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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Proportionality and units

This might be very easy, but I'm not 100% sure how it's done. Lets say I have this equation: $$R = R_{0} \cdot \left[1 - \frac{P_{0}R_{0}}{GM_{0}\rho_{0}}\right]^{-1},$$ where I know $P_{0}$, ...
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31 views

Operations on physical quantities [closed]

I know what quantities like meter and second are, they are a certain quantity of one-dimensional space and a certain duration of time respectively. And I know what a measurement of a quantity using a ...
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2answers
48 views

What is the Planck quantity of an expression?

I don't know what a Planck quantity is (I tried google), but someone at school gave me this problem. As you know, I have no idea how to approach this due to the weird terminology. Find the Planck ...
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2answers
37 views

How do I make an equation to be dimensionally consistent? [closed]

Velocity is related to acceleration and distance by the following expression: $$v^2=2ax^P$$ Find the power P that makes this equation dimensionally consistent. $$\frac{v^2}{2a}=x^P$$ ...
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50 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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In terms of scale, where does the concept of Reynold's number cease to have meaning?

The Reynolds number is classically described in terms of pipe geometries but its use has also been usefully extended to other more complex surface geometries to predict transitional flow behavior. But ...
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54 views

Choice of units when truncating Taylor series for physical quantities

It is common practice in physics to truncate Taylor series of (possibly) very complicated functions to obtain a good approximation of the relevant physical behaviour; for example, the Coulomb ...
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41 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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2answers
75 views

How can I understand how $\text{m}^2/\text{s}^2$ is related to $\text{J}/\text{kg}$?

$E=mc^2$ is the famous equation that states the equivalence of mass and energy, with a conversion factor in units of $\text{m}^2/\text{s}^2$. But in my naive mind, the conversion factor of mass and ...
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2answers
109 views

How to recover units?

Theorists frequently use convenient units like $\hbar=1$ or $m=2$ or whatever is useful to simplify the notation in the problem. And after all the calculations are done the units are recovered based ...
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1answer
61 views

Dimensional Analysis in Electromagnetism (SI vs Gaussian-cgs)

Looking at Konopinski's formula for conjugate momentum (in the comment after equation 3 of "What the Vector Potential Describes"): p = M v + q A /c it is plain enough that M v is momentum, but if we ...
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49 views

Dimensions of wave equation

If you take the homogenous wave equation: $$-\Delta_x u(x,t) + \frac{1}{c^2} \frac{\partial^2 u}{\partial^2 t} (x,t) \ = \ 0 \ \ \mathrm{in} \ \Omega \times (0, \infty),$$ with some proper initial- ...
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Why is it natural to look for solutions involving dimensionless quantities?

While studying the Heat Equation, I got stuck in a statement in my book. It says: We have seen that the combination of variables $\displaystyle \frac{x}{\sqrt{Dt}}$ is not only invariant with ...
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1answer
71 views

Relationship between temperature and energy

What is the definition of temperature in relation to energy? I'm mostly interested in general dimensional terms. Is temperature the kinetic energy per mass? Or kinetic energy per volume?
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Expansion in Quantum Fluctuations of the Path Integral

In this post: Dimensionless Constants in Physics there is a discussion about dimensionful vs. dimensionless constants in physics. In the context of this discussion, I'm wondering about the ...
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2answers
180 views

Unit of gradient/slope?

So I have a graph: The value of the gradient/slope is $1.6±0.4$ and the value of the intercept is $0.9±0.4$. But what are the units of the graph? Is the unit of the gradient $v^2M^{-1}$? What about ...
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1answer
108 views

help with absolute pressure to gauge pressure derivation steps

I would like some help with the explicit math steps to go from equation 2 to 3. These equations are presented in a paper that I am reading. I will show where these equations came from and my attempt ...
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1answer
69 views

Distance and velocity question

I know that speed is the derivative of distance. So integrating speed should give you distance. Let's suppose we have a speed which obeys this function: $$ v(x) = 2^{2^x} $$ So at time 0 the speed ...
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Quantum Efficiency Estimation

Might there be a way to do a rough estimate of the quantum efficiency of a photo-detector like a CCD or CMOS sensor based only on a picture taken with it? I've read papers and guides (like this one: ...
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132 views

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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93 views

In general, could any ad-hoc relationship of constants be useful?

In general; if one creates an ad-hoc relationship of constants, can we use it to solve equations OR is it just an abstract/artificial math construct? I'm a grad student and as we all know, these ...
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1answer
75 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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why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units? [duplicate]

so this might have some duplicated inquiry that this question or this question had, and while i think i have some of my own opinion about it, i would like to ask the community here for more opinions. ...
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218 views

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 $F=-1$, ...
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1answer
72 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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120 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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125 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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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 ...
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37 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 ...
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1answer
154 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 $$\frac{x^2}{at}=E$$ $$\frac{L^2}{aT}=ML^2T^{-2}$$ $$a=M^{-1}T^{1}$$ ...
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3answers
109 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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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.
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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 ...
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187 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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339 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 ...
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2answers
410 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?
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61 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 $K$?
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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 ...
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139 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 ...
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2answers
227 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 ...
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1answer
94 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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127 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 $x^\alpha$ ...
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120 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
123 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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31 views

Justifying order of magnitude reasoning

So in the context of a set of notes I am reading about acoustics I get to equation (23) in this paper. Basically it comes down to showing that (note the dots above the a's meaning time derivative!) ...
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Why dimensionality of the Electric Charge varies with the spacetime dimensions?

The point is: We can find via dimensional analysis that the electric charge dimensionality varies with the dimension of space-time. $$[\text{charge}] = eV^{(3-D)/2}$$(You can see below the way I did ...
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60 views

Order of magnitude estimation for some intriguing questions

The physics TA showed us a few examples in which one can estimate many things from first principles and sound logic based and scaling arguments. This led usually to understanding of why some numbers ...
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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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124 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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160 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 ...