Questions tagged [dimensional-analysis]

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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19
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4answers
3k views

Is dimensional analysis valid for integrals

Can we apply dimensional analysis for variables inside integrals? Ex: if we have integral $$\int \frac{\text{d}x}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{-1} \left(\frac{a}{x}\right),$$ the LHS has no ...
-1
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1answer
132 views

Converting GPa and TPa to N/m

I have a problem with converting units, in some papers, ultimate tensile strength has been shown with GPa or TPa, but in some papers, it has been presented with N/m. (not newton per square meter) As ...
2
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1answer
464 views

Mass Dimension of derivative in a Lagrangian

What is the mass dimension of the derivative $\partial$ in a Lagrangian? I am really confused about this. I read somewhere it is 1 and another place I saw it is -1. Please could someone clear this ...
3
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2answers
134 views

Stokes's law proportionality to radius

Is there a logical explanation why Stokes's drag $F_d=6\pi R \eta v$ is proportional to the radius, $R$ of the sphere? Naively I would have expected that it is proportional to the cross section, i....
1
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1answer
154 views

Are the 7 fundamental SI units able to differentiate between all elementary particles?

More specifically, can the 7 base SI units express qualities like quark strangeness (of quarks) and quark color? How do these SI units differentiate between different quarks (charm, up, top...)?
2
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2answers
78 views

Dimensions of a distance time relation

Recently I came accross a question which was:- Suppose the velocity of a moving particle varies with time as $$v=50t^2.$$ And we have to find out the acceleration at $t = 10s.$ I know that I can use ...
0
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1answer
136 views

Reformulate quantum harmonic oscillator Schrödinger equation using dimensionless quantities

I am trying to rewrite a Schrödinger equation using dimensionless quantities but here, the potential is perturbed by $\lambda x^4$: \begin{equation}V(x) = \frac{m\omega^2}{2}x^2 + \lambda x^4\end{...
11
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2answers
2k views

Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration?

The incompressible Navier-Stokes equations widely used in hydrodynamics don't have the gravitational acceleration. $$ \begin{align} \frac{\partial u_i}{\partial x_i} & = 0, \\ \frac{\partial u_i}{\...
0
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1answer
83 views

Rewriting Schroedinger's equation with $\hbar c$?

I am solving a special case of the radial component of Schroedinger's equation numerically. The equation looks like this: $$ u''(r) = \frac{2\mu}{\hbar^2}(V(r) - E)u(r). $$ $V(r)$ is a potential and $...
1
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1answer
102 views

Buckingham pi theorem: alternative pi terms and orthogonality

A question on the buckingham pi theorem: It provides one with the socalled pi terms forming linearly independent quantities based on the relevant dimensions occuring in the problem. They are a ...
1
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0answers
27 views

Dimension scaling of a model hamiltonian system?

I have a model 2dof double well potential Hamiltonian given as $$H = P1*P1/2 + P2*P2/2 + V(Q1,Q2) ,$$ where $$V(Q1,Q2) = -2*Q1^2 + Q1^4 - Q2^2 +Q2^4 +0.4*Q1*Q1*Q2*Q2 ,$$ there is no units are ...
2
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3answers
142 views

The $c^2$ factor is just a conversion factor from mass to energy?

I have some questions: The $c^2$ factor is just a conversion factor from mass to energy. What does it mean? I know it's about $E=mc^2$ and so on, but really I need deep understanding of it. I know ...
1
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0answers
96 views

Einstein's Field Equation via Buckingham Pi Theorem

I am trying to obtain the field equation using the Buckingham Pi theorem. I was wondering if there is a way to obtain the field equation using dimensions alone from just $G,c,g_{\mu\nu}$, and the ...
2
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2answers
235 views

Multiplication of units

I can understand the operation of dividing two units for example: 1 m/s it means that the object covers the distance of 1m in one second but really I can not understand the operation of multiplication ...
0
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1answer
34 views

How to refute the claim that action at a distance force can be a polynomial function of $r$?

Suppose that we have a body $B$ exerting action at a distance force on body $A$. Now if someone says that the dependence of force on distance between the bodies ($r$) is: $f(r)=\dfrac{1}{r^2}+\dfrac{...
0
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1answer
62 views

Dimensions in the Second Quantization of an Operator

Consider the one-particle operator $\hat A_{1p}$. As given in e.g. (Altland and Simons, 2nd ed, 2010; pg47) the second quantized version of this is given by: $$\hat A=\sum_{\mu,\nu} \left< \mu \...
0
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1answer
56 views

Can the logarithm quotient rule be used when numerator and denominator both have units?

