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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Is it possible to change units in order to simplify the value of an exponential?

I have the equation $$F=e^{E_0 i \pi}, $$ where $E_0$ is the time-independent electric field, and $F$ is just some important value I am trying to calculate. Obviously, it would be better if $F=-1$, ...
3
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2answers
196 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 ...
3
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3answers
290 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
140 views

Why are laws of physics always of product forms?

A first observation is that all the extant laws of physics are of product forms. This phenomenon is somewhat intriguing. The question is: why do law of physics always take, instead of a sum of two ...
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3answers
811 views

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 ...
3
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1answer
66 views

Deborah Number for harmonic excitation

I think I do not understand well the concept of Deborah number. It is presented in the sources available to me as the ratio between the relaxation time of a fluid and a characteristic time scale of ...
0
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1answer
158 views

Find out the dimension of $\frac{a}{b}$ [closed]

$E=b-\frac{x^2}{at}$ [x=distance,t=time, E=energy] I have tried following but don't know whether I am correct or not $$\frac{x^2}{at}=E$$ $$\frac{L^2}{aT}=ML^2T^{-2}$$ $$a=M^{-1}T^{1}$$ ...
3
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3answers
204 views

Is quantity a dimension? [closed]

We believe that time is a dimension and that $x$,$ y$, $z$ are dimensions in space. Is quantity a dimension like these? And if not, how do we have dimensionless numbers (like $e$, $\pi$ etc.)?
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1answer
139 views

The position of a particle at any time $t$ is given by $S = V0/a [1-e^{-at}]$. What are the dimensions of $a$ and $V_0$?

To find the dimensions of and V0, I must know the dimension of S and e. So I want to know it.
3
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1answer
132 views

How to interpret $t^2$? [closed]

I can't think of the meaning of squaring the Time (multiplying it by itself). It makes sense in Mathematics. But how can you figure it out in nature (or physics). As an example, the formula ...
0
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1answer
112 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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3answers
210 views

Apparent dimensional mismatch after taking derivative

Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them. ...
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4answers
811 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 ...
0
votes
2answers
3k views

Taking force, mass and length as base units, find the dimensional formula of velocity [closed]

My doubt is that how can force be considered as a base quantity. Is that possible? How can I represent the dimension of velocity using it?
0
votes
1answer
80 views

Finding dimensional formula

$$y(x,t)=2A\sin(Kx)\cos(\omega t)$$ $A$ and $x$ are in metre, $\omega$ is angular frequency. Then find dimensions of $A$ and $K$. In this equation how can I find the dimension of $K$?
2
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3answers
97 views

Is dimensional analysis always sufficient to establish equivalence of quantities?

In dealing with the Biot-Savart law, it was argued that $$ q\frac{d\vec{s}}{dt}\equiv Id\vec{s} $$ using the fact that the units are equal. Does this kind of argument always work? It seems too ...
2
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3answers
647 views

What is a proportionality constant? (Planck's constant)

I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. There are 2 things that im trying to verify: isn't the number that Planck's constant ...
5
votes
2answers
936 views

Exponential or logarithm of a dimensionful quantity?

I have a unit measure, say, seconds, $s$. Furthermore let's say I have a dimensionful quantity $r$ that is measure in seconds, $s$. What is the unit measure of $e^r$? ($1/r$ is in $Hz$.) My question ...
0
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1answer
148 views

Dimensional analysis

I was recently introduced to dimensional analysis and I wanted good references for learning the ideas behind it and representation of the natural world. I'm a grad student in biology. I don't have ...
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2answers
177 views

Why do we set $x^0 = ct$ instead of $x^0 = t$?

When we deal with Special Relativity and we start considering spacetime instead of space and time each at once, we usually see books saying that we consider a space with four coordinate $x^\alpha$ ...
0
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1answer
689 views

What does really mean by- power of a number or an exponential function is dimensionless? [duplicate]

Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples.
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1answer
530 views

Dimensional Analysis to Determine a Formula

The kinetic energy of a particle confined to a spherical region with a uniform internal potential depends on its mass, the radius of the sphere, and the Planck constant. An electron, confined to such ...
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2answers
520 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 ...
4
votes
1answer
164 views

How can Planck units be consistent with conflicting dimensions of mass?

