What is Dimensional analysis: Definition and 260 Discussions

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or SI system than in others, due to the regular 10-base in all units. Dimensional analysis, or more specifically the factor-label method, also known as the unit-factor method, is a widely used technique for such conversions using the rules of algebra.Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are originally expressed in differing units of measure, e.g. yards and metres, pounds (mass) and kilograms, seconds and years. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are originally expressed in, e.g. meters and kilograms, seconds and kilograms, meters and seconds. For example, asking whether a kilogram is larger than an hour is meaningless.
Any physically meaningful equation, or inequality, must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation.
The concept of physical dimension, and of dimensional analysis, was introduced by Joseph Fourier in 1822.

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  1. Filip Larsen

    Universal wing- and fin-beat frequency scaling

    Good to know the power of dimensional analysis is still able to find gold: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0303834
  2. cianfa72

    I ##L^2## square integrable function Hilbert space

    Hi, I'm aware of the ##L^2## space of square integrable functions is an Hilbert space. I believe the condition to be ##L^2## square-integrable actually refers to the notion of Lebesgue integral, i.e. a measurable space ##(X,\Sigma)## is tacitly understood. Using properties of Lebesgue integral...
  3. T

    How to write Equations and Symbols in the Homework forum?

    Homework Statement: . Relevant Equations: . How do I write equations and symbols in the homework statement and relevant equations sections?
  4. J

    I Dimensional analysis in the formula for gyration frequency

    I have a question about dimensional analysis in the formula of the gyration frequency of synchrotron radiation of a relativistic particle (electron of charge e and mass m) in a magnetic field B. Leaving aside the adimensional Lorentz factor γ and numerical factors, the formula reads v ~ (e * B)...
  5. A

    How Does Changing Fundamental Units Impact Scientific Measurements?

    Hello folks, I am Alex, a guy in his 20s with autism who enjoys science and physics. After a search, I have joined, cos I want to be able to talk with like minded individuals who share the same interest in science as I do. Also, I am currently working on figuring out the inner processes of a...
  6. L

    A Going from Cauchy Stress Tensor to GR's Energy Momentum Tensor

    Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units? Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR? If so, What...
  7. etile

    Solve Angular Velocity & Acceleration - Dimensional Analysis

    Hi guys, Please see attached image - it's the part highlighted yellow that I'm stuck on. Here is what I got for linear velocity and angular velocity. (requested by mod) Thanks! c) Angular to velocity Dia=0.8m .. rad=0.4m v (linear velocity) = r (radius of circle) * omega (angular velocity)...
  8. H

    Dimensional Analysis: Forster Energy Transfer equation

    I'm trying to calculate Forster's Resonance Energy Transfer rate, but I just can't seem to get the units right. I'm trying to teach my students how to calculate them. Here is the (relatively) original technical note of FRET equation, made by the original author...
  9. simplemind

    B Question about a physical entity (dimensional analysis)

    i try to understand what physical entity mass or massdensity divided by speedquadrat could be? Any ideas? thanks
  10. G

    I Is "nondimensionalization" a misnomer?

    While doing some unit conversions for a task at hand, I understood that, even though I can do the conversions without any problems, there is something I find hugely confusing about terminology. By and large, the confusion is related to the interpretation of "nondimensionalization". As an...
  11. C

    I Using Dimensional Analysis to derive an equation (Walter Lewin video)

    Hello everyone. I was watching the Walter Lewin lectures and I noticed in the talk he used something called dimension analysis to study the time it takes for an object to drop based on differing heights. I'm 22:07 minutes into the video. With some guess work on what's proportional to what...
  12. Safinaz

    I The Units of the Cosmological Constant: eV^2

    In natural units, it’s known that the unit of the cosmological constant is ##eV^2##. I don‘t get why in this paper : https://arxiv.org/pdf/2201.09016.pdf page (1), it says the value of ##\Lambda \sim meV^4##, this means ##\Lambda \sim (10^6 ~ eV)^4 \sim 10^{24} eV^4 ##, shoud not the unit ##eV...
  13. D

