Physics formulas of the form $A \cdot x^r$ and $r$ not integer

Question

For a math class I am looking for some examples from physics for formulas of the form

$$ f(x) = A \cdot x^r $$ and $r$ non-integer.

There are many simple examples where $r$ is an integer, but I didn't find so much so far for noninteger $r$. Only the period of a mathematical pendulum $T = 2\pi \sqrt{\frac{l}{g}}$ where $r$ would be $0.5$ (or $-0.5$ depending on wether you want do take $l$ or $g$ the role of $x$ above) and also adiabants in thermodynamics.

So are there other (simple) examples?

You might want to look at the physics of phase-transition theory, things like Critical exponents and universality classes... — Ismasou, Feb 08 '23 at 15:27
Check the binding energy in nuclear physics for the liquid drop model https://www.nuclear-power.com/nuclear-power/fission/liquid-drop-model/ — DanielC, Feb 08 '23 at 15:39
Area of a circle $~A=\pi,r^2\quad\Rightarrow~r\propto A^{1/2}~$ Sphere volume $~r\propto V^{1/3}$ — Eli, Feb 08 '23 at 15:46
Beat me to it. Similar for $d = 1/2 at^2$. There must be many such examples. — mmesser314, Feb 08 '23 at 15:48
@mmesser314 that's with integer $r$. Ok, you could solve it for $t$. — Julia, Feb 08 '23 at 15:52
@Julia what is the difference to your example ? $~\frac lg~$ is constant ? — Eli, Feb 08 '23 at 15:55
do you want analytical expressions (difficult: not easily integratable...) or just phenomenological examples from physics ? — rfl, Feb 08 '23 at 16:39
List-of-stuff questions are off-topic on this site. Questions should have a specific answer. — Ghoster, Feb 08 '23 at 17:14
Related: https://physics.stackexchange.com/q/162883/2451 and links therein. — Qmechanic, Feb 08 '23 at 17:55
Check out equations (5.48) and (5.50) here, with exponents 12/19, 870/2299, -67/19, and 1181/2299 arising from one of the simplest physical systems: two gravitating point masses. — Ghoster, Feb 08 '23 at 17:58
The unnumbered equation after (5.48) has the simple $Ax^r$ form, with $r=12/19$. — Ghoster, Feb 08 '23 at 18:15

Fynn · Answer 1 · 2023-02-08T16:07:30.173

2

Third Kepler's law: An object's orbital period $T$ squared is proportional to the the semi-major axis of its orbit $a$ cubed: $$T^2=\frac{4\pi^2}{G(M+m)}a^3$$

edited Feb 08 '23 at 16:07

answered Feb 08 '23 at 16:04

Fynn

21

Thanks, good example for $r = 1,5$ – Julia Feb 08 '23 at 17:14

Physics formulas of the form $A \cdot x^r$ and $r$ not integer

1 Answers1