I suspect I'm missing something obvious, but I'm coming up blank. I've gotten pretty comfortable with so-called natural units over the years: in doing quantum mechanics/QFT, it's common to set $c = ...
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2answers
503 views

What is dimensional units/quantity and dimensional state

First, I am not a native English-speaking student so I am not good at physics definitions in English. I participated in the MIT e-learning course on classical physics. The 1st lesson is about 3 ...
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2answers
2k views

Units of the Stokes-Einstein rotational diffusion coefficient

The Stokes-Einstein rotational diffusion relation tells us that we can write down a rotational diffusion coefficient for a sphere as: $$D_r \approx \frac{k_B T}{\zeta_f} \approx \frac{k_B T}{(8 \pi ...
3
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4answers
15k views

Definition of Ohm in SI basic units in words

One way Wikipedia defines Ohm is (this is also teached in school): $$1\Omega =1{\dfrac {{\mbox{V}}}{{\mbox{A}}}}$$ They describe this definition in words, too: The ohm is defined as a resistance ...
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8answers
6k views

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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1answer
284 views

Is there a physical quantity which is the reciprocal/multiplicative inverse of time?

Is there a physical quantity which is the reciprocal/multiplicative inverse of time? If time =distance/speed what is speed/distance. Please forgive my ignorance if there is a really simple ...
2
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1answer
121 views

Dimensional Analysis : Thermodynamics

I was coming across some notes online for phase transitions. In one of the places, the author has written the Claussius-Clayperon equation in this form, $$ \frac{d(ln P)}{d(ln T)} = \frac{T\Delta ...
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0answers
475 views

What prediction led to the vacuum catastrophe?

The disagreement between predicted and measured energy density of the vacuum is one of the great unsolved problems of science. According to this book the predicted energy density was obtained as ...
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8answers
3k views

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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3answers
2k views

What does the decay constant mean?

In my curriculum, the decay constant is "the probability of decay per unit time" To me, this seems non-sensical, as the decay constant can be greater than one, which would imply that a particle has a ...
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2answers
3k views

Integrating equations with units

I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. Example 6.1 During a collision with a ...
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5answers
373 views

Could velocity be taken as fundamental instead of time?

In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as ...
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1answer
166 views

Surface gravity for a rotating charged black hole

I have that the surface gravity (at the outer event horizon) for a Kerr-Newman black hole is $$ K_+ = \frac{r_+-r_-}{2(r_+^2+(J/M)^2)} = ...
5
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3answers
470 views

Units INSIDE of a Dirac Delta function

I know that the units of a Dirac Delta function are inverse of it's argument, for example the units of $\delta(x)$ if $x$ is measured in meters is $\frac{1}{meters}$. But, my question is what are ...
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0answers
363 views

Dimensional analysis to estimate order of magnitude of quantities

In the Coursera course From the Big Bang to Dark Energy on several occasions dimensional analysis was used to estimate the scale of quantities. This almost seems like a contradiction in terms to me, ...
4
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1answer
113 views

Problems with dimensions when solving an ODE

I'd like to solve the following differential equation: $$\frac{dQ}{dt}=\frac{k_BT}{m}-\frac{\alpha Q}{m}$$ where $Q$ has units of $\text{m}^2\text{s}^{-1}$, $k_B$ is Boltzmann's constant, $T$ is ...
3
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2answers
1k views

Mass dimension of coupling constants in various dimensions

Just a quick question: Suppose I want to consider QED or YM in 4 dimensions we always say that the coupling constants are dimensionless and that the field then has a specific mass dimension. What ...
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3answers
300 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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2answers
246 views

Confusion With How Dimensions Work

Form what I understand if you have an equation such as: $$v = v_0 + at$$ then the dimensions must match on both sides i.e. $L/T = L/T$ (which is true in this case), but I have seen equations such as ...
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2answers
5k views

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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5answers
498 views

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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0answers
204 views

Scale-invariant differential operator

For example, the differential operator Laplacian is $$\nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ My questions are: Is it scale-invariant? what is ...
0
votes
1answer
100 views

Non-dimensionalized Laplacian of Gaussian

From the perspective of dimensional analysis, in the Laplacian of Gaussian operator $$LoG(x,y,\sigma)=\frac{\partial^2g}{\partial x^2} +\frac{\partial^2g}{\partial y^2}.$$ I think $x,y$ are variables ...
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3answers
325 views

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) $$\langle x|\psi\rangle = [\frac{1}{\sqrt{Length}}]$$ ...
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2answers
371 views

Dimensional Analysis on Maximum speed of Sailboat [closed]

I'm doing the MIT Physics 1 : Classical Mechanics course, offered by OpenCourseware. I'm watching the first lecture and reviewing the slides, and don't seem to understand this question on Dimensional ...
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3answers
2k views

Frequency of small oscillation of particle under gravity constrained to move in curve $y=ax^4$

How to find the frequency of small oscillation of a particle under gravity that moves along curve $y = a x^4$ where $y$ is vertical height and $(a>0)$ is constant? I tried comparing $V(x) = \frac ...
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0answers
59 views

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; ...