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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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General question about normalizing differential equations

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
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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
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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
57 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
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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
42 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
55 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
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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
821 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
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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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2answers
106 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
45 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
68 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
50 views

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^...
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0answers
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Terms in photometry

Difference between luminous energy, illumination , luminous flux ,luminous intensity. Also describe how these terms are related with each other and please specify units as well.
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3answers
101 views

$\rm kg$ is to symbol, as kilogram is to unit as mass is to what?

I'm sure this has a very obvious answer but I can't even work out how to properly phrase the question. How do you collectively define what mass, temperature, and area actually are? For example, $\rm ...
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1answer
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Question on units for voltage equation of a voltage at a distance z from a single line charge

I have a very basic question on the units for the equation of the potential at a distance: $z$ from a single line charge. The electric potential due to a very long line charge at some distance $z$ is ...
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3answers
58 views

What are the units of time when planck's constant is equal to 1?

If I express a Hamiltonian $H$ in units of Hz by dividing the energy terms in the Hamiltonian by hbar $\tilde{H}=\dfrac{H}{\hbar}$ which means you set $\hbar =1$. Then what are the units of time? Also ...
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1answer
50 views

Bohr's atoms model

Why might Bohr have been especially curious about the possible values of the angular momenta of electrons in quantum mechanics? What looks special about them?
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1answer
39 views

Unit vector in displacement

When we use vectors in physics why does the unit vector (for displacement) equals magnitude of 1 or magnitude of 1m?
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2answers
100 views

How to determine the units of my integral? [closed]

Given that I have some coefficient(i.e. a number) which is to be determined from a radial integral: $$b_{n00} = \frac{(2\pi)^{1/4}}{\sigma^{3/2}} \frac{1}{\sqrt{3}} [C(000|000)]^2 \int^{\infty}_{r = ...
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4answers
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Why is the speed of light used to define the fourth axis of spacetime?

The four axes of spacetime are $x, y, z$ and $ct$, where $c$ is the speed of light, and $t$ is time. Why is the speed of light (not any other speed) used to define the fourth axis of spacetime? If ...
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1answer
38 views

Converting $\mathrm{ps/nm}$ to $\mathrm{ps}^2$

I have a dataset in the unit $\mathrm{ps/nm}$ for many different $\lambda$ which I want to convert to $\mathrm{ps}^2$. I guess I can assume that I only deal with Gaussian bandwidths such that $1\ \...
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0answers
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Dimensional Formulae

Reading Bridgman's Applications to Theoretical Physics, I found a discussion on dimensional analysis where the charge of an electron is expressed in terms of mass, time, and length. What did he mean ...
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0answers
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What are the units for state vectors in QM? [duplicate]

Specifically, it seems that a state vector should be dimensionless because: $| \langle \psi | \psi \rangle |^2$ is a probability which is dimensionless However, I find myself doing manipulations ...
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1answer
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Meaning of units in the Heisenberg Uncertainty Principle

So we have the inequality $\Delta x\,\Delta p\geq \hbar/2$ Why is the uncertainty of position in meters? Isn't uncertainty measured in probabiliies? Does it make sense to say that the uncertainty in ...
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4answers
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Is a unit vector really unitless and dimensionless?

According to my textbooks, a unit vector has no units and no dimensions, but is only used to specify direction. It only shows the orientation of a corresponding vector in space. I think it's true, or ...
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2answers
3k views

Are the units of energy the same in higher dimensions?

In 3 spatial dimensions, $$[E] = [ML^2 T^{-2}]$$ Would it change in higher dimensions? If yes, then what would be the dimensions for 4 spatial dimensions?
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1answer
68 views

In the value of gas constant $R$, what kind of mole is it?

Wikipedia gives $$R=8.3144598(48)\, \rm J\,mol^{−1}\,K^{−1}$$ But I want to ask how is it possible to write $\mathrm{mol}$ without a suffix? I mean, we must determine if we have $1\ \mathrm{mol}$ of ...
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1answer
53 views

What do we mean, concretely, by the unit $\rm N\:m$ (newton $\cdot$ meter)? [duplicate]

What do we mean, concretely, by the unit $\rm N\:m$ (newton $\cdot$ meter)? For example, $1 \:\rm m/s$ mean that each second, we make one meter. Also, $1\:\rm m/s^2$ mean that each second the speed ...
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1answer
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Dimensional correctness of equations [duplicate]

I have studied that a dimensionally correct formula/equation may or may not be correct. But in order for a formula or an equation to be correct, it must be dimensionally correct, according to the ...
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1answer
34 views

Issue in conversions from force to mass units

I have a problem converting force to mass units, in this example for 9.81kgf: To get the mass, dividing by gravity, I get 1kg ...
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1answer
57 views

Conversion between mass units

I am enrolled in a course related to physics, and converting between mass units is so tricky for me. I always treat in my work only in SI units and maybe MKS units, at level of weights, I find it easy....
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1answer
70 views

mathematically explain why add operators with different physical unit is wrong?

