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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Why is $(4.9v*t^2)/4$ equivalent to $1.225v*t^2$? [on hold]

In this khan academy video: https://www.khanacademy.org/science/physics/one-dimensional-motion/kinematic-formulas/v/deriving-max-projectile-displacement-given-time At 6:58 he derives the second ...
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7answers
3k views

Why are radians more natural than any other angle unit?

I'm convinced that radians are, at the very least, the most convenient unit for angles in mathematics and physics. In addition to this I suspect that they are the most fundamentally natural unit for ...
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3answers
165 views

Putting the Schrödinger equation in dimensionless form [closed]

The Schrödinger equation in spherical coordinates after separation of variables, for the hydrogen atom, is given by $$ \frac{-\hbar^{2}}{2m} \biggl[ \frac{1}{r^{2}} \frac{\partial}{\partial r} \biggl(...
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4answers
99 views

Is there an official list of independent units of measurements?

When I say 'independent units', I mean those which cannot be broken down anymore, and simultaneously forms the basis for any more, complex measurements. For example, height, length, and width can all ...
122
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14answers
10k views

Are units of angle really dimensionless?

I know mathematically the answer to this question is yes, and it's very obvious to see that the dimensions of a ratio cancel out, leaving behind a mathematically dimensionless quantity. However, I've ...
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2answers
372 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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2answers
5k views

converting power spectrum to photon flux density

So I'm having trouble converting units and was hoping somebody could point out where I've gone wrong... It seems I'm missing something fundamental. a Power Spectrum has units $kW/m^2-\mu m$ for the y-...
4
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1answer
339 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 ...
16
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4answers
859 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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3answers
57 views

Is “power to mass ratio” in fact independent to mass?

I was trying to find power to mass ratio of sun (work done per second at each unit mass at average), but I found the unit is quite straight: $ W=kg \space m^2 s^{-3}$ Then $W/kg= m^2 s^{-3}$ Which ...
3
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1answer
129 views

Is meters per second equivalent to seconds per meter?

I know this question is probably ridiculous, but bear with me for a moment. This thought emerged while I was converting between nm and wave numbers ($\rm cm^{-1}$). In order to prove this conversion, ...
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1answer
51 views

Is this differential equation (for damped & driven physical pendulum) physically valid?

Following is the equation of motion for a physical pendulum which is damped and driven by a force of frequency $f$: $$\frac{d^2 \theta}{dt^2} + b \frac{d\theta}{dt} + sin(\theta) = Tsin(2\pi ft)$$ ...
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1answer
43 views

What are dimensions of co-ordinates which are used to define an electric field?

A possible electrostatic field is: $ E_x = 6xy$ $ E_y = 3x^2-3y^2$ $ E_z = 0$ Suppose we are using SI system. So unit for components of field is volts/meter. Then what are dimensions of $x$ and $...
3
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5answers
430 views

Could velocity be taken as fundamental instead of time?

In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as ...
0
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0answers
33 views

Decay rate revisited

According to Peskin&Schroeder (pp. 107), we have $$ d\Gamma=\frac{1}{2m_A}\left(\prod_f\frac{d^3p_f}{(2\pi)^3}\frac{1}{2E_f}\right)|\mathcal{M}(m_A)\rightarrow {p_f}|^2(2\pi)^4\delta^{(4)}(p_A\...
0
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1answer
74 views

Why does $k^z/E$ have dimensions of inverse velocity rather than velocity?

I'm studying quantum field theory and I want to prove the cross section. In Peskin's book, equation 4.77 says that: $$ \frac{1}{\left | \frac{k_{A}^{z}}{E_{A}}-\frac{k_{B}^{z}}{E_{B}}\right |}=\frac{...
4
votes
4answers
597 views

Are quantum operators dimensionless?

I'm slightly confused as to whether quantum (hermitian) operators, which we get by promoting observables to operators, are dimensionless or not? Clearly the Hamiltonian of the system, say of the ...
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8answers
6k views

What justifies dimensional analysis?