    I Dimensional analysis seems wrong for this equation: z = -1/(x^2+y^2)

    Hi I've just done a question regarding a marble moving on a surface given by z = -1/(x2+y2) In this case what happens with dimensional analysis ? x and y have dimensions of length while z has dimensions of 1/(length)2. Is this question badly written or do i just accept that i won't be able to...
  14. P

    Dimensional Analysis: Solving Confusing Steps

    I tried to use dimensional analysis, there is variable for part i) m= P,A and Y also parameter is used in analysis is n=3(M,L,T). So m-n=0 number of dimensionless analysis group. I am confused at this step however I did this calculation to reach solution: i) P=MLT-2 (Ice Force) m-n=4-3=1(number...
  15. B

    I How to ensure an equation is dimensionless when it includes "Debye"

    I am trying to check if an expression is dimensionless. If it is, then I have done things correctly. However, I am stuck on how to deal with a (Debye^2) term. How can I break it down to find out if it cancels out with the other units I have left? I know this is probably a trivial question...
  16. P

    Temperature difference by Dimensional analysis.

    Attempt at solution: I wanted to try and solve this with dimensional analysis. I reasoned that I would chose the following dependent variables: - [V] : Volume ( of the block) - [Q] : Heat ( the radioactive decay would cause some heating of the water) - [R]: Radiation - [Cv]: Heat capacity...
  17. G

    Dimensional Analysis and units of time

    time = x(min) distance = 1(y) y = unknown unitsI think the answer should be 1(y)/ Min. This is not correct becase 1(y) is unknown. Any help? I have the answer but am confused
  18. W

    Dimensional analysis assumption

    Problem: Source: Halliday et al Physics 4e, p9, sample problem 4. Why is it valid to assume that $$F\alpha v^b$$ as the author does here, and not some more complex equation e.g. $$F=av^{e_1}+bm^{e_2}+cr^{e_3}$$, or $$F=av^{e_1}+bv^{e_2}+\cdots $$ (or some other equation)? A very similar...
  19. T

    Dimensional analysis of the Froude Number

    So, I have calculated that to dimmensional analyse this equation;
  20. Leo Liu

    I What are the applications of dimensional analysis and why does it work?

    I know that dimensional analysis is a rough way to check the correctness of the solution to a homework problem. However, what exactly is it and what are the other applications of it? Why does it work? Thanks.
  21. tomdodd4598

    I Renormalisation scale and running of the φ^3 coupling constant

    I am still rather new to renormalising QFT, still using the cut-off scheme with counterterms, and have only looked at the ##\varphi^4## model to one loop order (in 4D). In that case, I can renormalise with a counterterm to the one-loop four-point 1PI diagram at a certain energy scale. I can...
  22. B

    Dimensional Analysis for Multi-Layer Flow Equations

    Hello, New to the forum. I am looking for some help with some work. It is dimensional Analysis, i have done Dimensional analysis with the analysis bridge before. But i am stuck with the question below and trying to learn it. Any help much appreciated. I have the equation q = U A dT. But i am...
  23. rachelmaddiee

    Chemistry Converting 100 Yards to Centimeters: Dimensional Analysis Explained

    I’m trying to solve this question using the exact method in my book but I’m having trouble with setting this up. Here is my work** The length of a football field (100 yards) needs to be converted to centimetersLength of paperclip = 3.2 cm long36 inches = 1 yardhow many paperclips in 100 yds =...
  24. BillTre

    The Higher Dimensional Analysis of Embryonic Development

    I recently read this article in Science magazine: "The 200-year effort to see the embryo." by John B. Wallingford. Sadly it is probably not open access. This article reviews the history increasingly detailed information available on developing embryos. Recent technological advances in optical...
  25. D

    B What exactly is Dimensional Analysis?

