In school, we learn that $T = Fd$, where $T$ is the torque, $F$ is the force perpendicular to the moment arm, and $d$ is the length of the moment arm. If I was the first physicist to come up with this model for torque, what might my train of thought be? In other words, how does one come up with this model?
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3The rate of change of angular momentum is directly proportional to the net unbalanced torque acting on it for a rigid body. $$L = mvr$$ $$\tau = \frac{dL}{dt} = m(v\frac{dr}{dt} + r\frac{dv}{dt}) = mr\frac{dv}{dt} = mra = r ma = rF$$ – Yashas Jun 09 '17 at 17:04
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Newton's laws of motion... – valerio Jun 09 '17 at 17:51
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6Possible duplicate of How was the torque-force relationship discovered – sammy gerbil Jun 10 '17 at 08:30
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The way you worded the question sounds to me like you're asking more for an intuitive understanding of why someone thought Fd would be a useful enough quantity to give it its own name, why it's taught to kids as a sort of fundamental equation rather than just another step in a derivation, and also where it fits into the big picture.
If that's the case, however, I can't tell you what the actual line of thinking really was, but I can at least offer you one view of why we consider it to be an important/fundamental quantity.
You can think of torque as being the rotational analog of linear force. We observe some simple relationships such as: $$\omega_{\text{avg}}=\frac{v_{\text{tangential}}}{r}$$ $$\alpha_{\text{avg}}=\frac{a_{\text{tangential}}}{r}$$ and one might wonder, if we simply multiply the $a_{\text{tangential}}$ by m, will the result give us something akin to rotational force?
Well it turns out if you do you get:
$m\alpha_{\text{avg}}=\frac{ma}{r}$
Of course the ma quantity is our familiar linear force, however if you multiply both sides by $r^{2}$ you will find "moment of inertia" on the left:
$(mr^{2})\alpha_{\text{avg}}=\frac{F}{r}(r^{2})=Fr$
therefore it is reasonable to think of $(mr^{2})\alpha$ as some sort of "angular force" which we've named "torque" usually given the symbol $\tau$, and to think of $mr^{2}$ almost as a sort of "angular mass" which we have given the name "Moment of Inertia" with the symbol I.
ison
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