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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Drag force - dimensional analysis

I have tried the following example from the link: MIT OCW 8.012 PS1 It is about dimensional analysis. Derive an expression for the drag force on a ball of radius $R$ and mass $M$ moving with ...
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When can two quantities be added together?

Whenever two things are to be added together, one typically needs to check whether this actually makes sense, and an addition is said to make sense, in principle, when the units match up. Yet, ...
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1answer
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Expanding physical quantities in dimensionless parameters [on hold]

I have a system with two dimensionful parameters, say, chemical potential ($ \mu $) and temperature ($ T $). Now I want to write down an ansatz for any physical quantity (e.g, Greens function) at ...
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1answer
36 views

Rayleigh dimensional analysis [closed]

Rayleigh dimensional analysis is often used by scientists to find formulas through their physical dimensions. How does it work and can you provide an example?
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6answers
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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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4answers
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Fundamental question about dimensional analysis

In dimensional analysis, it does not make sense to, for instance, add together two numbers with different units together. Nor does it make sense to exponentiate two numbers with different units (or ...
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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 ...
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1answer
47 views

physical meaning of dimensionless parameter

What does it mean when there is nor not a dimensionless parameter in my model? In quantum harmonic oscillator, we don't have dimensionless parameter while in hydrogen atom case we have one which is ...
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8answers
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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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1answer
46 views

What is the difference between unit and dimensions?

If I say Height of a block = 2m, then would "Height" be called as a dimension
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2answers
659 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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11answers
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Why are angles dimensionless and quantities such as length not?

So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths. Ok so far, so good. Then came the question: ...
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1answer
143 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
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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 ...
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3answers
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Must the product of the two complementary quantities in an uncertainty relation have unit $\text{Js}$?

I know that the uncertainty principle is: $$\Delta p\Delta q \ge \frac{\hbar}{2}.$$. But do the units on the left-hand side of the equation always have to equal $\text{Js}$, i.e. $\text{energy} ...
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1answer
168 views

Drag - Dimensional Analysis / Buckingham Pi

Dimensional Analysis / Buckingham Pi Theorem I'm working on dimensional analysis and I'm having trouble. Here's a problem from my book I'm working on for practice. I'm suppose to consider a small ...
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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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3answers
160 views

Why are log scales so common?

I am currently reading a very nice book on scales in physics. There is a discussion on the different physical scales which are based on the effect of the corresponding phenomena, the given examples ...
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2answers
150 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 ...
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1answer
89 views

Relating period, volume, surface area and the velocity of sound by dimensional analysis

The question is:- There is a dimensional relation between period T, volume V, surface area A and the velocity of sound C. Assume that period increases with volume and decrease with increase in area. ...
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3answers
216 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
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Do bras and kets have dimensions?

I'm trying to understand more intuitively what bras and kets are, but some aspects of them remain a mystery to me. We usually think of $\psi (x)$ as having dimension of $[1/\sqrt{L}]$ so that ...
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0answers
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Exercises with solutions in dimensional analysis - reference request

I am currently trying to brush up on my skills in dimensional analysis, and computing with units. Is there a good source of worked examples, and exercises with solutions? I'd prefer to have solutions ...
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2answers
118 views

What does this equation mean? [closed]

So I have just entered 11th grade and started limits on my own but my Physics textbook has an equation which I don't understand, I suspect it uses integration which I haven't learned yet. So can ...
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0answers
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Intuitive understanding of multiplied units [duplicate]

I intuitively grasp division of units as "per", or "the amount you get out compared to what you get in," but when units are multiplied together, as in Newton-meters or whatever, I'm not really capable ...
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2answers
97 views

Dimensional Analysis Question [closed]

First of, I would like to say that I have tried this question, and have my answer as well, just not sure such a method of obtaining the answer is valid or not, therefore trying to look for help here. ...
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1answer
80 views

Buckingham-$\Pi$ theorem application: the case of only 0 or 1 dimensionless groups?

In dimensional analysis, we might consider a problem like: $$ f(q_1, q_2, ..., q_{n-1}) = q_{n} $$ where $q_i$ are physical quantities of interest. In order to determine how many $\Pi$ groups can be ...
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1answer
101 views

Is the dimension “number of particles” a fundamental, or derived dimension (based on mass), or does it depend on the context, or is it dimensionless?

I consider "fundamental quantities" to be those that have dimensions that are are like length, mass, time, temperature, and so on. "Derived quantities" have dimensions that can be written in terms of ...
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What does mathematical equivalence means here?

