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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Units inside a logarithm

I have troubles understanding a seemingly simple integral in a physical context. Take a look at $\int_{V_1}^{V_2} \frac{\mathrm{d}V}{V}$ which appears in isothermal expansions (V being the volume of a ...
2
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1answer
56 views

Deriving (dimensionless) physical constants from theory

The Wikipedia entry on Physical Constants says: With the development of quantum chemistry in the 20th century, however, a vast number of previously inexplicable dimensionless physical constants ...
10
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4answers
514 views

How to indicate that a unit is dimensionless [duplicate]

For my dissertation I am preparing a list of symbols used in the text, which basically is a table that consists of the symbol, a short explanation and the dimension it has as indicated below: ...
0
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1answer
52 views

I read, some time ago about a dimensionless constant in physics [closed]

and, my terminology is probably off, but, I think I can explain with an example. Take a Newton, which can be described as a KG*Meter/Second^2 - which frankly, written that way, looks confusing to me, ...
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0answers
10 views

Non-dimensionalizing the “bead on a rotating hoop, with viscous damping” problem

This is not a homework question. Rather, this is an exercise I have taken up on myself. In particular, I am trying to find an algorithmic way to non-dimensionalize known equations, using the ...
2
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0answers
13 views

Do the non-dimensionalizing equations that result from the Buckingham-$\pi$ algorithm necessarily have a unique solution?

Consider the Buckingham-$\pi$ algorithm: Let us say that we have $n+1$ relevant variables: $\{Q_0, ..., Q_n\}$. Let us say that we can define their dimensions in terms of $k$ basic dimensions. So, ...
4
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2answers
50 views

Complex dimensional analysis

Does complex numbers have physical dimensions? Is it sensible to talk about the dimensional analysis of $Z$ where $Z$ is the impedance of a mechanical oscillatory system? Or is it the $|Z|$ which has ...
0
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1answer
37 views

Scaling of an eigenvalue with the coupling constant

Consider the Hamiltonian $H = - \frac{d^2}{dx^2}+gx^{2N}$. Scaling out the coupling constant $g$, the eigenvalues scale as $\lambda \propto g^{\frac{2}{N+2}}$. So, we can drop the g dependence and ...
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3answers
85 views

How is it that Force = Mass $\times$ Length / Time ^2?

I understand how $F=ma$ but what I am looking for is a diagram, idiom or concept that explains how force can be explained (in a partial layman's terms) as a combination of the dimensions; Length, Mass ...
1
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1answer
56 views

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 ...
0
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0answers
24 views

What is the “Pidebuck” generalization of Buckingham π about?

The site http://www.oasification.com/PideBuckingham_en.htm links to two articles that seem to propose a generalization of the (Vaschy?-)Buckingham Pi Theorem which looks quite interesting, but ...
2
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3answers
862 views

Centripetal acceleration units

I perform some basic calculation for circular motion. Formulas we get from school are: $v$ - Linear speed with the units m/s $r$ - Radius of curve in meters $\omega$ - Angular speed with the units ...
12
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6answers
2k views

Can you find the length of a pencil without a ruler or clock?

Edit: Assume you do have access to all sorts of instruments, but they are all shrunk in proportion. My real question is: If you are shrunken (or expanded) by a constant factor and put in a room ...
22
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5answers
2k views

What do units like joule * seconds imply?

I can easily understand what divisive units imply, but not what multiplicative units imply. What I mean is, when I read "$12 \:\mathrm{eggs/carton}$", I mentally convert it to, "There are 12 eggs ...
-1
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1answer
59 views

Why is the first term of First Law of Black-Hole Thermodynamics in other unit than in joule? [closed]

http://en.wikipedia.org/wiki/Black_hole_thermodynamics#The_First_Law http://www.physics.umd.edu/grt/taj/776b/lectures.pdf (p.13) The 2 sources have various forms of the same law. I found both are ...
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3answers
69 views

For the Uncertainty Principle, Do the Units of the Two Complementary Quantities have to Equal Js?

I know that the Uncertainty Principle is: $△P•△Q=ħ/2$. But do the units on the Left Hand Side of the equation always have to equal 'Js', i.e. Energy x Time (the same is the Plank Constant, $h$) or is ...
0
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0answers
102 views

How can q=mcp(deltaT) be made dimensionless?

Specifically, how can I make $m C_p$ dimensionless? I've tried using the Reynolds number and Peclet number definitions to plug into there but the closest I've gotten to was: $q=\pi D Re Pe ...
-2
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1answer
36 views

Dimensional analysis of sums [closed]

Quick question: What is the dimension of the following fraction ($\sum_{i=1}^{n}a_i b_i )/ \sum_{i}^{n}a_i$ where $a_1,a_2,...,a_n$ are in kg and $b_1,b_2,...,b_n$ are in Newton?
0
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2answers
95 views

Is it possible to prove that units can be manipulated algebraically?

With expressions such as $$4\ \mathrm{\frac{m}{s}} \times 2\ \mathrm{kg} = 8\ \mathrm{\frac{m}{s}} \times 1\ \mathrm{kg}$$ We can justify that a $2\ \mathrm{kg}$ mass moving at $4\ \mathrm{m/s}$ has ...
1
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1answer
134 views

Why to write the Navier-Stokes equation with dimensionless quantities?

The Navier-Stokes equation is $$\rho \dfrac{D\mathbf{u}}{Dt} = -\nabla p+(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u})+\mu\nabla^2\mathbf{u}$$ Then if the flow is incomprresible, and the fluid is ...
5
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1answer
281 views

Why should it be allowed to set the einbein to unity?

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by $$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$ where ...
1
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2answers
144 views

Can Newton's Law of gravity be deduced using dimensional analysis?

I tried using dimensional analysis to deduce Newton's law of gravity but I wasn't able to do so as one of the equations were $0=-2$ which is a contradiction. But I thought that we can't do that ...
1
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1answer
35 views

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 $P_{0}$, ...
-1
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1answer
46 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 ...
0
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2answers
56 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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1answer
171 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: ...
7
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3answers
82 views

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 ...
0
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0answers
65 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 ...
1
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0answers
113 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 ...
1
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2answers
79 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 ...
2
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2answers
126 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 ...
1
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1answer
99 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 ...
1
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1answer
55 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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6answers
1k views

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 ...
0
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1answer
79 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?
1
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1answer
92 views

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
1k 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 ...
0
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1answer
145 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 ...
1
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1answer
73 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 ...
2
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0answers
28 views

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: ...
1
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1answer
435 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 ...
0
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1answer
102 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 ...
4
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1answer
108 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 ...
4
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3answers
205 views

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. ...
3
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3answers
230 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 $F=-1$, ...
3
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1answer
87 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 ...
4
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4answers
194 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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1answer
127 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 ...
6
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3answers
129 views

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 ...
3
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1answer
44 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 ...