Physics Stack Exchange
Physics Stack Exchange

Questions tagged [dimensional-analysis]

Dimensional analysis is the process of obtaining results by analysing the units and dimensions in questions, equations, and so on using The Principle of Homogeneity. Note: DO NOT USE THIS TAG if your question is about degrees of freedom or spatial dimensions.

Dimensional analysis means to obtain results by analyzing the units in question, etc. DO NOT USE THIS TAG if your question is about dimensions as in degrees of freedom.

An example of dimensional analysis would be constructing the Planck time. The Planck time is to involve the Planck constant, the gravitational constant, and the speed of light. So, we let the Planck time be products of these constants raised to unknown exponents. Then by requiring that this Planck constant has units of time, we calculate these unknown exponents.

Other instances of the usefulness of dimensional analysis include:

  • Calculating other Planck units; e.g. Planck length, Planck mass, etc.
1399 questions
104
votes
11 answers

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 formula cannot possibly be the correct expression…
Jack M
  • 1,979
56
votes
12 answers

Why are angles dimensionless and quantities such as length not?

So my friend asked me why angles are dimensionless, to which I replied that it's because they can be expressed as the ratio of two quantities -- lengths. Ok so far, so good. Then came the question: "In that sense even length is a ratio. Of length…
xrisk
  • 679
39
votes
5 answers

Why do we assume, in dimensional analysis, that the remaining constant is dimensionless?

Walter Lewin's first lecture (at 22:16) analyzes the time $t$ for an apple to fall to the ground, using dimensional analysis. His reasoning goes like this: It's natural to suppose that height of the apple to the ground ($h$), mass of the apple…
Fine Man
  • 1,493
15
votes
5 answers

Dimensional analysis restricted to rational exponents?

After some reading on dimensional analysis, it seems to me that only rational exponents are considered. To be more precise, it seems that dimensional values form a vector space over the rationals. My question is, why do we restrict ourselves to…
EuYu
  • 1,244
  • 1
  • 14
  • 22
13
votes
4 answers

Is it possible for a valid equation to have different dimension in both sides?

Suppose we have the following equation: $$ \frac{1}{r}\mathrm{d}r=\frac{1}{T}\mathrm{d}T $$ where $r$ be the distance and $\dim r=L^1$, and T can be the temperature with $\dim T =\Theta^1$. In this equation, both the LHS and the RHS are…
Fadelis Hu
  • 171
9
votes
1 answer

Do standard treatments of the Buckingham Pi Theorem gloss over this subtlety regarding signs?

Standard treatments of the Buckingham Pi Theorem seem to imply that given a dimensionless function $f$ of variables $q_1, q_2, \dots, q_n$ with associated dimension matrix having rank $r$, there exists a function $\phi$ of $\nu = n-r$ variables and…
joshphysics
  • 57,120
6
votes
5 answers

Is dimensional analysis wrong?

In many physics textbooks dimensional analysis is introduced as a valid method for deducing physical equations. For instance, it is usually claimed that the period of a pendulum cannot possibly depend on its mass because if it did the units would…
Gamabunto
  • 125
  • 6
6
votes
2 answers

What is the justification for Rayleigh's method of dimensional analysis?

It seems to me that although dimensional analysis is useful for demonstrating when a physics equation is wrong (when the dimensions are inconsistent), there is little justification for how it is often used to show that a given equation is correct.…
Wade Hodson
  • 664
5
votes
1 answer

why we can use dimensional analysis to discover formulas?

if I want to discover the time duration ($t$) of a object falling from an heigh ($h$) to the ground in a gravitational field ($g$), I can guess that $t$ is proportional to $h, g$ and the mass ($m$). then, with some $K$ as an adimensional, we have: $…
Guilherme Correa Teixeira
  • 75
5
votes
2 answers

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/second, T T-1, also known as time drift. It's the…
Aistis Raulinaitis
  • 51
4
votes
3 answers

Why does dimensional analysis to find a characteristic length, time, etc work?

For example, let's say you have an equation with planck's constant h, some mass m, and some velocity v. When you say "the characteristic length of the system is defined as h,m, and v to some powers that dimensionally result in a length," why does…
Mike Delmonaco
  • 163
4
votes
1 answer

Why can we only add or subtract things of the same dimension (meters/second, newtons, etc)?

Dimension analysis is a nice tool to create functions using physics dimensions that are desirable for our lives. I know, as an axiom, of sorts to me, that addition and subtraction must conserve units, and thus are actually relatively uncommon in…
sangstar
  • 3,192
4
votes
2 answers

Dimensional analysis, dimensionless quantities and ratios

What justifies the "canceling out" of the same units? I have difficulty understanding the point of dimensionless quantities. Usually, when you have a concept like mass over volume, which is density, you can state any division as: On every unit of…
Utter Confusion
  • 41
4
votes
2 answers

Functions and Length Scales

Regretfully I have to start with an apology as I fear I might be unable to express the question rigorously. Often reading physics papers the concept of "length scale" is used, in statements such as "over this length scale, the phenomenon can be…
Smerdjakov
  • 493
  • 3
  • 16
4
votes
1 answer

Buckingham-$\Pi$ theorem application: the case of only 0 or 1 dimensionless groups?

In dimensional analysis, we might consider a problem like: $$ f(q_1, q_2, ..., q_{n-1}) = q_{n} $$ where $q_i$ are physical quantities of interest. In order to determine how many $\Pi$ groups can be formed, we consider how many "fundamental"…
bzm3r
  • 227
1
2 3 4 5 6