Consider the example of trying to estimate exponential growth rate $\gamma$ from some series of measurements with units vs. time $y(t)$ using the model $y = y_0 \exp{(\gamma t)}$. Solving the model ...
-4
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1answer
66 views

Is the seconds cancel out in equation? [closed]

Do the seconds cancel out in this equation leaving seconds instead of seconds squared?
2
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1answer
126 views

Simple 2D Curl example with units?

I'm studying Calculus, not physics, but was curious how units work when we consider curl over a 2D velocity field. Given $\mathbf{F} = M(x,y)\mathbf{i} + N(x,y)\mathbf{j}$ the curl is defined to be $(\...
1
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0answers
69 views

What is the meaning of doing dimensional analysis in Condensed Matter physics?

In QFT, we do dimensional analysis because the superficial degree of divergence is related to the dimension of the dimension of coupling constant, but in Condensed Matter physics, the aim of ...
0
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1answer
58 views

Show that for some $f$ it holds that $\dot{\gamma}_0 = \dfrac{U^2\rho}{\eta}f\bigg(\dfrac{Ux\rho}{\eta}\bigg)$

I'm not a physicist and I'm having some trouble understanding the following problem: We model the ground by a horizontal flat plate (standing still) with the air or water flowing over it, assuming ...
2
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2answers
90 views

Problem related to dimensional analysis [closed]

In dimensional analysis, why is $\pi$ not considered a base quantity (length)? Why is it considered a magnitude?
0
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2answers
117 views

Is there a natural scale associated with polynomials?

This question is related to a previous question asked here. Power laws are scale invariant. They don't have a built-in or characteristic scale associated with them. Exponentials such as $e^{-x/\xi}$ ...
0
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0answers
148 views

What are the units of a scalar field if I only impose $c=1$?

I know that a scalar field in 4d in natural units ($\hbar = c =1$) has mass dimension 1. We can see this by requiring that the kinetic term in the action $$ \int \text{d}x^4 \: \partial_{\mu} \phi \: \...
0
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0answers
29 views

Can I morph dimensionful physical constants meaningfully? [duplicate]

Discussions in the literature (e.g. https://arxiv.org/pdf/1412.2040.pdf) say in no uncertain terms that it is meaningless to consider time variations of fundamental dimensionful parameters such as c, ...
3
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1answer
147 views

Sciama's paper, On the origin of inertia

In his paper "On the origin of inertia", Sciama identifies $\frac{\Phi + \phi}{c^2} = -\frac{1}{G}$ This identity has confused me because I wonder how the right hand side arises since $\frac{\phi}{c^...
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1answer
66 views

Given a Force $F$ in newtons, what are the appropriate units for a scalar $Q$ so that $F = Q \times 2.00\rm \:N = 20.0\:N$? [closed]

If: $$ \mathrm{force} = Q \times 2.00\rm \:N = 20.0\:N$$ then what does $Q$ equal? What are the appropriate units for $Q$ so the value of the force comes out with the correct units of newtons?
2
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0answers
126 views

Time rescaling in overdamped Langevin simulation

I'm simulating a system according to the Langevin equation (with inertia), however my friction coefficient is high enough that I am essentially in the overdamped regime on the timescales of one ...
1
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2answers
81 views

Dimensional analysis: a particular problem I don't know how to solve [closed]

I have the following configuration: in which a viscous fluid with dynamical viscosity $\mu$ and density $\rho$ slides down the inclined plane due to gravity $g$. After having solved the Navier-...
1
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2answers
599 views

Complete list of fundamental properties [closed]

What are all of the fundamental properties, that is all measurable quantities which are not derived from anything else? Many quantities are derived e.g. area is length squared, velocity is length per ...
0
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3answers
82 views

Is energy always measured in units of mass × length$^2$/time$^2$ in physics?

Is it always $ M\,L^2/T^2$? Is special relativity different from general relativity regarding the units of energy?
3
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5answers
657 views

Is dimensional analysis wrong?

In many physics textbooks dimensional analysis is introduced as a valid method for deducing physical equations. For instance, it is usually claimed that the period of a pendulum cannot possibly depend ...
1
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1answer
100 views

Units of the metric tensor or how to get the unit right for the line element

In this answer it is stated that the metric tensor elements have no physical unit, i.e. $[g_{\mu\nu}] = 1$. What is the convention to get the physical unit of the line element $ds = g_{\mu\nu}dx^\mu ...
0
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1answer
66 views

What are the mass dimensions of doublets and singlets?

