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The formula for moment about a point is:

$$M = Fd$$

After looking at other answers on stackexchange, I'm still not convinced with the 'intuitive' explanations that are given. I understand the cross product relationship between F and d and how to compute the moment, but I'm not searching for a mathematical explanation. I would prefer an explanation based purely on the explanation of concepts intuitively.

I'm not sure how the formula was decided to express the 'turning effect intensity' of a certain force. Why specifically this formula and not some other form? Maybe there's an explanation using rigid bodies?

XXb8
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1 Answers1

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Consider a level of the second type with load $m$ at distance $r$ and a force perpendicular to a lever acting at a distance $d$. The system is in zero gravity.

If the lever rotated by a small angle $\Delta\phi$, then the force has done the work $W=F\Delta s=Fd\Delta\phi$. On the other hand this energy was spent to increase the kinetic energy of the load: $$ \Delta T=mv\Delta v = m(r\omega)(r\Delta\omega) = m\left(r\frac{\Delta\phi}{\Delta t}\right)(r\Delta\omega) = mr^2\frac{\Delta\omega}{\Delta t}\Delta\phi $$ Since $W=\Delta T$, we can conclude that: $$ (mr^2)\frac{\Delta\omega}{\Delta t}=Fd. $$ Compare this to the Newton's law: $m\frac{\Delta v}{\Delta t}=F$. You can see that in “rotational world”, $I=mr^2$ plays role of inertia (mass), $\Delta\omega/\Delta t$ is angular acceleration, so “turning effect” $Fd$ is an analogue of force.

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Vasily Mitch
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