Dimensional analysis, and the notion that quantities with different units cannot be equal, is often used to justify very specific arguments, for example, you might use it to argue that a particular ...
3
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1answer
218 views

Gaussian integration and dimension argument

I made a mistake recently regarding the Gaussian density, by putting the determinant of the variance to the power $\frac{d}{2}$. Would the following argumentation be valid to highlight it should be to ...
4
votes
2answers
586 views

Check dimensions of the integral of a function

I and a colleague are arguing about the dimensions of: $$\int_0^x f(x) dx $$ in this particular case $[f(x)]=m^2/s^3$ and $[x]=m$. Does it follow that $[\int_0^x f(x) dx]=m^2/s^3$ or $[\int_0^x f(x)...
3
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1answer
154 views

Newton's Second Law of Motion

Newton originally wrote his second law as: "The rate of change of momentum of a body is directly proportional to the resultant force applied to the body, and is in the same direction as the force." ...
4
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2answers
78 views

Non-dimensionalizing incompressible Navier-Stokes

I have a question about non-dimensionalization of the incompressible Navier-Stokes (NS) equations. My understanding is that the purpose of non-dimensionalization is to "collapse" solutions onto one ...
2
votes
3answers
740 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 ...
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0answers
38 views

Finding the exponent of $\lambda$ in Wien's displacement law

I am reading this paper on a short history of the $T^4$ radiation law. In particular, on page 5, By assuming that the wavelength of radiation emitted by a molecule was a function only of its ...
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2answers
12k views

Dimensonal analysis of damping constant?

What are the units of the damping constant from the following equation by dimensional analysis? $$\zeta = \frac{c}{2\sqrt{mk}}$$ I'm assuming the units have to be s^-1, as the damping constant is ...
0
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0answers
27 views

units of measurement in Bloch equations

I am trying to understand the Bloch equations for understanding MRI and the equation describes the change of magnetization during NMR excitation and relaxation as: $$ \frac{dM}{dT} = \gamma M \times ...
1
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1answer
52 views

Are there physical law that are not unit-free?

One of the prerequisites of the Buckingham π theorem is that the physical law in question should be unit-free. I couldn't find an example of a physical law that is not unit-free. Is there such thing? ...
2
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4answers
65 views

Is the concept of work only defined in mechanics?

I'm studying energy and work, so far it looks like work only makes sense in kinematics (objects that move), but energy makes sense in many other ways (electric, thermodynamic, mechanic). Is work a ...
2
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1answer
47 views

How to identify the sign of a derived nondimensional parameter and its physical meaning?

I think that the nondimensional group is ordinarily defined to be positive value in a physical problem. But in some particular case, we probably need to decide the sign of a derived dimensionless ...
2
votes
3answers
120 views

How can geometrized units have more than one constant equal to 1?

I can understand how you could manipulate units to make a certain constant equal to $1$, like $c$ or $G$, et cetera. But how can you make it so two constants (in this case $c$ and $G$) are equal to $1$...
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1answer
97 views

When can the constant of proportionality in an eq be set equal to 1 and when not? [duplicate]

In $F=kma$, $k=1$ but in $F=kx$, $k$ is not equal to 1?So what are the conditions for the constant of proportionality to be set 1?
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2answers
171 views

What is the dimensional formula of angular velocity?

I have problem to determine the dimensional formula of angular velocity. My friend said that the dimensional formula of angular velocity is $T^{-1}$. It's come from rad/s, rad is dimensionless, the ...
0
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0answers
67 views

What does it mean to take a derivative with respect to $\hbar$?