    Can someone explain it to me exactly how dimensional analysis works (perhaps gives some examples.) What use does it have? Is it a convenient way to check your solutions? My teacher showed this example E ~ G, M, R [E] = k[G^α * M^β *R^γ] Maybe someone can explain this.
  26. VVS2000

    I In what cases does dimensional analysis fail?

    It possible a diagram would be really really helpful
  27. K

    Dimensional analysis (Speed of sound)

    I've been going round in circle and am stuggling as maths is not my strong suit. Any ideas? Up to the point where i equate the powers and then confuse myself, any help would be greatly appreciated.
  28. D

    Dimensional analysis of the equation u=prg

    u=prg LT^-1 = MLT^-2 ML^-3 LT^-2 = (ML^-2T^-2)^a (ML^-3)^b (LT^-2)c L^2 LT^-1 = M^a L^-2a T^-2a M^b L^-3b L^c T^-2c =M^(a+b) L^(-2a+3b+c) T(-2a+2c) M; a+b=0 L;-2a+3b+c=1 + T^-1 Either the formula is mixed up or...
  29. nineteen

    B What are the dimensions of kWh?

    What are the dimensions of kilo Watt hours? Is is M L2T-2?? If yes, why is that? If no, please teach me about what the right dimensions are and please be kind enough to provide a good explanation. Thank you in advance. P.S. I am wondering why it doesn't have the dimensions ML2T-3...
  30. D

    Determining the flow rate for a cooling system

    MODERATOR'S NOTE: HOMEWORK INCORRECTLY POSTED TO CLASSICAL FORUM, SO NO TEMPLATE I need help with the following question: Please have a look at the question and my attempt at the solution. Alternative cooling systems are considered for a large computing centre requiring 1 MW of cooling...
  31. JD_PM

    Dimensional analysis on an Ising model solution

    Homework Statement The mean field solution for the Ising model is: $$m = tanh[\beta (mJz + H)]$$ I wanted to carry out a dimensional analysis in order to verify the equation. Homework Equations $$m = tanh[\beta (mJz + H)]$$ The Attempt at a Solution Knowing that: $$[m] = \frac{A}{L}$$...
  32. JD_PM

    Dimensional analysis of an equation of motion

    Homework Statement The evolution of the density in a system of attractive spheres can be described by the following dynamic equation. $$\frac{\partial}{\partial t} \rho (r,t) = D_o [\nabla^2 \rho (r,t) + \beta \nabla \rho (r,t) \int dr' [\nabla V (|r-r'|)] \rho (r',t) g(r,r',t)]$$ a)...
  33. Auto-Didact

    Naturalness: dimensionless ratios

    The concept of naturalness as dimensionless ratios of parameters of order unity has recently come under criticism, most obviously because Sabine Hossenfelder wrote a book (Lost In Math) criticizing it. Very recently however, Peter Shor and Lee Smolin had a discussion about this over at Peter...
  34. J

    Calculating Total Energy with Dimensional Analysis | Screenshot Included

    Homework Statement Please look at the screenshot. Homework Equations dimensional analysis The Attempt at a Solution Since the heat capacity is given as 11.3 kJ/(C*g), and energy is measured in Joules (or kJ), I thought to multiply 11.3 by the change in temp (7.3 C) and also 1.50 g of...
  35. G

    I A question about units in equation: Dimensional analysis

    This question is about the matching the unites in an equation. The equation for an electron mobility is given by m^2/V.S. However substituting the units in the equation, doesn't yield the unit of mobility [m^2/V.S]. I have done the simplification using mks units and didn't come up to m^2/V.S...
  36. F

    Dimensional analysis-how one variable scales w/ another

    Homework Statement In PSet 3, Prob. 1(e), you determined the thickness h of a thin film of fluid (of viscosity µ and density ρ) flowing down an incline of angle θ, driven by gravity (acceleration g), via a supplied upstream flow rate Q per unit width (i.e., [Q] = (length)2/time). (a) Using...
  37. W

