# How was the torque-force relationship discovered

I understand the equations of angular motion and how they relate to one another, however it is not clear to me where the torque equation comes from.

We have $F=ma$. By solving for the angular interpretation of linear acceleration, we have $$a=\frac{dv}{dt}r=\frac{d\omega}{dt}r=\alpha r$$ and thus $F=m\alpha r$.

With $\tau=Fr$, $\tau=m\alpha r^2$. When we have multiple torques we can add them up because $\tau$ is linearly related to force and force is a sum. By defining $I=\sum{m_ir_i^2}$, we have $\tau=I\alpha$

I understand that the concept of inertia is really an abstraction. We see the discrete $\sum{m_ir_i^2}$ or $\int{m_ir_i^2}$ forms frequently when dealing with equations of angular dynamics and thus decide to name the quantity moment of inertia based on the common role it plays in the equations (eg: for kinetic energy)

My question, a clarification

However, I'm unclear about how we know $\tau=Fr$. Is $\overrightarrow{\tau}=rF$ a wholly mathematical corollary of newton's laws of motion, (without creating a tautology), or a later development backed by empirical research.

• It can be traced back to the Law of Levers by Archimedes. – user36790 Nov 3 '16 at 12:49
• So it is an idea external of newtons laws of motion? – theideasmith Nov 3 '16 at 12:51
• Possible duplicate: physics.stackexchange.com/q/244655/50583 – ACuriousMind Nov 3 '16 at 13:03