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.

learn more… | top users | synonyms

1
vote
0answers
58 views

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 ...
4
votes
4answers
312 views

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 ...
1
vote
1answer
116 views

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} + \...
10
votes
4answers
814 views

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 ...
1
vote
1answer
82 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?
1
vote
2answers
89 views

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 ...
0
votes
1answer
98 views

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 ...
13
votes
2answers
706 views

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 ...
1
vote
5answers
256 views

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 $E=mc^3$?
1
vote
1answer
344 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 ...
0
votes
1answer
82 views

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 $...
2
votes
3answers
185 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 ...
1
vote
1answer
83 views

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 ...
0
votes
3answers
2k views

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?
0
votes
1answer
48 views

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 ...
4
votes
2answers
151 views

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: seconds/...
1
vote
0answers
78 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 ...
-1
votes
1answer
131 views

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 =...
0
votes
1answer
71 views

Setting constants equal to 1 conditions

I have the following expression for the entropy of an ideal gas in a microcanonical ensemble, $$S=Nk_B\ln \left[ \frac{Ve}{N}\left(\frac{4\pi m e E}{3Nh_0^2}\right)\right] $$ Ideally I would like to ...
2
votes
2answers
503 views

Checking units for equation with degree symbol

Using the following equation: $$ U = \left(\frac{B \times L \times \sin(\theta)}{C}\right)^{1/3} $$ I can calcukate the velocity of a flow traveling down a slope. I would like to check that the ...
5
votes
2answers
378 views

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
votes
0answers
123 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
votes
4answers
999 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
votes
1answer
88 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, ...
1
vote
0answers
48 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
votes
0answers
31 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
votes
2answers
115 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
votes
1answer
55 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 ...
1
vote
3answers
1k 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
vote
2answers
231 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
votes
1answer
192 views

Moment of Inertia and Linear Dimensions

For similar solid bodies made from constant density, how does the Moment of Inertia about a particular axis vary with linear dimensions? This is from an school textbook. I have covered all of MI ...
3
votes
3answers
7k 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
votes
6answers
3k 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 ...
29
votes
5answers
3k 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
votes
1answer
71 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 ...
1
vote
3answers
131 views

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} \...
1
vote
2answers
361 views

Why is the Fermi coupling constant always expressed in units of $(\hbar c)^3$?

Everywhere I've looked so far (such as NIST) the Fermi coupling constant $G_F$ is always expressed as $$\frac{G_F}{(\hbar c)^3} = 1.166 364(5) \times 10^{-5} \textrm{ GeV}^{-2}$$ never as just ...
0
votes
0answers
365 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 \frac{k}{...
-2
votes
1answer
57 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
votes
2answers
179 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
vote
1answer
648 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 ...
7
votes
1answer
411 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 $e$...
2
votes
3answers
703 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
vote
1answer
138 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 $P_{0}$, $M_{0}$...
-1
votes
1answer
75 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
votes
2answers
2k 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. $$\frac{v^2}{2a}=x^P$$ $$P={\log_x ...
2
votes
1answer
466 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: \begin{equation}...
7
votes
3answers
183 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
votes
0answers
104 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 ...
4
votes
1answer
330 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 ...