ...
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dimensional analysis for gravitational radiation expression

on this paper, please refer to equation 2.117 for the power emitted for a rotating mass system: $$ P = - \frac{128}{5} G M^2 R^4 \Omega^6 $$ power in cgs should be (g is grams, m is meter, s is ...
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Understanding units and the units of the derivative operator

Suppose that $f$ is a function from unit $A$ to $B$, then what is the unit of $f'(x)$?. We can do $f'(x)\Delta x$ to get an estimate of $f(x + \Delta x)$. Since the latter has unit $B$, so has the ...
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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 ...