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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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Capacitance of coplanar capacitor

I wanted to calculate the capacitance of capacitor with coplanar plates. How to do that? The dimensions of plate are width w, length l and the gap between plates is a.
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0answers
18 views

How to introduce dimensionality in a dimensionless framework?

This question is an extension of this one. I have been told that to introduce dimensionality in a dimensionless quantity I need to multiply with suitable parameters. For instance, for velocity I have ...
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0answers
39 views

An empirical formula for cosmological constant [on hold]

I found an interesting empirical formula to get a very close value to the cosmological constant in reduced Planck units. $$\Lambda \approx \frac{2^2}{3^{2^{2^3}}} = 2.88\times 10^{-122}$$ The ...
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2answers
72 views

How to find the corrsponding expression after working with natural units $\hbar=c=1$?

If one does long calculations in natural units how does one find the right expression in let's say SI units in the end? I know that natural units make the calculations easier and also help to show ...
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0answers
41 views

Ladder Operators - Quantum Mechanics [closed]

In my lectures notes, we define the raising operator as $$L_+ \Phi_{l,m_l} = \hbar \sqrt{l(l+1) - m_l (m_l + 1)} \hspace{1mm} \Phi_{l, m_l + 1}$$ Then we later look at spin-orbit coupling using ...
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1answer
70 views

Dimensional analysis of electron-positron spectrum

The theoretical formula for the numbers of particles per energy $\varepsilon$ with colliding photons with energies $\omega_{1}$ and $\omega_{2}$ is given by following expression (reaction $\gamma\...
2
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1answer
43 views

Electrostatic in 2D: dimensional analysis

After reading this very interesting post about the electric field and the electric potential of a point charge in 2D and 1D, I've understood that, for the $2D-$case, the following formulas hold: $$ \...
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1answer
21 views

How to deal with motion on a 2-D lattice in terms of dimension?

I am reading a paper titled: Random walks of molecular motors arising from diffusional encounters with immobilized filaments. There the authors consider the molecular motor moving on a 1-D protein ...
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3answers
130 views

Why does $\sqrt{\frac km}$ represent angular velocity and not frequency?

When I break down $\omega = \sqrt{\frac km}$ (angular velocity for a simple harmonic oscillator) into its units, I get: $$\omega = \sqrt{\frac{kg * \frac {m}{s^2}}{kg *m}}$$ which simplifies to: $$\...
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1answer
30 views

Is cross-sectional area over length classified separately from length?

The ratio of cross-sectional area to length (or its reciprocal) appears in several formulas, including those for electrical resistance and capacitance in terms of the resistivity and permittivity. ...
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0answers
25 views

Hubble parameter in radiation dominated era and dimensional analysis

Starting from dimensional arguments, is it possible to give the dependence of Hubble parameter on temperature $T$ in the radiation dominated (RD) era? In the RD era, the only relevant mass/energy ...
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4answers
81 views

Do smaller aircraft have lower take-off speeds?

Assuming there are two aircraft, each of the same density and each the same shape, am I correct in understanding that the smaller aircraft would have a lower take-off speed? I have explained how I ...
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0answers
66 views

What is the difference between the radian and the steradian? [closed]

In the SI, the radian is a dimensionless unit for the plane angle. The steradian is the unit for a solid angle where 1 sr = 1 rad$^2$, which means the steradian is also dimensionless in the SI. ...
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2answers
88 views

Are $2$ and $1/2$ universal constants? [closed]

For example, if the equation for energy were: $$E = mc^{2.713397972993}$$ clearly $2.713397972993$ would be a universal constant. And in the Einstein field equation: $$R_{\mu \nu} - \tfrac{1}{2}R \...
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0answers
38 views

The Implication of the Gravitational Constant in the Planck Mass [duplicate]

Why is it considered acceptable to derive an expression for the Planck mass [mP] using the gravitational constant G simply because the resulting units are of 'mass' when it is far from clear what the ...
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1answer
60 views

Are constants derived or calculated?

I am currently writing up a lab report on the determination of Planck's constant using x-ray diffraction and atomic spectra. In my introduction, I am talking about the history of Planck's constant, ...
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2answers
54 views

Why the quantities are dimensionless in curves plots?