On Motls blog, http://motls.blogspot.com/2012/06/on-importance-of-conformal-field.html, while I was trying to understand what dimensional transmutation means, he said: I said that by omitting the ...
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1answer
101 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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2answers
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Natural unit conversion

I'm a bit confused about different notions of "natural units" that I encounter occasionally. I'm familiar with Planck units, and in particular I can understand the conversion between, say, metres and ...
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4answers
1k views

Simple Harmonic Motion - What are the units for $\omega_0$?

I'm trying to understand the units in: $$mx''+kx=0$$ And the general solution is $$x(t)=A \cos(\omega_0 t)+B \sin(\omega_0 t).$$ Let $\omega_0 =\sqrt{\frac{k}{m}}$ - the unit for the spring ...
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1answer
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What are the units of the creation and annihilation operators?

The creation and annihilation operators - also known as ladder operators are; $ \hat{a}^\dagger$ and $\hat{a}$ respectively. Using the equation $\hat{H} = \hbarω\left(\hat{a}^\dagger \hat{a} + ...
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Can dimension analysis be used in developing more advanced physics equations?

It is obvious that dimensional analysis can be used to derive many classical mechanics equations (excluding constants). As long as all the dependent quantities are known. My question is whether this ...
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1answer
123 views

Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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1answer
73 views

Does the path integral measure have dimension?

For example, in the field functional integral: $$\int D\phi \ e^{S[\phi]} $$ Does the $D\phi$ here have dimensions?
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What does a unit like $C^{1/5}$ or $kg^{1/2}$ physically mean?

I'm more of a math guy than a physics guy so bear with me.... In fractal geometry, fractals are considered to have fractional dimension. For instance an object such as the Koch curve has a fractal ...
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1answer
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Dimensionless numbers in relativistic theory

Dimensionless numbers allow physicists and engineers to extend the physical modeling landscape by reducing otherwise complex mathematics to a simple proportional relationship. For example by assuming ...
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5answers
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What is the significance of the “squared” in $E =mc^2$? [duplicate]

If $c$ is just an arbitrary constant, why don't we say $E=mc$ and define the value of $c$ to be $\sqrt{299 792 458} \approx 17314$ meters per second? Or, why not use $E=mc^3$?
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Is $0\,\mathrm{m}$ dimensionless?

Is $0 \, \mathrm m = 0 \, \mathrm s = 0 \,\mathrm {kg} = 0$? How do we define $[0 \, \mathrm m]$? I once was given an assignment where I was asked to deduce and write down some physical quantity. It ...
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1answer
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Converting $F^2$ to $C^2$

I'm trying to convert Fahrenheit squared $F^2$ into Celsius squared $C^2$. I know how to convert a value $x$ in $F$ into $C$ with: $\frac{5}{9}(x - 32)$ I also know how to convert a value $x$ in ...
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Connection between the cosmological constant $\Lambda$ and the cutoff scale $\Lambda$

I'm trying to understand the connection between the $\Lambda$ from cosmology and the $\Lambda$ from QFT. Cosmology: The cosmological constant enters the Einstein equations. In the special case of the ...
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2answers
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Is a vector and a unit vector dimensionless

Lets say I have a position vector $\vec r$. Is it dimensionless or does it have a dimension of length i.e $[L]$. Also does the unit vector $\hat r$ have a dimension?
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1answer
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Dimensional analysis explanation and teacher issues! [closed]

This is going to sound stupid but anyways. I am currently in a physics class and my teacher likes us to use dimensional analysis which I do not understand how to use or what to do with it! So firstly ...
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2answers
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Dimensional analysis, valid reductions of dimensions, and their physical interpretation

So I have been thinking about dimensional analysis and I have been thinking about quantities with components that have negative and positive exponents in the same expression. Two examples: ...
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0answers
59 views

Black Hole Entropy Calculation

I was watching "Leonard Susskind on The World As Hologram" ( youtube ). At one point he describes the way Bekenstein calculates the entropy of a black hole. Paraphrasing: Take a minimally sized black ...
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1answer
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Confusion in understanding wave number

The wave number is the number of complete wave cycles in a meter. So, $$K = \frac{1m}{\lambda}$$ and also, $$K = \frac{2\pi}{\lambda}$$ so according to both of the above equation how is $$2\pi ...
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1answer
243 views

making an equation dimensionless

I have a balance of energy equation as following (for a spherical particle that colliding with a spherical fluid droplet) Left hand side is for before collision and RHS for after that: ...
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1answer
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Formula for Electrical Arc Length

I was playing with some High-Voltage the other day, when a question popped into my head. Can you calculate length of an electrical arc? It probably would be proportional to :- 1. Voltage of the source ...
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2answers
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Units in gravitational $N$ body simulations

I am trying to write a code in Python to simulate $N$ bodies interacting through gravity. In particular I am trying to see whether a system of particles with random initial positions and zero velocity ...