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We are all taught that the torque $\boldsymbol{\tau}$ is given by $\boldsymbol{\tau} = \mathbf{r}\times\mathbf{F}$ so that torque increases with the lever arm length. What is the physical intuition behind this fact? Is there some compounding of certain inter-molecular forces in the lever arm that leads to a compounding of forces?

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The torque is defined to be $r \times F$, so that it will have some analogy with the mathematical construction of the force. For instance the force $ F = \frac{dp}{dt}$ and the corresponding analogy is that the torque $\tau = \frac{dL}{dt}$, where p is the linear momentum and L is the angular momentum. These definitions will also follow some symmetry (Noether's theorem) facts such as, if the system has translational symmetry then the momentum is conserve and if there is some rotational symmetry then the angular momentum is conserve.

Now to the question as of why the definition leads to the fact that torque increases with the lever arm length. It just a matter of consequences to how the torque and angular momentum is constructed so that it has the same mathematical construction with force and linear momentum.

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No. It follows directly from ${\bf F}=m{\bf a}$. Just multiply both sides of that equation by by ${\bf r}\times...$

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  • $\begingroup$ That's undoubtedly clear, but is there any intuitive reason why it should be the case? It's hard to believe that it's a fluke of classical mechanics that I can lift an object more easily with a longer lever than a shorter one. $\endgroup$ Feb 9, 2023 at 12:12
  • $\begingroup$ That's undoubtedly clear, but is there any intuitive reason why it should be the case? It's hard to believe that it's a fluke of classical mechanics that I can lift an object more easily with a longer lever than a shorter one. but the key thing is that you cannot lift it as far. $\endgroup$
    – Farcher
    Feb 9, 2023 at 13:13
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There is this thing called Noether's Theorem, from which it follows that there exists a quantity called angular momentum $\vec{L}$ that is conserved for isotropic problems. Now if you are interested in the dynamics of $\vec{L}$, it turns out $\dot{\vec{L}}=\vec{\tau}$, where $\vec{\tau}$ happens to be exactly what you wrote. So the choice of an expression for torque is not arbitrary, it is what it must be do describe the time evolution of angular momentum.

This is why torque is defined this way and not another. But your question might instead be "why do we get a "force amplification" in a lever?". If so, check the first volume of Feynman lectures where talks about energy conservation.

Intuitively you have a similar situation in a hydraulic press. Here the force is "amplification" is also related to energy conservation, but microscopically it occurs because more fluid molecules hit per second over a larger area.

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Torque does not create forces, but rather forces can create torque.

With this point of view, it takes more torque to resist a force with a bigger moment arm. Keep the force vector the same, and move its point of application to understand what the equipollent torque $$ \boldsymbol{\tau} = \boldsymbol{r} \times \boldsymbol{F}$$

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The most intuitive way I know relates to cause and effect on a resistant rotating structure. Consider a very massive sticky bolt being turned with a wrench. Pushing the wrench toward the bolt contributes nothing to turning it. Only force perpendicular to the wrench works to turn the bolt. Energy goes into the bolt when you turn it through a specific angle $\theta$, possibly required due to friction. When you pull at the end of the handle, radius $R$, the force pulls over an arc of length $R\theta$ to do the work. If you are at half the radius, the force pulls over only half the arc length to rotate the bolt through the same angle. This requires twice the force to produce the same work. This makes radius times perpendicular force the quantity that decides how much work is done when turning something through the certain angle. Radius times perpendicular force is torque.

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