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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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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
178 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
45 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
55 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?
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1answer
102 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
141 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
48 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
69 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
34 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
44 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
16 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
18 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
81 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
59 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?
3
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1answer
30 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
31 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
93 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
65 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
164 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
102 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
56 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
48 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 ...
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1answer
50 views

Dimensional analysis on diffusion equation

I was studying the equation of motion for the probability density function of the position coordinates of the Brownian particles, also known as the Smoluchowski Equation (SE). Particularly, I came ...
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1answer
85 views

Mass dimension of an $n$-particle scattering amplitude in 4D

For the 4-dimensional case, and using the cross-section formula, how can we show that the mass dimensions of an $n$-particle amplitude must be $$[A_n] = 4-n~?\tag{2.99}$$ My understanding is that the ...
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1answer
70 views

Harmonic oscillator energy difference between $(n+\frac{1}{2})h \omega$ and $(n+\frac{1}{2})\hbar \omega$

When I was studying the Harmonic Oscillator using the Schrödinger equation, I was told in lectures to pay attention to the units. There were 2 different equations given for the Energy of a Harmonic ...
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1answer
33 views

Non-relativistic limit of the cosmological constant

Usually, when we apply the non-relativistic limit ($c \rightarrow \infty$) to relativistic equations, the cosmological constant $\Lambda \sim \mathrm{L}^{-2}$ is simply offhandedly neglected by ...
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3answers
112 views

Why is angular frequency $\omega = \sqrt(k/m)$ dimensionally correct?

So I'm learning about simple harmonic motion, and I came to the part where the differential equation $$\frac{\mathrm d^2x}{\mathrm dt^2} = -\frac{k}{m} x$$ is solved and simplified to $$x(t) = A\...
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2answers
42 views

Dimensional analysis on kinematic equation [closed]

I have tried to do dimensional analysis on the equation $v=u+at$. It has resulted in $v=2\,\mathrm{ms}^{-1}$. However, the units of velocity are clearly just $\mathrm{ms}^{-1}$. What have I done wrong?...
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4answers
117 views

$E=mc^2$: Why does the speed of light constant affect the Energy or Mass of an object? [closed]

So this is really just for fun. I often talk to my friend who studied some Physics degree (or similar) and he simply cannot accept the possibility it could be wrong in any way. To the point where he ...
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3answers
99 views

What is $\text{kg}\cdot \text{m}^2$ concretely? (multiplication of units)

I have problem to understand what $\text{kg}\cdot \text{m}^2$ (moment of inertia) is. So, for example a force does a work of $3\text{J}=3\text{Nm}$ means that the force can displace a weight of $3\...
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2answers
46 views

Dimension of an Exponential with Dimension

My thermodynamics professor gives a formula for ideal gas like this: $$s _ { 2 } - s _ { 1 } = c _ \mathrm{ v , a v g } \ln \frac { T _ { 2 } } { T _ { 1 } } + R \ln \frac { v _ { 2 } } { v _ { 1 } }.$...
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5answers
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Couldn't we always redefine units so that inertial mass and gravitational mass are equal?

It is a known fact that inertial and gravitational masses are the same thing, and therefore are numerically equal. This is not an obvious thing, since there are even experiments trying to find a ...
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1answer
27 views

Units of Klein-Gordon equation

I'm looking at the units of the Klein-Gordon equation $$u_{tt} - c^2\Delta u = -\frac{m^2c^2}{\hbar^2}u. $$ Disregarding the units of $u$, which are the same everywhere and so cancel, I get $seconds^{-...
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1answer
81 views

Getting the density of states for photons

I know that the density of states $g(\epsilon)d\epsilon$ is the number of states in the energy range $[\epsilon, \epsilon + d\epsilon]$. I considered a system of non-interacting free photons in 3 ...
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0answers
43 views

General question about making differential equations dimensionless

Suppose you have a set of differential equations that you wish to normalize/make dimensionless. From what I've seen, you can usually use dimensional analysis to figure out a good choice of constants ...
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2answers
80 views

Can we set cosmological constant equal to one?

People often say that the cosmological constant is too small. $\Lambda=10^{-120}$ in Planck units. Can we set $\Lambda=\hbar=c=1$ ? If so what would this give for $G$, the gravitational constant in ...
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1answer
48 views

Is there a general algorithm for conversion of units?

I'm not exactly sure where the best place to put this, as it's more of a general question about dimensional analysis. I decided I was tired of having to convert units all of the time, and was not ...
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1answer
62 views

Derive formula for air resistance $F = \frac{1}{2}CAdv^2$ through dimensional analysis

I have an assignment, where I’m required to derive the formula for air resistance for a falling object. $$ F = \frac{1}{2}CAdv^2 $$ where $C$ is drag coefficient, $A$ is the cross-sectional area, $...
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2answers
49 views

How to do dimensional analysis?

$$mgh = \frac{mc^2}{\sqrt{1-(v/c)^2}}-mc^2.$$ In dimensional analysis do we just ignore the square root? Or do we solve what’s inside first then we do the square root? Do we say $(v/c)^2$ is 1 as ...
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1answer
48 views

Constants of proportionality in Force Equations/ Physics in General

I was in physics class and we were talking about the gravitational constant G (6.67 x 10^-11 Nm^2/Kg^2). The question came up: "Why does $F= (GMm)/r^2$ have a constant of proportionality and not $...
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1answer
59 views

Definition of a meter and Newtonian law of Gravity

Newtonian law of Gravity: $$F_g = \frac{m_1 m_2}{l^2} G$$ $$G = 6.7 * 10^{-11} \frac{m^3}{kg * s^2}$$ A meter is defined as: the length of the path traveled by light in a vacuum in $1/...
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1answer
84 views

Can a physical quantity be of different dimensions depending of the system of measurement?

When comparing the Wikipedia articles on the International System of Units, the Planck unit system, and the geometrized unit system one question arises: can a physical quantity be of different ...
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2answers
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Where do the Lennard-Jones dimensionless/reduced units come from?

I'm a little bit confused as to where the expressions for the Lennard-Jones Potential Dimensionless units shown in this Wikipedia chart come from. I know they're derived from the L-J force equation ...
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5answers
907 views

Why do we nondimensionalize the Schrödinger equation when solving the quantum harmonic oscillator?

I read about how to solve the Schrödinger equation for the quantum harmonic oscillator in one dimension. It started with the Schrödinger equation, $$ \frac{p^2}{2m}\psi(x, t)+\frac{1}{2}m\omega^2x^2\...
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1answer
82 views

Write the dimension of 1D wave function? [closed]

I want to know how to find the dimension or unit of one-dimensional wave function
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135 views

What are the units of probability density?

Units of probability density? If bound electron is thought of as a cloud of charge, and it's charge density is proportional to the probability density. Then coulombs /m3 proportional to?
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1answer
53 views

What does the Planck-like charge $Q=\sqrt{\frac{\pi}{c\hbar \varepsilon_0}}$ represent?

While experimenting something with equations I got a equation of charge which is not Planck's charge, dimensionally it is correct. I used dimensions method to get this. $$Q=\sqrt{\frac{\pi}{c\hbar \...
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1answer
69 views

Is spacetime defined mathematically without using $c$ speed?

Is there a mathematical definition of spacetime that does not use $c$ speed as a conversion factor or involve the spacetime interval? If not why?
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3answers
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How does relativity dimensional contraction affect point like particles such as the electron and neutrino?

I might be misunderstanding a basic concept here, so forgive me. I know that the faster an object gets, the more it's dimensions will contract according to the following equation: $${1\over D} = 1-{V^...