# Is it possible to deduce the Archimedes' law of the lever using only the laws of conservation of the physics?

Is it possible to deduce the Archimedes' law of the lever using only the laws of conservation of the classical mechanics? I never saw (which is strange), but I think that it's possible.

To be clear, let me quote the most general part of the law

Two magnitudes, whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes.

In other words, masses $A$ and $B$ balance if

$$m_A x_A = m_B x_B \tag{1}$$

where $m$ is the mass and $x$ the distance to the pivot.

You could prove it via conservation of angular momentum about the pivot of the lever. Multiplying $(1)$ by $g$, we see that the torque about the pivot is $0$, hence there is no change in angular momentum; the balance beam remains at rest.

You could also prove it using conservation of energy. If we let mass $A$ drop by a small distance $y_A$ and $B$ rise by $y_B$, we have the relation

$$\frac{x_A}{y_A} = \frac{x_B}{y_B}$$

combining it with $(1)$, we see that there is no change in total gravitational potential energy $m_Agy_A - m_Bgy_B$, hence it is not energetically favourable for the lever to tilt either way.

In general, conservation laws are the most fundamental laws of physics, and all of classical mechanics can be derived from them. In particular, Newton's laws can be derived from conservation of energy and momentum.

For the law of the lever, you can derive it from conservation of energy for the case where the forces at the ends are weights that have some potential energy. Once the law is found in this case, it can be applied to all other cases, since the equilibrium of the lever is independent of the types of forces applied to it.