Torque and its origin

What's the origin of the definition of torque, that is, the moment of force about the point in consideration? - (1)

Is there a proof?

Why should torque be r x F, and not r² or r³ x F for that matter? I have tried searching many books but none gave the proof as to why torque should be r x F only, and not any other combination of the two.

I was hoping if it'd be possible to design a thought experiment, that clearly validates equation (1)

Thanks!

By definition a moment in physics is the product of a physical quantity and a position. So the quantity $\vec{r}\times\vec{F}$ could by definition be considered a moment of force. The utility of this moment is found when we consider the moment of momentum, $\vec{L}=\vec{r}\times \vec{p}$ ($\vec{p}$ is momentum), also called the angular momentum. It's called this because moments are generally related to rotational or turning or angular behaviors.

If we accept that angular momentum is a conserved quantity, then an important idea to consider is the time rate of change of momentum, which we call torque, often symbolized by $\vec{\Gamma}$ or $\vec{\tau}$. Conservation of angular momentum says $$\vec{L}_{\mathrm{new}} = \vec{L}_{\mathrm{old}}+\int\Gamma\ \mathrm{d}t.$$

Calculating the time rate of change of $\vec{L}$ we get $$\frac{\mathrm{d}\vec{L}}{\mathrm{d}t}= \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\times \vec{p}+\vec{r}\times \frac{\mathrm{d}\vec{p}}{\mathrm{d}t}$$

The first cross product vanishes (I'll let you figure that out), and $\frac{\mathrm{d}\vec{p}}{\mathrm{d}t}=\vec{F}.$

So the time rate of change of momentum, which we name torque, is equal to the moment of a force. [Edit: added "equal to"]

• So there is no explanation as to why r x F was selected in the first place? I understand the fact that this selection is 'useful', though – arya_stark Sep 23 '17 at 4:35
• It wasn't selected. It appears from the math, and it was given a name. – Bill N Sep 23 '17 at 4:43
• I'm trying to connect this to physics, the physical ways of nature. Could you possibly explain it using physics, thought experiments rather than math? – arya_stark Sep 23 '17 at 4:53
• A moment is a mathematical concept. The conservation of angular momentum is a physics and mathematical concept (see en.wikipedia.org/wiki/Noether%27s_theorem ). You're not going to get anywhere productive with the physics if you ignore the math. You asked why not $r^2$ or $r^3$. Because they don't show up when you do the physics, using math language. – Bill N Sep 23 '17 at 5:02