In a lot of plots they use dimensionless quantities, why really we don't let quantities in their physical dimension and plot the curves normally.
5
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1answer
94 views

Why are angular velocity and angular frequency not measured in Hertz?

Recently, I was doing my homework and I found out that Angular Velocity and Angular Frequency can be calculated using $\omega=v/r$. This means the units of angular velocity and angular frequency are (...
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1answer
40 views

In the SHM equation $F= -kx$, $k =mw^2$ why not use $mf^2$ where $f$ is frequency $w$ here comes out to be $1/s$ not $\text{rad}/s$?

The reason I am stating this is because on calculating the units of w(omega) I found is equal to s^-1 not regular the rad/s. Proof: ...
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3answers
107 views

Do scalar quantities have magnitude only?

I've heard that vector quantities have both magnitude and direction but I've never heard that scalar quantities have magnitude only. Magnitude of vector quantities cannot be negative but what about ...
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1answer
60 views

Why do the units in the period of a mass-spring SHM not work out? [closed]

I am a high school physics teacher having students use the period of a mass-spring system with a known mass to determine the spring constant. We are practicing linearizing functions, so rather than ...
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1answer
23 views

Is (velocity=angular rotation*radius) dimensionally homogenous? [duplicate]

I have been driving myself mad trying to prove it one way or the other, I understand how it is derived and how to use it etc. but it still seems to me to be saying that (m/s)=(rad/s)*(m) which I don't ...
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1answer
29 views

Why is contact resistance measured in $\Omega~\mu m$?

In many papers the contact resistance of a metal in contact with a semiconductor is given in units of $\Omega~\mu m$, for example in the paper by Li et al. (Appl. Phys. Lett. 102 (2013), p. 183110): ...
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1answer
88 views

What makes a system of units self-consistent?

What are the rules governing the creation of a self-consistent system of units? To be clear, I'm not asking about making units universal or replicable, I'm asking about the mathematics governing what ...
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2answers
62 views

What is the difference between physical dimensions and physical quantities?

What is the difference between physical dimensions and physical quantities if the dimension of mass is also mass?
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1answer
37 views

Notation for feet and inches dimension

I am looking at a set of construction plans where all the dimensions read as x' - y". One example would be 4' - 6". I am confused by the dash in between the feet and inches. Is this supposed to mean ...
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2answers
66 views

Super confused over simple unit conversion from centimeters to cubic centimeters

I'm stuck in a really stupid question. It seems like the meaning of "centi" is chancing meaning. Because if: $ r = 5 cm = 5 \cdot 10^{-2} m $ so $c = 10^{-2}$ But say i need to take the cubic root ...
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4answers
169 views

Is the fine-structure constant related to the size of the observable universe?

The fine-structure constant $\alpha \approx 1/137$. In Planck units, this is also the charge of the electron squared, $e^2 = \alpha$ ($e \approx 0.085$). In Planck units the size of the observable ...
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4answers
195 views

How does natural unit make sense? [duplicate]

Both the fundamental constants $\hbar$ and $c$ have dimensions. In particular, $[\hbar]=ML^2T^{-1}$ and $[c]=LT^{-1}$. But in natural units, we make them dimensionless constants of equal magnitude. ...
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2answers
59 views

About the dimension of the SI units vector space

We know that the set of fundamental and derived physical units can be structured as a vector space over the rational numbers. In the International System of Units the dimension of this space is $7$ ( ...
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2answers
59 views

When is the order of magnitude not equal to the exponent of scientific notation?

Explain why the order of magnitude is sometimes not the same as the exponent in scientific notation. It is because of the units?
2
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1answer
110 views

When do you use Quantum Mechanics? [duplicate]

Given a problem, how does one know whether to use quantum mechanics or classical mechanics? Take for example electron scattering from a nucleus. The electrons are given a wavefunction in this case. ...
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3answers
151 views

Where is the line between Quantum and Relativity?

