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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.

theideasmith
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When a rigid body is rotating, the acceleration vector of each particle of the body is different from that of every other particle. So it is daunting to think of writing a force balance on each and every particle (as well as to consider its detailed interactions with adjacent particles) to quantify the motion. Fortunately, in the case of a rigid body, this can be all be circumvented and simplified by doing a moment balance on the body. The moment balance can be derived by taking the cross product of the differential momentum equation (aka, equation of motion, aka, stress equilibrium equation) with a position vector drawn from the center of mass, and integrating over the volume of the body.

Chet Miller
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