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Suppose you have a (nearly massless) lever you are using to lift something up. On the other end there is a $2\,\text{kg}$ object at $1\,\text{m}$ away from the pivot. Let's say I input some force on my end to lift the object some distance.

Now suppose I redo the procedure and I replace that object by a $1\,\text{kg}$ object at $2\,\text{m}$ away from the pivot.

Archimedes's law of the lever says that the amount of effort need to lift this new object is the same as in the previous procedure. However, the moment of inertia of the new setup is about double (consider $I = mr^{2}$ and take $m\rightarrow m/2$ and $r\rightarrow 2r$), so it seems that the amount of effort needed to lift the object is now greater.

Is the amount of effort the same or greater? How could we resolve this apparent contradiction?

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    $\begingroup$ One talks about the effort to balance the load and the other effort to rotationally accelerate. Those are completely different scenarios. $\endgroup$ Commented May 11, 2023 at 21:14
  • $\begingroup$ What you've discovered here is the working principle behind the counterweight trebuchet: the counterweight is placed on a short arm. Making it sufficiently heavy means you still get enough torque, but without the high moment of inertia that a long arm with lighter weight would incur, which would prevent it from rapidly speeding up the rotation to hurl the projectile away. $\endgroup$ Commented May 12, 2023 at 15:37
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    $\begingroup$ A large part of the confusion is that "effort" is not a scientific term. In one case, effort is "work" and in the other, effort is "force" $\endgroup$ Commented May 12, 2023 at 20:54

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You are making an interesting subtle version of the confusion between mass and weight.

Archimedes's law of the lever is equivalently a statement about torques. It is not about $$(2\text{ kg})(1\text m) = (1\text{ kg})(2\text m)$$ but rather $$(2g\ \text N)(1\text m)=(1g\ \text N)(2\text m)$$ whereas in the concept of moment of inertia, it really is the mass that is involved. Now, the rotational version of N2L is: $$\text{rN2L}:\qquad\sum\tau=I\alpha$$ In Archimedes's law of the lever, he is saying that the torque you need to apply on the left to work against the torque due to gravity is the same for the two scenarios. He did not say that the angular acceleration $\alpha$ in both scenarios will be the same. As you correctly pointed out, because the moment of inertia $I$ is twice as large in the longer case of 2m, that means its angular acceleration will be halved.

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Let's set aside lifting for now, and only consider static systems where the force applied to the lever is exactly enough to counteract gravity.

Your confusion is that the torque is the same in each system, and the angular acceleration is the same, but the moment of intertia is different. How can this be, when $\tau = I \alpha$?

In fact, the ostensible contradiction is resolved only because (net) torque and angular acceleration are both zero; $0 = I 0$. When considering dynamic systems, the two in question will behave differently, as discussed in naturallyInconsistent's answer.

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  • $\begingroup$ No, it can hold even in the case of imbalanced torque. The angular acceleration will just be smaller if the moment of inertia is bigger for a fixed size of the unbalanced torque. $\endgroup$ Commented May 11, 2023 at 13:54
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Lets put a mass $m$ a distance $l$ to the right of the pivot. The torque $\tau$ from gravity is then $mgl$. Lets apply a counter - force $F$ a distance $l'$ say, to the left of the pivot, so that its acting with a torque $\tau '$ that perfectly balances the lever (actually the magnitudes will be the same). But now instead, switch the mass on the right with a mass $\frac{m}{2}$ placed at a distance $2l$, but all the while keeping the same torque $\tau '$ on the left. The torque from gravity will of course be the same. How is it possible that the lever is still balanced now that the moment of intertia about the pivot is doubled? While $\tau '$ certainly has the increased moment of intertia to struggle with in this changed scenario, note that so does the torque from gravity. Looking at the lever whithout the force $F$ for a moment, its easier for gravity acting with the torque $\tau$ to rotate the lever when the mass is placed at $l$, than when half the mass is placed at $2l$. Since the increase in moment of intertia "affects" both torques the same, the lever will be balanced also in the second scenario.

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