Its often said QM is for the very small and GR for the very large. This brings to mind that there should be some limit at which one starts to apply and the other stops. Now I know there are more ...
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2answers
60 views

Dimensions of mass in terms of Length and Time

From Maxwell's Treaties on Electricity and Magnetism: For acceleration due to attraction of a mass m at a distance r is by the Newtonian Law m/(r^2). Suppose this attraction to act fro a very small ...
5
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1answer
75 views

Hydrodynamic interaction between two spheres in $Re\ll 1$ flow

I am studying the interaction between two spherical particles of radius $a$ in a low Reynolds number flow. Because of linearity, I know that their respective velocities will be linear in the forces ...
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1answer
48 views

A question about natural/geometrized units

I had a question about the following document- Natural units I understand the conversion factors. But if you look at the tables, they take an SI unit, say 1 kg, convert it into geometrized units, say ...
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2answers
48 views

Factors of $c$ when giving masses in natural units?

I am starting a course on particle physics, and have been introduced to natural units. I am slightly confused, because we are using 'natural units', and yet masses are stated as, for instance, $139....
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0answers
19 views

Dimensionless properties of turbulence with law of the wall

The velocity profile is defined as law of the wall is defined as $u^+ = f(y^+)$, where $u^+ = \frac{\bar{u}}{v^*}$; $y^+ = \frac{yv^*}{v}$ and $v^*=\sqrt{\frac{\tau_w}{\rho}}$. How would one then non ...
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1answer
19 views

Units of forcing function in the inhomogeneous wave equation

The units of the d'Alembertian are distance$^{-2}$. It should be the case that the inhomogeneous wave equation describing $$\square u = f$$ should have matching units on both sides. My understanding ...
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3answers
84 views

Confusion about Unit Systems

I was reading about Deuterium and came to know that its binding energy is $2.22$ MeV. I am curious whether this energy is in the natural unit system where $\hbar = c =1$. For instance, the energy ...
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2answers
60 views

Is there any constant with unit meter-second?

I wanted to know if there exists a constant with a unit meter-time or say length-time. Dimensionally [LT]. I have searched browsed a lot. Is there any quantity arising with such a unit?
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1answer
34 views

Length dimension in the Lane-Emden equation

I was deriving the Lane-Emden equation from the hydrostatic equation and the polytrope. I was following the procedure presented by Carroll & Ostlie's book. I was stuck on this part, it said that ...
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0answers
34 views

Laminar vs. turbulent boundary layer equations

The 2D boundary layer equations (continuity and momentum) are given by: Can these also be applied to turbulent boundary layers? These simply come from Navier-Stokes and are simplified with scaling ...
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2answers
104 views

Is length an extensive property?

From my experience, volume, surface and length are extensive properties. Indeed : the reunion of two cubes of 1 $m^3$ leads to a cube of 2 $m^3$ the reunion of two tiles of 1 $m^2$ leads to a tile ...
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2answers
69 views

Is gravitational constant a rational number? [duplicate]

The question is the title. But I'm quite doubtful if this question is meaningful or not. Since this constant is obtained by experiment, we can never know its exact value, unlike $π$ or $e$. Is it ...
3
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1answer
167 views

Non-dimensionalization and perturbative expansion

I need to expand an equation, of the form $$\dot{r} = \gamma(a,\mu) F_1 + g(\mu,\ell,h,R) F_2$$ in powers of $\epsilon = a/\ell$. So I thoughts I would non-dimensionalize it first. I know that $$\...
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3answers
115 views

Dimensional analysis - application to logarithms

I read some nice threads about this topic: physics StackExchange maths StackExchange stats StackExchange However, it still puzzles me that logarithm of some physical quantity has no units. Example, ...
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3answers
2k views

Why is torque sometimes reported in kg m, instead of the usual N m?

On various websites I see torque expressed as $\rm kg\: m$, but I was always thought that torque is $\rm N\:m$ or $\rm kg\: m^2/s^2$. These are clearly not the same, so why are they called the same, ...
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2answers
81 views

What does it mean for a unit vector to have a magnitude of 1?

Imagine a Cartesian coordinate system whose origin is associated with two unit vectors, ê and â, in a 2D-space. Now, let 0.5 cm be the unit of length in this coordinate system. The magnitude of a ...
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1answer
51 views

What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...