Problem 6.32 of Griffiths Introduction to Quantum Mechanics, 2ed is In part (b), we take a derivative with respect to $\hbar$. Since $\hbar$ is a constant, what does it mean to take a derivative ...
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3answers
8k 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 ...
1
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1answer
32 views

Differences in notation of momentum 4 vector

I have noticed three ways to write the 4 momentum vectors: $P = (E/c, \vec{p})$ $P = (E, \vec{p})$ $P = (E, c\vec{p})$ I know how to derive equation 1, and as far as I know, one can use the ...
0
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1answer
62 views

Issues of normalization & differential final state momenta in analysis of normalized differential quantum-field-theoretic probability of scattering

The normalized differential quantum-field-theoretic probability $dP$ of scattering is given by $$dP=\frac{|\langle f |S|i\rangle|^{2}}{\langle f|f\rangle\langle i|i\rangle}d\Pi,$$ where $|i\rangle$ ...
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0answers
42 views

Photon Propagator in QFT

Please forgive my temporary split-brain, but I'm a little thrown off by something when considering units at the moment. In QED (depending on the guage), the photon propogator is written as $<0|\...
0
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1answer
58 views

The true dimension of Dirac field

In natural units with $\hbar=1$ and $c=1$, as we know, the energy dimension of the Dirac field $\psi(x)$ in QED is $\frac{3}{2}$. But in cgs units, what is the true dimension of the Dirac field $\psi(...
0
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2answers
60 views

Proper units for physical quantities when $\hbar$=$1$

How to deal with the units of quantities if $\hbar=\tfrac{h}{2\pi}=1$? For example, the energy $E=\hbar\omega$: If I have chosen $\hbar=1$, how do I use the units to properly differentiate between ...
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0answers
28 views

Initial value in rescaling differential equation

I've re scaled the simple harmonic oscillator differential equation as below: original equation: $d^2x/dt^2+\omega^2x=0$ re scaling factor: $\omega t\to t'$ re scaled (dimensionless) equation: $d^2x/...
28
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12answers
6k views

Do all equations have identical units on the left- and right-hand sides?

Do all equations have $$\text{left hand side unit} = \text{right hand side unit}$$ for example, $$\text{velocity (m/s)} = \text{distance (m) / time (s)},$$ or is there an equation that has different ...
3
votes
0answers
22 views

Arguing on dimensions of logarithms and exponentials [duplicate]

Suppose you have some physical quantity $x$ of dimension $l$. We all know that the dimension of $x^2$, for example, will be $l^2$, and that of $\dfrac{1}{x}$ is $l^{-1}$. However, what will be the ...
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1answer
174 views

Difference between theoretical equations and empirical equations

Some equations are theoretical in the sense that they are derived from an underlying theory. Other equations are empirical in the sense that they were selected only because they fit experimental data ...
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6answers
157 views

Why do constants have dimensions?

I am just a beginner in dimensional analysis, and I see that $G$, the universal gravitational constant, is independent of everything. Speed, for example, depends on distance and time, but $G$ does not ...
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9answers
2k views

How to interpret the units of the dot or cross product of two vectors?

Suppose I have two vectors $a=\left(1,2,3\right)$ and $b=\left(4,5,6\right)$, both in meters. If I take their dot product with the algebraic definition, I get this: $$a \cdot b = 1\mathrm m \cdot 4\...
3
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1answer
98 views

The dimension of the energy-momentum tensor and the Einstein-Hilbert action

I have been thinking recently what will happen if one uses the energy momentum tensor of the Dirac field as a source in the Einstein Field equations. It is well known that in this case $$ T_{\mu\nu}=...
3
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1answer
365 views

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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0answers
51 views

What kind of unit is $m^2s^{-4}$ in terms of gyro/accel?

Background While working on something in the field of avionics, I have discovered the following unit as part of an inertial-physics equation... $$m^2s^{-4}$$ I am trying to figure out the formal ...
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5answers
171 views

What in nature causes Newton's gravitation constant to have its given value?

Does the value of Newton's universal gravitational constant $G$ remain a mystery? Why does it have the value that it has?
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1answer
67 views

A simple explanation of Kepler's Third Law

Is there a simple way to explain how Kepler's third law follows from the inverse square law that of gravity (and laws of motion) For example for Kepler's second law we can say it's because Gravity ...