Within the Standard Model (SM) each Lagrangian term has to have a mass dimension of [L] =4. While the mass dimensions of scalar fields $[\Phi] = 1$, Dirac fields $[\Psi] =3/2$ and Vector fields $A_{\...
0
votes
1answer
109 views

The Gauss's law for gravitational field and the unit system

Here $g$ is the gravitational field, $G$ is the gravitational constant, and $M$ is the total mass in the volume $V$. I wonder if this formula holds for any unit system. That is, does the coefficient $...
1
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2answers
180 views

Definition of stress-energy tensor

The image from the wiki article on the stress energy tensor gives $T_{00}$ as $1/c^2$ times the energy density. I believe this is incorrect and that the $1/c^2$ factor should be dropped. Am I ...
0
votes
1answer
135 views

Is frequency$\times$(time period) = 1 unit?

In my book, I have read that the frequency of sound is inversely proportional to the time period i.e., $1/T = \nu$. So does that mean $$\text{frequency} \times \text{time period} =1$$ i.e., is $\nu \...
0
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3answers
114 views

Dimensionless consistency and quantities

I am a chemical engineering student learning about dimensionless quantities. This is a practice question that I am trying. The Van der Waals equation of state can be used to predict the behaviour ...
0
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2answers
104 views

How does the gradient affect units in physics?

I intuitively understand the gradient in a mathematical sense, especially the fact that it points in the direction of maximum increase and easily tells us the function's sensitivity to change in each ...
1
vote
2answers
130 views

Minimal substitution, four-potential and units

When we make the minimal substitution \begin{equation*} p^\mu\rightarrow p^\mu+\frac{e}{c}A^\mu \end{equation*} the four-potential $A^\mu$ must be proportional to $1/e$ in order to ensure the whole ...
0
votes
1answer
83 views

Why is meters/second the same as meters per second? [closed]

In quantities such as speed where the derived (SI) unit is m/s, why do we pronounce it and interpret it as meters per second? My guess is that 1 m is associated with 1 second. Similarly, 5 m/s is ...
-1
votes
1answer
70 views

Dimensionality Dilemma: Dimensional Analysis Invalidates my Mathematical Model

I am trying to derive an equation that describes the rotational motion of an "auto-unravelling system": systems comprised of a material (string, chain, cloth etc.) wound around a cylinder and left to ...
2
votes
1answer
262 views

What's the matter with Planck mass $M_P$ in Einstein-Hilbert action?

The Einstein-Hilbert (EH) action is often written as $$S_{EH}=\frac{c^4}{16\pi G}\int d^4x \sqrt{-g}R\tag{1}$$ and often as $$S_{EH}=\int d^4x \sqrt{-g}M_P^2 R\tag{2}.$$ Comparing (1) and (2), one ...
-1
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1answer
110 views

How to convert a quantity in natural units

Suppose I am working with a system of units where $c = G = \hbar = 1$. I can then write e.g. a distance in units of kg by converting with a factor of $$ \frac{c^2}{G} $$ Now if I have an energy in ...
-1
votes
1answer
195 views

Why is $k$ taken as 1 in the derivation of $F=kma$? [duplicate]

In the derivation of F=ma, when we reach the point F=kma, we take k=1. Why can't we take 'k' as some other value?
0
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2answers
85 views

What allows us to treat physical units in algebra?

I have been thinking about this problem: $$Speed = \frac{Distance}{Time}$$ Following this, is makes sense that the units of speed is m/s. However, I do not follow why we are able to divide units to ...
-1
votes
1answer
233 views

Are ‘fundamental measures’ a thing?

The question I want to ask is: What measures are needed to describe the physical world and what are the fundamental ones of those, in the proper sense of the word fundamental? But that might be too ...
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1answer
88 views

Why is $E=\frac{nh} {2\pi}$ equal to the energy in the citation below, if h has the dimension of an action?

In this article on matrix mechanics in quantum theory you can read, in the subsection of the harmonic osscilator, that $$E=\frac{nh}{2\pi},$$ Where $E$ stands for the possible energies of the ...
0
votes
1answer
95 views

Why dimensional analysis is never off by more that $(2\pi)^{(\pm1)}$?

I've been reading about dimensions analysis and at one point it mentions that there could be constants that dimensional analysis fails to define and dimensional analysis is never off by more that $(2\...
0
votes
2answers
48 views

Confusion about airflow resistivity units

What is difference between $\frac{Pa·s}{m}$ and $\frac{Pa·s}{m^2}$? What does that "$2$" after "$m$" mean? I saw both versions,and I dont know if they are same thing or not,no idea what that $2$ is ...