Suppose I have two operators $J^2$ and$J_z$ where they represent the length of angular momentum and its $z$ component respectively. Sure, it's legal to write new operators like $J^2-J_z^2$ or $J_x+...
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1answer
39 views

Dilatations and action on derivatives of fields

How do derivatives of fields transform under dilatations? Specifically I am interested on what I misunderstand with the example: Consider a theory that has a field $A_\mu$ that transforms under ...
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2answers
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How can momentum and position be combined into a phase space when they have different units?

Elaboration of the question: What is the geometrical interpretation of units? As in, a unit of length is a choice of scaling of the coordinate systems i.e. it is a choice of diffeomorphism, but then ...
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2answers
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How do I add measurement units to a power-function relationship like $y = 0.17x^{0.52}$?

I have the equation $y = 0.17x^{0.52}$, where the units on $x$ are $\rm cm$ and the units on $y$ are seconds. I can't figure out how to put units on the constants so that they work out and I get the ...
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1answer
77 views

I'm confused about electric flux?

In a previous physics class, I learned that the electric flux was $\vec{E}\cdot\vec{A}$ (dot product), and hence the unit is $Nm^2/C$. But in my electromagnetics book, it says the unit is Coulomb, and ...
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4answers
176 views

Dimensional analysis of classical action

Hamilton's action for classical systems has the units Joule seconds ($\rm J\cdot s$), which in base units is $\rm kg\:m^2/s$. Does the $\rm m^2$ have anything to do with area? I'm having a hard ...
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1answer
50 views

Is this a valid argument for the dimensions of this integral? [closed]

Given the integral: $I = \int_{-\infty}^{\infty} x^{2n}e^{-\alpha x^2} dx$ If I say that if: $$[I] = [L]^{2n+1}$$ And the dimensions of alpha must be $\frac{1}{[L]^2}$ since the exponent must be ...
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1answer
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How to handle dimensional analysis under exponents

Say there is a river of saline solution and I want to measure the flux of salt. That is, I want to measure the miligrams (mg) of salt per second ($\sec$) passing through a square centimeter area (cm$^...
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2answers
70 views

What is the dimensional analysis of the position?

My teacher gives me an equation for the position of a particle to analyse the equation by the dimensional analysis method: The position of a particle moving under uniform acceleration is some ...
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1answer
42 views

Can this quantity defined by exponentiating lengths be meaningful?

I just read Accurate predictions of coexistence in natural systems require the inclusion of facilitative interactions and environmental dependency in which they defined the following property:$^1$ $$\...
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1answer
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Possible fundamental dimensions in Pi Buckingham theorem

While the Pi Buckingham is often taught in a fluid dynamics context, in which the relevant fundamental dimensions are generally : Mass, Length, Time and Temperature, it can obviously be applied to ...
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1answer
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Non-dimensionalization of Navier-Stokes equations multiphase flows

I am currently dealing with multiphase flows and have to use the non-dimensional form of the Navier-Stokes equations (NSE). In the scientific literature I found various formulations (and almost no ...
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3answers
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Whats the dimension of no of molecules per unit volume

I know its trivial but still i am confused,i know that no of molecules per unit volume (n) is dimentionless but my confusion is what if we describe it by total number of molecules by total volume(n=N/...
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0answers
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Non-spreading wave packets

I am reading an article (Nonspreading wave packets, by Michael Berry, Am. J. Phys. 47, 264 (1979), author eprint), and I have some questions about the maths within. First of all they take a wave ...
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2answers
1k views

Why are the dimensions of escape velocity correct?

How does this formula work, from a dimensional analysis perspective? $$ v_\text{escape} = \sqrt{\frac{2GM}{R}}$$ The way I'm thinking about it is that $G$ is in units $\text{N} \cdot \text{m}^2/\...
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2answers
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How to put $c$ back into relativistic equations?

Many books set the speed of light $c=1$ for convenience. For example, Weinberg in his textbook "Gravitation and Cosmology" (though $G$ is still left as a constant): $$\begin{align} \mathrm{d}\tau^2 &...