    I Dimensional analysis involving partial derivatives

    It is mentioned in Reif's book, statistical physics, that trough dimensional analysis it can be shown that: $$\frac{1}{\beta} = kT $$ where ##\beta## equals ##\frac{\partial \ln \Omega}{\partial E}## and k is the Boltzmann constant. I don't quite see how to reach this result, can anyone give me...
  38. Zack K

    Dimensional Analysis of an oscillation

    Homework Statement The period of oscillation of a nonlinear oscillator depends on the mass m, with dimensions M; a restoring force constant k with dimensions of ML-2T-2, and the Amplitude A, with dimensions L. Use dimensional analysis to show what the period of oscillation would be proportional...
  39. DracoMalfoy

    Dimensional Analysis: Period of a Pendulum to the length

    Homework Statement Use unit analysis to show that the constant, 2π, is unitless. Homework Equations T=2π√L/g[/B] The Attempt at a Solution T= [T] L= [L] g= a= [L]/[T]^2= [L T^-2] [T]= 2π√[L]/[L T^-2] [/B] Is this correct? I wasn't really sure how to do this. I'm using book examples to...
  40. S

    Simple dimensional analysis problem, thanks for any help

    Homework Statement It requires 10 microliters of a Cu(NO_3)_2 solution to produce a spot of 1cm in diameter. If the Cu(NO_3)_2 solution contains about 6g Cu^2+ per liter, then how many micrograms of Cu^2+ ions are there in one spot? Homework Equations simple dimensional analysis problem: 1st...
  41. M

    Interpret success-rate/time * $

    Homework Statement You are applying for a ##\$1000## scholarship and your time is worth ##\$10## an hour. If the chance of success is ##1 -(1/x)## from ##x## hours of writing, when should you stop? Homework Equations Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##...
  42. Pushoam

    Dimension of length using h,G,c

    Homework Statement Homework EquationsThe Attempt at a Solution Dimension of length using h,G,c [h] = [F r] ##[G] =[ \frac { Fr^2}{m^2} ] \\ [\frac { hG}c] = [L] ## So, the answer is option (b). Is this correct?
  43. M

    B Radian unit -- Why neglected in dimensional analysis?

    Why radians are usually neglected in dimensional analysis?
  44. scottdave

    Insights Make Units Work for You - Comments

    Greg Bernhardt submitted a new PF Insights post Make Units Work for You Continue reading the Original PF Insights Post.
  45. lichenguy

    Dimensional analysis problem

    Homework Statement A block of mass ##m = 1.00 kg## is being dragged through some viscous fluid by an external force ##F = 10.0 N##. The resistive force can be written as ##R = -bv##, where ##v## is the speed and ##b = 4.00 kg/s## is a phenomenological constant. You may ignore gravity (we...
  46. M

    Dimensions of Air Drag Constants and Terminal Speed Equation

    Homework Statement The object is falling vertically in a strange fluid, the magnitude of the air drag is best described by the following FD = bv+cv2 where v is the speed of the object and b and c are constants. A. What are the dimensions of b and c B. If the object has mass m find an algebraic...
  47. D

    Dimensional Analysis involving power

    Homework Statement The power required by a helicopter when hovering depends only upon the vertical thrust (a force) F provided by the blades, their length l, and the mass density of air, ρ. Establish an equation that relates the helicopters power requirement to these three quantities. By what...
  48. P

    I Limitations of Dimensional Analysis in Predicting Proportional Relationships

    Our professor introduced us to dimensional analysis and told us that we can use it to predict how some variables are proportional to others, for example: I have a ball at a certain height and i want to know the time it requires to touch the grond, i can make a guess that it will depend on the...
  49. R

    Dimensional Analysis Buckingham Pi Theorem

    This is the problem I'm currently working on: The pi groups I identified were h1, h2, d, D, g, t, and velocity, but when I looked at the solution, it selected Δh, D, t, ρ, d, ϒ, h1, with no explanation why those variables are needed. If I was solving with the Bernoulli equation, I wouldn't...