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dimensionless
diˈmensionless, a. [f. as prec. + -less.] 1. a. Without dimension or physical extension. b. Of no (appreciable) magnitude; extremely minute. c. Without dimensions: see dimension 3 a.1667 Milton P.L. xi. 17 To Heav'n thir prayers Flew up..in they pass'd Dimentionless through Heav'nly dores. 1752 Warb... Oxford English Dictionary
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Dimensionless quantity
Dimensionless quantity. A dimensionless quantity (also known as a bare quantity, pure quantity as well as quantity of dimension one) [1] is a quantity to which no physical dimension is assigned. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics.
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Dimensionless quantity
Dimensionless units are special names that serve as units of measurement for expressing other dimensionless quantities. Dimensionless physical constants (e.g., fine-structure constant) and dimensionless material constants (e.g., refractive index) are dimensionless quantities wikipedia.org
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Why radian is dimensionless? Can't understand why we say that radians are dimensionless. Actually, I understand why this is happening: theta = arc len / r `Meters/meters` are gone and we got this d...
A dimensionless quality is a measure without a physical dimension; a "pure" number without physical units. However, such qualities may be measured in terms of "dimensionless units", which are usually defined as a ratio of physical constants, or properties, such
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Dimensionless physical constant
In physics, a dimensionless physical constant is a physical constant that is dimensionless, i.e. a pure number having no units attached and having a numerical Dimensionless constants wikipedia.org
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How can I prove that the argument of a transcendental function must be dimensionless? We all know from school that arguments of transcendental functions such as exponential, trigonometric and logarithmic functions, or...
Without that, it wouldn't be possible to handle a change of unit with mere factors. For instance, $e^{2.54 x}$ can't be expressed in terms of $fe^x$ where $f$ would be a suitable conversion factor. This is by contrast with a power function like $x^3$, such that $(2.54x)^3=(2.54)^3x^3$.
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List of dimensionless quantities
This is a list of well-known dimensionless quantities illustrating their variety of forms and applications. The tables also include pure numbers, dimensionless ratios, or dimensionless physical constants; these topics are discussed in the article. wikipedia.org
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How to normalize (dimensionless) the relativistic Binet's equation for mercury $$\frac{\delta^2 u}{\delta\theta^2}+u=\frac{\mu}{h^2}+3\mu u^2. $$ This is Binet's relativistic equation. I was trying to make perihelion...
$\frac{d^2 u}{d\theta^2}+u=\frac{r_s c^2}{2h^2}+\frac{3r_s}2 u^2$$ with $$r_s=\frac{2G\mu}{c^2}$$ Now $u$ is inverse distance, so in order to make it dimensionless Then for $x=ur_s$, you multiply the first equation with $r_s$ to get $$\frac{d^2 x}{d\theta^2}+x=\frac{r_s^2 c^2}{2h^2}+\frac 32 x^2$$ All terms are now dimensionless
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Dimensionless numbers in fluid mechanics
List All numbers are dimensionless quantities. See other article for extensive list of dimensionless quantities. Certain dimensionless quantities of some importance to fluid mechanics are given below: References Fluid dynamics wikipedia.org
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Scale a population equation with Allee effect into dimensionless form. A population equation with Allee effect $$\dfrac{dN}{dt} = N[r-a(N-b)^2]$$ where $a$, $b$, $r$ are positive constants. Let $n=\dfrac{N}{c}$, $\t...
No need for a chain rule. Pure algebra. Since $c$ and $d$ are constants, you may safely pull them out of differentiation, that is, $$\frac{c}{d}\dfrac{d(n)}{d(\tau )} = nc[r-a(nc-b)^2]$$ Cancel $c$ and multiply by $d$: $$\dfrac{d(n)}{d(\tau )} = nd[r-a(nc-b)^2] = n[rd - adc^2(n - \frac{b}{c})^2]$$ N...
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Consider the predator prey model and make change of variables to put it in dimensionless form. I have this predator-prey model \begin{align*} \frac{dH}{dt} &= rH \left(1-\frac{H}{K} \right) - \alpha \frac{PH}{H+ \beta...
This is very easy. It's just replacing of variables and constants by new variables and constants, and the chain rule, of course. For example, $${dx\over d\tau}={1\over K}\>{dH\over d\tau}={1\over K}\>{dH/dt\over d\tau/dt}={1\over Kr}\>{dH\over dt}=\ldots\quad.$$ Now plug in the right side from your ...
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Development of Dimensionless Rod-bundle CHF Correlation Based on ...
At present, the empirical correlations of critical heat flux (CHF) of advanced PWR rod-bundles at home and abroad generally have the common problems of complex mathematical form, numerous independent variable coefficients and lack of physical significance. In this study, based on 485 rod-bundle CHF data points of 5×5 PWR rod-bundles selected from the rod bundle CHF database of American ... hdlgc.xml-journal.net
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How to write the following equation in dimensionless form? We have the following equation $$mg\sin(\theta)=kx\left(1-\frac{L}{\sqrt{x^2+a^2}}\right)$$ and we are asked to put it in the following dimensionless form: $$...
\frac{mg\sin\theta}{kx} & = \frac{L_0}{\sqrt{x^2 + a^2}} \\\ 1 - \frac{mg\sin\theta}{kx} & = \frac{L_0}{a\sqrt{(x/a)^2 + 1}} \end{align*} Define the dimensionless It is clear that $R$ is dimensionless since $L_0$ and $a$ are lengths.
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Is this equation nondimensionalized? the question as mentioned in title. i want to know how this equation is dimenssionless as my efforts didn't give me the answer the equation is: $$z=(L/ \hbar) \cdot \sqrt{2m \cdot...
Note: It seems $L$ is not angular momentum, but some length. For the SI units we have: $$ [L] = \text{m} \\\ [\hbar] = \text{J}\text{s} = (\text{kg}\cdot (\text{m}/\text{s})^2) s = \text{m}^2 \cdot \text{kg} / \text{s} $$ And $$ [m] = \text{kg} \\\ [E] = J = \text{kg}\cdot (\text{m}/\text{s})^2 \\\ ...
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How to nondimensionalize the equation $dN/dt=N[s - m(N - a)^2]$? I have a differential equation for population growth, which is $$dN/dt=N[s - m(N - a)^2]$$ How to nondimensionalize the equation so it can depends on...
Define $N(t)=a x $ and $\tau=s t$. Then the differential equation becomes $$\frac{\mbox{d}x}{\mbox{d}\tau} = x [1-k^2 (x-1)^2]$$ where $k^2=m a^2/s$ as desired.
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