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We all know that force and torque are physical quantities which we have defined according to our purpose of work. Now we are all aware that Newton's Second Law from his Laws of Motion states how to measure force i.e. what is the magnitude of force yields the answer $\vec F=m\vec a$. This is a law stated by Newton, and like all laws, it requires no proof and no deduction.

But which law or postulate or corollary states that torque should be equal to $\vec \tau=\vec r \times \vec F$? If not, then why are we sure that the magnitude of torque is actually that only and nothing else? Anyone could have calculated torque by some other formula.

Any help is welcome.

EDIT: According to Resnick-Halliday-Krane, there is something called Newton's Second Law of Rotation which states that $\vec \tau=\vec r \times \vec F$. Is it right?

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  • $\begingroup$ It started probably from the law of lever $$F_1l_1 = F_2l_2.$$ Each product, what Archimedis called leverage , is actually the simplest definition of torque. $\endgroup$ – user36790 Nov 13 '15 at 10:47
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    $\begingroup$ $\vec \tau := \vec r \times \vec F$ is the definition of torque. Are you asking why that's a sensible definition or who first wrote it down? (The latter would be more suited at History of Science and Mathematics) $\endgroup$ – ACuriousMind Nov 13 '15 at 10:49
  • $\begingroup$ @ACuriousMind The first one is my question. $\endgroup$ – SchrodingersCat Nov 13 '15 at 10:50
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Let you apply force $\bf F$ at point $P$ the coordinate of which is $ \bf r$ measured from a specific point $O$ - the point about which you want to rotate. Let $\bf r$ and $\bf F$ be in the same plane.

Now, if you were to rotate $P$ about $O$, it would rotate around some axis perpendicular to the plane in which the force and the point lies; if anti-clockwise, the axis would point upwards & if clockwise, then it points downwards. Now, this could be excellent if this direction could be associated with the property of torque; as if it were some vector.

We know from the Law of Lever that only force perpendicular to the lever can exert torque about the pivot; that means only the perpendicular component of $\bf F$ can impart rotation.

Let $\bf F$ makes angle $\phi$ with $\bf r.$ Then the perpendicular component with respect to $\bf r$ is $rF\sin\phi$ from the definition of torque given by the Law of Lever, that is torque is equal to $\text{Force}\cdot \text{Length of the lever from the pivot where the force is applied}.$

Now, this is what we know as cross-product. So,

$$\bf{\tau}= \bf{r\times F}\;.$$ This also gives the desired direction about which torque acts. Hence, torque is a vector where the direction specifies about which axis the rotation takes place.

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